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Sede Amministrativa: Universita degli Studi di Padova
SCUOLA DI DOTTORATO DI RICERCA IN: Scienze
Matematiche
INDIRIZZO: Matematica
CICLO: XXIV
Operators in Sobolev
Morrey spaces
Direttore della Scuola: Ch.mo Prof. Paolo Dai Pra
Coordinatore d’indirizzo: Ch.mo Prof. Franco Cardin
Supervisore: Ch.mo Prof. Victor I. Burenkov
Ch.mo Prof. Massimo Lanza de Cristoforis
Dottoranda: Nurgul Kydyrmina
To my Father and Mother
i
Abstract
Morrey spaces were introduced by Charles Morrey in 1938. They are a useful
tool in the regularity theory of partial differential equations, in real analysis
and in mathematical physics.
In the nineties of the XX century an active study of general Morrey-type
spaces characterized by a functional parameter has started to develop. A
number of results on boundedness of classical operators in general Morrey-
type spaces were obtained.
At the beginning of the XXI century there were new active developments in
this area. In the last decade many mathematicians do research on smoothness
spaces related to Morrey spaces. Among these spaces the Sobolev-type spaces
play an important role.
In the thesis Sobolev spaces built on Morrey spaces are studied, which are
also referred to as Sobolev Morrey spaces. These are spaces of functions which
have derivatives up to certain order in Morrey spaces.
We analyze some basic properties of Morrey spaces and of Sobolev Mor-
rey spaces. Then we consider the embedding and multiplication operators
in Sobolev Morrey spaces. Finally, the dissertation provides a study of the
composition operator in Sobolev Morrey spaces.
The results presented in the thesis have been obtained under supervision
of Professors V.I. Burenkov and M. Lanza de Cristoforis.
iii
Sunto
Gli spazi di Morrey sono stati introdotti da Charles Morrey nel 1938. Essi sono
uno strumento utile nella teoria della regolarita per equazioni differenziali alle
derivate parziali, in analisi reale ed in fisica matematica.
Negli anni novanta del XX secolo ha iniziato a svilupparsi un attivo stu-
dio degli spazi di Morrey di tipo generalizzato che sono caratterizzati da un
parametro funzionale. E stato ottenuto un cero numero di risultati sulla limi-
tatezza degli operatori classici negli spazi di Morrey di tipo generalizzato.
All’inizio del XXI secolo ci sono stati nuovi e attivi sviluppi in questa area.
Nell’ultima decade molti matematici hanno svolto ricerche su spazi funzionali
relativi agli spazi di Morrey. Tra questi spazi gli spazi di tipo Sobolev giocano
un ruolo importante.
Nella tesi si studiano Spazi di Sobolev costruiti su spazi di Morrey, anche
detti spazi di Sobolev Morrey. Questi sono spazi di funzioni che hanno derivate
fino ad un certo ordine negli spazi di Morrey.
Si analizzano alcune proprieta di base degli spazi di Morrey e degli spazi
di Sobolev-Morrey. Poi si considerano operatori di immersione e di moltipli-
cazione negli spazi di Sobolev Morrey. La terza parte della tesi presenta uno
studio degli operatori di composizione negli spazi di Sobolev Morrey.
I risultati presentati nella tesi sono stati ottenuti sotto la supervisione dei
Professori V.I. Burenkov and M. Lanza de Cristoforis.
v
Acknowledgements
First and foremost, I would like to express my sincere gratitude to both of
my supervisors Prof. V.I. Burenkov and Prof. M. Lanza de Cristoforis for
the continuous support of my Ph.D study and research, for their patience,
motivation, and enthusiasm. Their guidance helped me in all the time of
research and writing of this thesis. It has been an honor for me to be their
Ph.D. student.
A special gratitude I give to Prof. E.S. Smailov who have always been a
constant source of encouragement and inspiration during my graduate study.
His unconditional support has been essential all these years.
I acknowledge the University of Padova for giving me the opportunity to
study in Italy and for providing financial assistance.
Last but not the least, I would like to thank my family for their unflagging
love and encouragement throughout my life. I thank my parents, Alim and
Bagila, for their faith in me, for their prayers and supporting me whenever I
needed it. I love you all dearly.
vii
Contents
Introduction 1
Notation 5
1 Morrey and Sobolev Morrey spaces 9
1.1 General Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Approximation by C∞ functions in Morrey spaces . . . . . . . . 20
1.3 Preliminaries on integral operators . . . . . . . . . . . . . . . . 28
1.4 Sobolev spaces built on Morrey spaces . . . . . . . . . . . . . . 36
2 The embedding and multiplication operators in Sobolev Mor-
rey spaces 43
2.1 The Sobolev Embedding Theorem . . . . . . . . . . . . . . . . . 43
2.2 Approximation by C∞ functions in Sobolev Morrey spaces . . . 48
2.3 Multiplication Theorems for Sobolev Morrey spaces . . . . . . . 50
3 The composition operator in Sobolev Morrey spaces 61
3.1 Composition operator in Morrey spaces . . . . . . . . . . . . . . 61
3.2 Composition operator in Sobolev Morrey spaces . . . . . . . . . 63
3.3 Continuity of the composition operator in Sobolev Morrey spaces 68
3.4 Lipschitz continuity of the composition operator in Sobolev Mor-
rey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Differentiability properties of the composition operator in Sobolev
Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Bibliography 79
ix
Introduction
This dissertation is devoted to Sobolev spaces built on Morrey spaces, also
referred to as Sobolev Morrey spaces, i.e., to the spaces of functions which
have derivatives up to a certain order in Morrey spaces.
In the first part of the dissertation we analyze some basic properties of
Morrey spaces and of Sobolev Morrey spaces. In particular,
(i) We characterize the functions in a Morrey space which can be approxi-
mated by smooth functions, as the functions which belong to a specific
subspace of the Morrey space, which we call the ‘little’ Morrey space.
(ii) Contrary to the classical Sobolev spaces built on the Lp spaces with
p < ∞, the Sobolev spaces built on Morrey spaces are not separable
spaces even if p <∞ and we cannot expect that the set of C∞ functions
of a Sobolev Morrey space be dense in a Sobolev Morrey space. However,
we show that the functions in a Sobolev space built on little Morrey
spaces can be approximated by C∞ functions.
In the second part of the dissertation we consider the embedding and mul-
tiplication operators in Sobolev Morrey spaces. Namely,
(i) We prove a Sobolev Embedding Theorem for Sobolev Morrey spaces.
The proof is based on the Sobolev Integral Representation Theorem and
on a recent results on Riesz potentials in generalized Morrey spaces of
Burenkov, Gogatishvili, Guliyev, Mustafaev [14] and on estimates on the
Riesz potentials contained in the dissertation. We mention that a Sobolev
Embedding Theorem for Sobolev Morrey spaces had been proved by
Campanato [19, Thm. II.2, p. 75], for a subspace of our Sobolev Morrey
space which corresponds to the closure of the set of smooth functions in
1
Introduction
our Sobolev Morrey space. The methods of the present dissertation are
considerably different from those of Campanato.
(ii) We prove a multiplication Theorem for Sobolev Morrey spaces which ex-
tends to Sobolev Morrey spaces a known result of Zolesio [61] for classical
Sobolev space (see also Valent [58], Runst and Sickel [51]).
Both the Sobolev Imbedding Theorem of (i) and the multiplication Theo-
rem of (ii) have been proved for bounded domains with the cone property. We
believe that one could prove the same type of results for unbounded domains
with the cone property and in particular for the entire space.
Then in the third part of the dissertation, we consider the composition
operator in Sobolev Morrey spaces, and as a first step we do so for Sobolev
Morrey spaces of the first order. Let Ω be a bounded open subset of Rn with
the cone property. Let W 1,λp (Ω) be the Sobolev space of functions with deriva-
tives up to order 1 in the Morrey space Mλp (Ω) with exponents λ ∈ [0, n/p],
p ∈ [1,+∞].
Let Ω1 be a bounded open subset of R. Let W 1,λp (Ω,Ω1) denote the set of
functions of W 1,λp (Ω) which map Ω to Ω1.
Let C0,1(Ω1) denote the space of Lipschitz continuous functions from Ω1
to R. Let r be a natural number. Let Cr(Ω1) denote the space of r times
continuously differentiable functions from Ω1 to R.
Then we prove the following results.
(j) We prove that if f ∈ C0,1(Ω1) and if g ∈ W 1,λp (Ω) has values in Ω1, then
the composite function f g belongs to W 1,λp (Ω) and the norm of f g
can be estimated in terms of the norms of f and of g. We note that in
case λ = 0, which corresponds to a classical Sobolev space such a result
is well known (see Marcus and Mizel [35]).
(jj) We exploit an abstract scheme of Lanza de Cristoforis [30] and prove
that if (1 + λ) > n/p, then the composition map T from Cr+1(Ω1) ×
W 1,λp (Ω,Ω1) which takes a pair (f, g) to the composite function f g is
r-times continuously Frechet differentiable. We note that in case λ = 0
the result of the present dissertation improves a corresponding result of
Valent [58] for case r = 1.
2
Introduction
(jjj) We prove that if f ∈ C1,1loc (R) and if (1 + λ) > n/p, then the map which
takes g to fg is Lipschitz continuous on the bounded subsets ofW 1,λp (Ω).
For a related result in the Besov space setting, we refer to Bourdaud and
Lanza de Cristoforis [9].
We believe that our sufficient conditions on f of (j), (jj), (jjj) are optimal,
just as they have been shown to be optimal in the frame of Sobolev spaces,
which corresponds to case λ = 0 (see Appell and Zabreiko [4, Ch. 9], Runst
and Sickel [51, Ch. 5], Bourdaud and Lanza de Cristoforis [9].)
We believe that by proving the Sobolev Imbedding Theorem of (i) and the
multiplication Theorem of (ii) above for unbounded domains with the cone
property, one could prove also the results of (j)–(jjj) for unbounded domains
with the cone property and in particular for Ω = Rn.
The composition operator has been considered by several authors. For
extensive references, we refer to the monographs of Appell and Zabreiko [4,
Ch. 9], of Runst and Sickel [51], of Dudley and Norvaisa [23], and to the
recent survey paper Bourdaud and Sickel [11]. In particular, the continuity,
the Lipschitz continuity and the higher order differentiability of f g has a
function of both f and g has long been investigated.
In the Sobolev space setting, we mention in particular Marcus and
Mizel [34]–[40], Adams [3], Szigeti [56], [57], Valent [58], [59], Gol’dshtein and
Reshetnyak [26], Drabek and Runst [22], Musina [41], Bourdaud and Meyer
[10], Bourdaud [6], [7], Bourdaud and Kateb [8], Sickel [53]. As far as con-
sidering the differentiability of the composition operator when both the func-
tions f and g belong to a Sobolev space, we mention a paper of Brokate and
Colonuis [12], and of Lanza de Cristoforis [32].
The results of this dissertation will appear as joint work with the supervi-
sors V.I. Burenkov and M. Lanza de Cristoforis.
3
Notation
N denotes the set of all natural numbers including 0. Throughout the paper,
n is an element of N \ 0.
As usual, R is the set of all real numbers, Rn is the n-dimensional Euclidean
space, and
Nn = N× · · · × N︸ ︷︷ ︸n
is the set of multi-indices.
P(Rn) – the linear space of polynomials with real coefficients and n real
variables.
B(x, r) – the open ball of radius r > 0 centered at the point x ∈ Rn.
vn – the volume of the unit ball in Rn.
supp f – the support of a function f .
f←(D) – f -preimage of a set D.
Dα ≡ ∂α1+···+αnf
∂xα11 ... ∂xαn
n– the (ordinary) derivative of the function f of order α,
and
Dαw ≡
(∂α1+···+αnf
∂xα11 ... ∂xαn
n
)w– the weak derivative of the function f of order α.
For an arbitrary nonempty set Ω ⊂ Rn we shall denote by:
diamΩ – the diameter of Ω,
Ω or cl(Ω) – the closure of Ω,
χΩ – the characteristic function of Ω, i.e. χΩ(ξ) = 1 if ξ ∈ Ω and χΩ(ξ) = 0
if ξ ∈ Rn \ Ω,
C0(Ω) – the space of functions continuous on Ω,
C0b (Ω) – the Banach space of functions continuous and bounded on Ω with
the sup norm in Ω,
C0ub(Ω) – the Banach space of uniformly continuous and bounded functions
5
Notation
on Ω with the sup norm in Ω,
Cm(Ω) (m ∈ N) – the Banach space of m-times continuously differentiable
functions on Ω,
C∞(Ω) =∩
m∈NCm(Ω) – the space of infinitely continuously differentiable
functions on Ω,
C∞c (Ω) – the space of functions in C∞(Ω) with compact support.
For a measurable nonempty set Ω ⊂ Rn we shall denote by:
Lp(Ω) (1 ≤ p < ∞) – the Banach space of functions f measurable on Ω
such that the norm
∥f∥Lp(Ω) =
∫Ω
|f(x)|pdx
1p
<∞.
L∞(Ω) – the Banach space of functions f measurable on Ω such that the
norm
∥f∥L∞(Ω) = ess supx∈Ω
|f(x)| <∞.
For an open nonempty set Ω ⊂ Rn we shall denote by:
Llocp (Ω) (1 ≤ p ≤ ∞) – the set of functions f defined on Ω such that for
each compact K ⊂ Ω f ∈ Lp(K),
B(Ω) ≡ f : Ω → R : f is bounded,
M(Ω) ≡ f : Ω → R : f is measurable,
M(Ω) – the factor spaceM(Ω)/Θ(Ω), where Θ(Ω) is the set of all functions
defined on Ω which are equal to 0 almost everywhere on Ω,
Cm(Ω) – the subspace of Cm(Ω) of functions f such that f and its deriva-
tives Dαf of order |α| ≤ m can be extended with continuity to Ω,
C∞(Ω) – the set of functions f from Ω to R such that there exist an open
neighborhood U of Ω and a function F ∈ C∞(U) such that the restriction of
F to Ω coincides with f .
Definition 0.1. Let Ω be a bounded open subset of Rn. We denote by Cm,α(Ω)
the subspace of Cm(Ω) whose functions have mth order derivatives that are
Holder continuous with exponent α ∈ (0, 1].
6
Notation
Definition 0.2. By definition, a function f belongs to C∞(Ω) if f ∈ C∞(Ω)
and for all x ∈ ∂Ω and for all α ∈ Nn there exists the limit
limy→x
y∈ΩDαf(y).
(By definition Dαf(x) = limy→x
y∈ΩDαf(y).)
Definition 0.3. Let p ∈ [1,+∞]. Let λ > 0, and m, k be two natural numbers
such that m < λ < m + k. Then f ∈ Hλp (Nikol’skii space) if and only if
f ∈ Lp and
sup|α|=m, |h|=0
|h|m−λ∥khD
αf∥Lp < +∞.
This space does not depend on the choice of the integers k,m satisfying
the inequality m < λ < m+ k. We recall that Hλp is also known as the Besov
space Bλp,∞.
Definition 0.4. Let V,Ω be open subsets of Rn. We write
V ⊂⊂ Ω
if V ⊂ V ⊂ Ω and V is compact, and say that V is compactly embedded in Ω.
Definition 0.5. Let X and Y be normed space. By L(X ,Y) we denote the
normed space of the continuous linear maps of X to Y equipped with the topol-
ogy of uniform convergence on the unit sphere of X .
7
Chapter 1
Morrey and Sobolev Morrey
spaces
1.1 General Morrey spaces
Definition 1.1. Let Ω be a Lebesgue measurable subset of Rn. Let 0 < p ≤
+∞ and let w be a measurable function from ]0,+∞[ to ]0,+∞[. Denote by
Mw(·)p (Ω) the space of all real-valued measurable functions on Ω for which
∥f∥Mw(·)p (Ω)
= supx∈Ω
∥w(ρ)∥f∥Lp(B(x,ρ)∩Ω)∥L∞(0,∞) <∞.
Definition 1.2. Let 0 < p ≤ +∞. Denote by Λp,∞ the set of all measurable
functions w from ]0,+∞[ to ]0,+∞[ which are not equivalent to 0 such that
∥w(ρ)∥L∞(1,∞) <∞, ∥w(ρ)ρnp ∥L∞(0,1) <∞.
In [15], [18] it is proved that, if w is a non-negative measurable function
from ]0,+∞[ to ]0,+∞[ which are not equivalent to 0, then the spaceMw(·)p (Ω)
is non-trivial, i.e. consists not only of functions f equivalent to 0 on Ω if, and
only if, w ∈ Λp,∞.
Definition 1.3. If wλ(ρ) =
ρ−λ, ρ ∈]0, 1],
1, ρ ≥ 1,, then we set
Mλp (Ω) ≡ Mwλ
p (Ω)
and the condition wλ ∈ Λp,∞ means that 0 ≤ λ ≤ np.
9
1. Morrey and Sobolev Morrey spaces
Note that
Lemma 1.4.
∥f∥Mλp (Ω) = max
supx∈Ω
sup0<ρ<1
ρ−λ∥f∥Lp(B(x,ρ)∩Ω), ∥f∥Lp(Ω)
. (1.1)
We find convenient to set
|f |ρ,w,p,Ω ≡ supx∈Ω
∥∥w(r)∥f∥Lp(B(x,r)∩Ω)
∥∥L∞(0,ρ)
∀ρ ∈]0,+∞[,
and
|f |ρ,λ,p,Ω ≡ |f |ρ,wλ,p,Ω
for all measurable functions f from Ω to ]0,+∞[ and for all functions w from
]0,+∞[ to ]0,+∞[. Clearly, |f |ρ,w,p,Ω ∈ [0,+∞].
Definition 1.5. Let Ω be an open subset of Rn. Let p ∈ [1,+∞].
(i) Let w be a function from ]0,+∞[ to ]0,+∞[. Then we define as gener-
alized little Morrey space with weight w and exponent p the subspace
Mw,0p (Ω) ≡
f ∈ Mw
p (Ω) : limρ→0
|f |ρ,w,p,Ω = 0
of Mw
p (Ω).
(ii) Let λ ∈ [0,+∞[. Then, in particular, the little Morrey space with expo-
nents λ, p is the subspace
Mλ,0p (Ω) ≡
f ∈Mλ
p (Ω) : limρ→0
|f |ρ,λ,p,Ω = 0
of Mλ
p (Ω).
Example 1.6. Let 0 < λ < n/p. Then
1) The function |x|α ∈Mλp (B(0, 1)) if and only if α ≥ λ− n
p.
2) The function |x|α ∈Mλ,0p (B(0, 1)) if and only if α > λ− n
p.
Lemma 1.7. Let Ω be an open subset of Rn. Let p ∈ [1,+∞]. Let w ∈ Λp,∞.
Then Mw,0p (Ω) is a closed proper subspace of Mw
p (Ω).
Proof. Let f ∈ Mwp (Ω). Let fjj∈N be a sequence inMw,0
p (Ω) which converges
to f in Mwp (Ω). We want to prove that f ∈ Mw,0
p (Ω).
10
1.1 General Morrey spaces
Let ε > 0. Since fj → f as j → ∞, there exists N1 ∈ N such that
|f − fk|+∞,w,p,Ω <ε
2∀ k ≥ N1.
By definition 1.5, there exists δ > 0 such that if ρ ∈]0, δ[, then
|fN1 |ρ,w,p,Ω <ε
2.
Since f = f − fN1 + fN1 , we have
|f |ρ,w,p,Ω ≤ |f − fN1 |+∞,w,p,Ω + |fN1 |ρ,w,p,Ω <ε
2+ε
2= ε,
for all ρ ∈]0, δ[.
Lemma 1.8. Let Ω be a bounded open subset of Rn. Let p ∈ [1,+∞]. Then
Mr−λ
p (Ω) =Mλp (Ω). Moreover, the quasi-norm
∥f∥Mρ−λ
p (Ω)= sup
x∈Ωρ>0
ρ−λ∥f∥Lp(B(x,ρ)∩Ω) ∀f ∈ Mρ−λ
p (Ω)
is equivalent to the quasi-norm
∥f∥Mλp (Ω) = sup
x∈Ω∥wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω)∥L∞(0,∞) =
= supx∈Ωρ>0
wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω) ∀f ∈Mλp (Ω).
Proof. A simple calculation shows that
∥f∥Mρ−λ
p (Ω)≤ ∥f∥Mλ
p (Ω).
Indeed, ρ−λ ≤ wλ(ρ) for all ρ ∈]0,+∞[ and thus
∥f∥Mρ−λ
p (Ω)= sup
(x,ρ)∈Ω×]0,+∞[
ρ−λ∥f∥Lp(B(x,ρ)∩Ω) ≤
≤ sup(x,ρ)∈Ω×]0,+∞[
wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω) = ∥f∥Mλp (Ω).
Conversely,
∥f∥Mλp (Ω) = sup
(x,ρ)∈Ω×]0,+∞[
wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω) ≤
≤ sup
sup
(x,ρ)∈Ω×]0,1]ρ−λ∥f∥Lp(B(x,ρ)∩Ω), sup
(x,ρ)∈Ω×]1,+∞[
1 · ∥f∥Lp(B(x,ρ)∩Ω)
=
11
1. Morrey and Sobolev Morrey spaces
= sup
sup
(x,ρ)∈Ω×]0,1]ρ−λ∥f∥Lp(B(x,ρ)∩Ω), sup
(x,ρ)∈Ω×]1,+∞[
∥f∥Lp(B(x,ρ)∩Ω)
.
Now we estimate
sup(x,ρ)∈Ω×]1,+∞[
∥f∥Lp(B(x,ρ)∩Ω).
If 1 < ρ < diamΩ, then
∥f∥Lp(B(x,ρ)∩Ω) ≤ (diamΩ)λ(diamΩ)−λ∥f∥Lp(B(x,ρ)∩Ω) ≤
≤ (diamΩ)λ sup(x,ρ)∈Ω×]1, diamΩ[
ρ−λ∥f∥Lp(B(x,ρ)∩Ω) ≤
≤ (diamΩ)λ sup(x,ρ)∈Ω×]0,+∞[
ρ−λ∥f∥Lp(B(x,ρ)∩Ω) = (diamΩ)λ∥f∥Mρ−λ
p (Ω).
If ρ ≥ sup1, diamΩ, then
∥f∥Lp(B(x,ρ)∩Ω) = ∥f∥Lp(Ω) ≤
≤ (diamΩ)λ(diamΩ)−λ∥f∥Lp(B(x, diamΩ)∩Ω) ≤
≤ (diamΩ)λ sup(x,ρ)∈Ω×]0,+∞[
ρ−λ∥f∥Lp(B(x,ρ)∩Ω) = (diamΩ)λ∥f∥Mρ−λ
p (Ω).
Therefore,
∥f∥Mλp (Ω) ≤ max1, (diamΩ)λ∥f∥
Mρ−λp (Ω)
,
and proof is complete.
Lemma 1.9. Let Ω be an open subset of Rn. Let 1 ≤ p ≤ +∞, 0 < λ < n/p.
Then Mλp (Ω) ⊆ Lp(Ω).
Proof. Let f ∈Mλp (Ω). We note that
wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω) ≤ ∥f∥Mλp (Ω) <∞ for all (x, ρ) ∈ Ω×]0,+∞[.
Since wλ(ρ) = 1 for ρ ≥ 1, we obtain
∥f∥Lp(B(x,ρ)∩Ω) ≤ ∥f∥Mλp (Ω) <∞ for all (x, ρ) ∈ Ω× [1,+∞[.
By taking supremum in ρ ≥ 1 we get
∥f∥Lp(Ω) ≤ ∥f∥Mλp (Ω).
12
1.1 General Morrey spaces
Lemma 1.10. Let Ω be an open subset of Rn. Let 0 < p < ∞, 0 < λ < np.
Then
Mλp (Ω) * Lloc
q (Ω)
for any q > p.
Proof. Without loss of generality, we can assume that 0 ∈ Ω and B(0, 1) ⊂ Ω.
Let
f(x) =
(2kkn−1)1p , x ∈ B
(0, 1
k
)\B
(0, 1
k− 2−kk−pλ−1
), k ∈ N,
0, otherwise.
Note that∣∣∣∣B(0, 1k)\B
(0,
1
k− 2−kk−pλ−1
)∣∣∣∣ = vn
((1
k
)n
−(1
k− 2−kk−λp−1
)n)≤
≤ n vn
(1
k
)n−1
2−kk−λp−1 = σn2−kk−n−λp−2, (1.2)
where vn, σn respectively, is the volume, the surface area respectively, of the
unit ball in Rn. Similarly, since 1k− 2−kk−λp−1 > 1
2k,∣∣∣∣B(0, 1k
)\B
(0,
1
k− 2−kk−λp−1
)∣∣∣∣ ≥ 21−nσn2−kk−n−λp−2. (1.3)
By using inequality (1.2) and the inequality
∑k≥a
1
kα+1≤(1 +
1
α
)1
aα, where α > 0, a ≥ 1,
we get that for any 0 < r ≤ 1 and x ∈ Rn
∥f∥pLp(B(x,r)) ≤ ∥f∥pLp(B(0,r)) =
=∑
1k−2−kk−λp−1≤r
2kkn−1∣∣∣∣B(0, 1k
)\B
(0,
1
k− 2−kk−λp−1
)∣∣∣∣ ≤≤ σn
∑k≥ 1
2r
1
kλp+1≤ σn
(1 +
1
λp
)2λprλp.
If r ≥ 1 and x ∈ Rn, then
∥f∥pLp(B(x,r)) ≤ ∥f∥pLp(B(0,r)) = ∥f∥pLp(B(0,1)) = σn
(1 +
1
λp
)2λp.
Therefore, f ∈Mλp (Ω).
13
1. Morrey and Sobolev Morrey spaces
On the other hand, by (1.3) for any q > p
∥f∥qLq(B(0,r)) ≥∑1k≤r
(2kkn−1)qp
∣∣∣∣B(0, 1k)\B
(0,
1
k− 2−kk−λp−1
)∣∣∣∣ ≥≥ 21−nσn
∑k≥ 1
r
2k(qp−1)k(n−1)
qp−n−λp−2 = ∞.
Hence, f /∈ Llocq (Ω).
Corollary 1.11. Let Ω be an open subset of Rn. Let 0 < p < ∞, 0 < λ < np.
Then
Hλp (Ω) ⊂Mλ
p (Ω)
and this inclusion is strict.
Proof. The above inclusion was proved in [29] for n = 1 and in [49] for n > 1.
The strictness of the inclusion follows since by the embedding theorem [48]
Hλp (Ω) ⊂ Lloc
q (Ω)
with q = npn−λp > p. Hence, the function f constructed in the proof of the
previous Lemma belongs to Mλp (Ω) but does not belong to Hλ
p (Ω).
Lemma 1.12. Let Ω be a Lebesgue measurable subset of Rn, mn(Ω) < ∞.
Let p ∈ [1,+∞[. Then the following statements hold.
(i) If λ ≤ np, then L∞(Ω) ⊆Mλ
p (Ω).
(ii) If λ < np, then L∞(Ω) ⊆Mλ,0
p (Ω).
Proof. (i) Let f ∈ L∞(Ω). Then we note that
∥f∥Lp(B(x,r)∩Ω) ≤ (mn(B(x, r) ∩ Ω))1p∥f∥L∞(Ω)
and
∥f∥Mλp (Ω) = sup
x∈Ωsupρ>0
wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω) ≤
≤ max
supx∈Ω
sup0<ρ≤1
ρ−λ(vnρn)
1p∥f∥L∞(Ω),
supx∈Ω
supρ>1
(mn(Ω))1p∥f∥L∞(Ω)
=
= max
v
1pn , (mn(Ω))
1p
∥f∥L∞(Ω).
14
1.1 General Morrey spaces
(ii) Let f ∈ L∞(Ω). Then for all ρ ∈]0, 1] we consider the norm
|f |ρ,wλ,p,Ω = sup(x,r)∈Ω×]0,ρ[
wλ(r)∥f∥Lp(B(x,r)∩Ω) ≤
≤ sup(x,r)∈Ω×]0,ρ[
wλ(r)(mn(B(x, r) ∩ Ω))1p∥f∥L∞(Ω) ≤
≤ sup(x,r)∈Ω×]0,ρ[
r−λ(mn(B(x, r)))1p∥f∥L∞(Ω) ≤
≤ sup(x,r)∈Ω×]0,ρ[
rnp−λv
1pn ∥f∥L∞(Ω) → 0 as ρ→ 0.
Hence, we obtain that f ∈Mλ,0p (Ω).
Corollary 1.13. Let Ω be a Lebesgue measurable subset of Rn. Let p ∈
[1,+∞[, λ ∈ [0, n/p]. If f ∈ L∞(Ω) and supp f is compact, then f ∈Mλp (Ω).
Proof. Let f ∈ L∞(Ω). Then
∥f∥Mλp (Ω) = sup
x∈Ωsupρ>0
wλ(ρ)∥f∥Lp(B(x,ρ)∩Ω) ≤
≤ supx∈Ω
supρ>0
wλ(ρ)(mn(B(x, r) ∩ supp f))1p∥f∥L∞(Ω) ≤
≤ max
supx∈Ω
sup0<ρ≤1
ρ−λ(vnρn)
1p∥f∥L∞(Ω),
supx∈Ω
supρ>1
(mn(supp f))1p∥f∥L∞(Ω)
=
= max
v
1pn , (mn(supp f))
1p
∥f∥L∞(Ω).
Corollary 1.14. Let Ω be an open subset of Rn. Let p ∈ [1,+∞[, λ ∈ [0, n/p].
Then C∞c (Ω) ⊂Mλp (Ω).
Next we state a known result for Morrey spaces. For the sake of complete-
ness we also give proofs.
Theorem 1.15. Let Ω be a bounded open subset of Rn. If 1 ≤ p < +∞ then
the following statements hold.
(i) M0p (Ω) = Lp(Ω);
(ii) If λ = np, then Mλ
p (Ω) = L∞(Ω) both algebraically and topologically;
15
1. Morrey and Sobolev Morrey spaces
(iii) If np< λ ≤ +∞, 1 ≤ p < +∞, then Mλ
p (Ω) = 0.
(iv) Let 0 < p ≤ q ≤ +∞ and 0 ≤ λ ≤ np, 0 ≤ ν ≤ n
q. If n
q− ν ≤ n
p− λ, then
M νq (Ω) →Mλ
p (Ω).
Proof. (i) Let us take f ∈M0p (Ω) and consider its norm in this space
∥f∥M0p (Ω) = sup
x∈Ωρ>0
w0(ρ)∥f∥Lp(B(x,ρ)∩Ω) = supx∈Ωρ>0
∥f∥Lp(B(x,ρ)∩Ω) = ∥f∥Lp(Ω).
(ii) If λ = np, then by the Lebesgue Theorem
∥f∥M
npp (Ω)
= supx∈Ωρ>0
ρ−np ∥f∥Lp(B(x,ρ)∩Ω) = υ
1pn sup
x∈Ωρ>0
∫
B(x,ρ)∩Ω|f(y)|pdy
υnρn
1p
≥
≥ υ1pn ess sup |f(x)| = υ
1pn ∥f∥L∞(Ω),
where υn is the volume of the unit ball in the space Rn.
Suppose now that f ∈ Mnpp (Ω) and f /∈ L∞(Ω). Since f /∈ L∞(Ω), i.e.
∥f∥L∞(Ω) = ∞, then for every K > 0 the set
S(f,K) = x ∈ Ω: |f(x)| > K
has positive measure.
Denote by A the set of all points x ∈ Ω for which
|f(x)|p = limρ→0+
1
mn(B(x, ρ))
∫B(x,ρ)
|f(y)|pdy.
Thus, A ∩ S(f,K) has a positive measure.
If x ∈ A ∩ S(f,K), then
limρ→0+
1
mn(B(x, ρ))
∫B(x,ρ)∩Ω
|f(y)|pdy = |f(x)|p > Kp.
From this fact it easy to see that for every K > 0 there exists ρ such
that1
mn(B(x, ρ))
∫B(x,ρ)∩Ω
|f(y)|pdy > K,
16
1.1 General Morrey spaces
and, thus,
K1p < v
− 1p
n ρ−np
∫B(x,ρ)∩Ω
|f(y)|pdy
1p
≤ v− 1
pn sup
x∈Ωρ>0
ρ−np ∥f∥Lp(B(x,ρ)∩Ω).
So we have ∥f∥M
npp (Ω)
= ∞. This contradicts the assumptions that
f ∈ Mnpp (Ω).
Now let f ∈ L∞(Ω). For every ordered pair (x, ρ) ∈ Ω×]0,+∞[ we have
ρ−np
∫B(x,ρ)∩Ω
|f(y)|pdy
1p
≤ ρ−np
∫B(x,ρ)∩Ω
dy
1p
∥f∥L∞(Ω) ≤
≤ ρ−np υ
1pn ρ
np ∥f∥L∞(Ω) = υ
1pn ∥f∥L∞(Ω) ∀ρ ∈]0,+∞[.
Thus,
∥f∥M
npp (Ω)
≤ υ1pn ∥f∥L∞(Ω).
From this inequality follows the continuity of the identity operator
I : L∞(Ω) →Mnpp (Ω).
(iii) Let f ∈Mλp (Ω), then exploiting the Lebesgue Theorem we obtain
∥f∥Mλp (Ω) = sup
x∈Ωρ>0
ρ−λ∥f∥Lp(B(x,ρ)∩Ω) =
= supx∈Ωρ>0
ρ−λ+np
∫
B(x,ρ)∩Ω|f |pdx
ρn
1p
< +∞ ⇔ f(x) = 0 a.e.
17
1. Morrey and Sobolev Morrey spaces
(iv) Let f ∈M νq (Ω), then
∥f∥Mρ−λ
p (Ω)= sup
x∈Ωρ>0
ρ−λ∥f∥Lp(B(x,ρ)∩Ω) ≤
≤ supx∈Ωρ>0
ρ−λ[mn(B(x, ρ) ∩ Ω)]1p− 1
q ∥f∥Lq(B(x,ρ)∩Ω) ≤
≤ max
sup
(x,ρ)∈Ω×]0,1]ρ−λ[mn(B(x, ρ))]
1p− 1
q ∥f∥Lq(B(x,ρ)∩Ω),
sup(x,ρ)∈Ω×[1,+∞[
ρ−λ[mn(Ω)]1p− 1
q ∥f∥Lq(B(x,ρ)∩Ω)
≤
≤ max
sup
(x,ρ)∈Ω×]0,1]υ
1p− 1
qn ρ−λ+
np−n
q ∥f∥Lq(B(x,ρ)∩Ω),
[mn(Ω)]1p− 1
q sup(x,ρ)∈Ω×[1,+∞[
wν(ρ)∥f∥Lq(B(x,ρ)∩Ω)
≤
≤ max
υ
1p− 1
qn , [mn(Ω)]
1p− 1
q
∥f∥Mν
q (Ω),
where υn is the volume of the unit ball in the space Rn.
Theorem 1.16. Let Ω be an open subset of Rn. Let p1, p2 ∈ [1,+∞] be such
that 1p1
+ 1p2
= 1p. Let λ1, λ2 ∈ [0,+∞[, λ = λ1 + λ2. Then the pointwise
multiplication is bilinear and continuous from Mλ1p1(Ω) × Mλ2
p2(Ω) to Mλ
p (Ω)
and maps Mλ1,0p1
(Ω)×Mλ2p2(Ω) to Mλ,0
p (Ω) and Mλ1p1(Ω)×Mλ2,0
p2(Ω) to Mλ,0
p (Ω)
Remark 1.17. This statement proves the Holder inequality for Morrey space
Mλp (Ω):
∥fg∥Mλp (Ω) ≤ ∥f∥
Mλ1p1
(Ω)∥g∥
Mλ2p2
(Ω)∀ (f, g) ∈Mλ1
p1(Ω)×Mλ2
p2(Ω).
Proof. Note that
wλ(ρ) =
ρ−λ, ρ ∈]0, 1],
1, ρ ≥ 1,=
ρ−λ1−λ2 , ρ ∈]0, 1],
1, ρ ≥ 1,= wλ1(ρ)wλ2(ρ).
Then, by Holder inequality, we have
|fg|ρ,λ,p,Ω = sup(x,r)∈Ω×]0,ρ[
wλ(r)∥fg∥Lp(B(x,r)∩Ω) ≤
≤ sup(x,r)∈Ω×]0,ρ[
wλ(r)∥f∥Lp1 (B(x,r)∩Ω)∥g∥Lp2 (B(x,r)∩Ω) ≤
18
1.1 General Morrey spaces
≤ sup(x,r)∈Ω×]0,ρ[
wλ1(r)∥f∥Lp1 (B(x,r)∩Ω) sup(x,r)∈Ω×]0,ρ[
wλ2(r)∥g∥Lp2 (B(x,r)∩Ω) =
= |f |ρ,λ1,p1,Ω|g|ρ,λ2,p2,Ω for all ρ ∈]0,+∞].
Therefore, by taking ρ = +∞, we deduce that fg ∈ Mλp (Ω) when
(f, g) ∈ Mλ1p1(Ω)×Mλ2
p2(Ω).
By letting ρ→ 0, we deduce that fg ∈Mλ,0p (Ω) when (f, g) ∈ Mλ1,0
p1(Ω)×
Mλ2p2(Ω).
The case when f ∈Mλ1p1(Ω) and g ∈Mλ2,0
p2(Ω) can be analyzed in the same
way.
Theorem 1.18. Let Ω be an open subset of Rn. Let p ∈ [1,+∞]. Let
λ ∈ [0, n/p]. Then the pointwise multiplication is bilinear and continuous
from Mλp (Ω)× L∞(Ω) to M
λp (Ω) and maps Mλ,0
p (Ω)× L∞(Ω) to Mλ,0p (Ω).
Proof. We show that if f ∈Mλp (Ω), g ∈ L∞(Ω), then
|fg|ρ,λ,p,Ω = sup(x,r)∈Ω×]0,ρ[
wλ(r)∥fg∥Lp(B(x,r)∩Ω) ≤
≤ ∥g∥L∞(Ω) sup(x,r)∈Ω×]0,ρ[
wλ(ρ)∥f∥Lp(B(x,r)∩Ω) =
= ∥g∥L∞(Ω)|f |ρ,λ,p,Ω for all ρ ∈]0,+∞].
Hence, by taking ρ = +∞, we deduce that fg ∈ Mλp (Ω) when
(f, g) ∈ Mλp (Ω)× L∞(Ω).
By letting ρ→ 0, we deduce that fg ∈Mλ,0p (Ω) when (f, g) ∈ Mλ,0
p (Ω)×
L∞(Ω).
Lemma 1.19. Let A ⊂ Rm be a measurable set. Let Ω be an open subset of Rn.
Let p ∈ [1,+∞]. Let λ ∈ [0, n/p]. Suppose that f is a measurable from A× Ω
to R. Let f(·, y) ∈Mλp (Ω) for almost all y ∈ A and
∫A
∥f(·, y)∥Mλp (Ω) dy < +∞.
Then for almost all x ∈ Ω the integral∫A
f(x, y)dy makes sense and Minkowski’s
inequality for Morrey spaces∥∥∥∥∥∥∫A
f(·, y)dy
∥∥∥∥∥∥Mλ
p (Ω)
≤∫A
∥f(·, y)∥Mλp (Ω) dy
holds.
19
1. Morrey and Sobolev Morrey spaces
Proof. By Minkowski’s inequality for the Lebesgue spaces and by the imbed-
ding of Mλp (Ω) into Lp(Ω) we know that for almost all x ∈ Ω the integral∫
A
f(x, y)dy makes sense and defines almost everywhere a function of Lp(Ω).
Then by applying the Minkowski’s inequality for the Lebesgue spaces in
(B(x, ρ) ∩ Ω)× A for all ρ ∈]0,+∞[, we obtain the following inequality∥∥∥∥∥∥∫A
f(·, y)dy
∥∥∥∥∥∥Mλ
p (Ω)
= sup(x,ρ)∈Ω×]0,+∞[
wλ(ρ)
∥∥∥∥∥∥∫A
f(·, y)dy
∥∥∥∥∥∥Lp(B(x,ρ)∩Ω)
≤
≤ sup(x,ρ)∈Ω×]0,+∞[
wλ(ρ)
∫A
∥f(·, y)∥Lp(B(x,ρ)∩Ω) dy =
=
∫A
sup(x,ρ)∈Ω×]0,+∞[
wλ(ρ) ∥f(·, y)∥Lp(B(x,ρ)∩Ω) dy =
∫A
∥f(·, y)∥Mλp (Ω) dy.
1.2 Approximation by C∞ functions in Morrey
spaces
Definition 1.20. If ϕ ∈ L1(Rn) and t ∈]0,+∞[, we denote by ϕt(·) the func-
tion from Rn to R defined by
ϕt(x) ≡ t−nϕ(x/t) ∀ x ∈ Rn.
By the formula of change of variables in integrals, we conclude that∫Rn
ϕt(x)dx =
∫Rn
ϕ(x)dx ∀ t ∈]0,+∞[,
whenever ϕ ∈ L1(Rn).
Lemma 1.21. Let p ∈ [1,+∞[, 0 ≤ λ < npand f ∈Mλ,0
p (Rn). Then
limk→∞
fχB(0,k) = f in Mλp (Rn).
Proof. Consider for 0 < ρ ≤ 1 the norm
∥fχB(0,k) − f∥Mλp (Rn) = sup
x∈Rn
∥wλ(r)∥fχB(0,k) − f∥Lp(B(x,r))∥L∞(0,∞) =
= supx∈Rn
∥wλ(r)∥f∥Lp(B(x,r)\B(0,k))∥L∞(0,∞) ≤
20
1.2 Approximation by C∞ functions in Morrey spaces
≤ supx∈Rn
∥wλ(r)∥f∥Lp(B(x,r))∥L∞(0,ρ) +maxρ−λ, 1∥f∥Lp(Rn\B(0,k)) =
= |f |ρ,λ,p,Rn + ρ−λ∥f∥Lp(Rn\B(0,k)).
Let k → ∞, then since f ∈ Lp(Rn) (by Lemma 1.9) and p <∞ we have
limk→∞
∥f∥Lp(Rn\B(0,k)) = 0.
Therefore, for all 0 < ρ ≤ 1
limk→∞
∥fχB(0,k) − f∥Mλp (Rn) ≤ |f |ρ,λ,p,Rn .
By passing to the limit as ρ→ 0, since f ∈Mλ,0p (Rn), we have
limk→∞
∥fχB(0,k) − f∥Mλp (Rn) = 0,
or equivalently
limk→∞
∥fχB(0,k) − f∥Mλp (Rn) = 0.
Then we have the following result of approximation by convolution.
Theorem 1.22. Let ϕ ∈ C∞c (Rn),∫Rn
ϕ(x)dx = 1. Then the following state-
ments hold.
(i) Let p ∈ [1,+∞], λ ∈[0, n
p
]. If f ∈Mλ
p (Rn) and ε > 0, then the function
f ∗ ϕε from Rn to R defined by
f ∗ ϕε ≡∫Rn
f(x− y)ϕε(y)dy ∀ x ∈ Rn
belongs to Mλp (Rn) ∩ C∞(Rn) and
∥f ∗ ϕε∥Mλp (Rn) ≤ ∥ϕ∥L1(Rn)∥f∥Mλ
p (Rn) ∀ f ∈Mλp (Rn).
(ii) Let p ∈ [1,+∞], λ ∈[0, n
p
]. If f ∈ Mλ,0
p (Rn) and ε > 0, then f ∗ ϕε
belongs to Mλ,0p (Rn) ∩ C∞(Rn).
(iii) Let p ∈ [1,+∞[. If f ∈ Mλ,0p (Rn), then f ∗ ϕε belongs to Mλ,0
p (Rn) ∩
C∞(Rn) ∩ C0ub(Rn) for all ε ∈]0,+∞[ and
limε→0
f ∗ ϕε = f in Mλp (Rn). (1.4)
21
1. Morrey and Sobolev Morrey spaces
(iv) Let p ∈ [1,+∞[. Then
clMλp (Rn)C
∞c (Rn) =Mλ,0
p (Rn). (1.5)
Proof. (i) Let f ∈Mλp (Rn) and ε > 0, then
wλ(ρ)∥f ∗ ϕε∥Lp(B(x,ρ)) ≤ wλ(ρ)
∥∥∥∥∥∥∫Rn
f(ξ − y)ϕε(y)dy
∥∥∥∥∥∥Lp,ξ(B(x,ρ))
≤
≤ wλ(ρ)
∫B(0,ε)
∥f(ξ − y)∥Lp,ξ(B(x,ρ))|ϕε(y)|dy =
= wλ(ρ)
∫Rn
|ϕε(y)|dy
supy∈B(0,ε)
∥f(ξ − y)∥Lp,ξ(B(x,ρ)) =
= wλ(ρ)
∫Rn
|ϕ(y)|dy
supy∈B(0,ε)
∥f(z)∥Lp(B(x−y,ρ)) =
= wλ(ρ)
∫Rn
|ϕ(y)|dy
supz∈B(x,ε)
∥f∥Lp(B(z,ρ)) ≤
≤ wλ(ρ)
∫Rn
|ϕ(y)|dy
supz∈Rn
∥f∥Lp(B(z,ρ)) for all ρ ∈]0,+∞[.
Thus, we have
∥f ∗ ϕε∥Mλp (Rn) ≤
∫Rn
|ϕ|dx
∥f∥Mλp (Rn) ∀ f ∈Mλ
p (Rn).
(ii) Let f ∈ Mλ,0p (Rn) and ε > 0. In the proof of statement (i) we have
proved that
wλ(r)∥f ∗ ϕε∥Lp(B(x,r)) ≤
wλ(r)
∫Rn
|ϕ(x)|dx
supx∈Rn
∥f∥Lp(B(x,r)) for all r ∈]0,+∞[.
Moreover,
wλ(r)∥f∥Lp(B(x,r)) ≤ |f |ρ,λ,p,Rn ∀x ∈ Rn, r ∈]0, ρ[,
and thus, by taking the supremum on x ∈ Rn, we have
wλ(r) supx∈Rn
∥f∥Lp(B(x,r)) ≤ |f |ρ,λ,p,Rn ∀ r ∈]0, ρ[.
22
1.2 Approximation by C∞ functions in Morrey spaces
Hence,
supr∈]0,ρ[
wλ(r)∥f ∗ ϕε∥Lp(B(x,r)) ≤
∫Rn
|ϕ(x)|dx
|f |ρ,w,p,Rn ,
and
limρ→0
|f ∗ ϕε|ρ,λ,p,Rn = 0 ∀ f ∈Mλ,0p (Rn).
(iii) Let η > 0. Since f ∈Mλ,0p , there exists ρη > 0 such that
|f |ρ,λ,p,Rn ≤ η
1 +
∫Rn
|ϕ(y)|dy
−1 ∀ ρ ∈]0, ρη].
Then we have
sup(x,r)∈Rn×]0,ρη [
wλ(r)∥f − f ∗ ϕε∥Lp(B(x,r)) ≤
≤ sup(x,r)∈Rn×]0,ρη [
wλ(r)∥f∥Lp(B(x,r)) + sup(x,r)∈Rn×]0,ρη [
wλ(r)∥f ∗ ϕε∥Lp(B(x,r)) ≤
≤ |f |ρη ,λ,p,Rn +
∫Rn
|ϕ(y)|dy
|f |ρη ,λ,p,Rn =
= |f |ρη ,λ,p,Rn
1 +
∫Rn
|ϕ(y)|dy
≤ η, (1.6)
for all ε ∈]0,+∞[.
Further, we have
sup(x,r)∈Rn×[ρη ,+∞[
wλ(r)∥f − f ∗ ϕε∥Lp(B(x,r)) ≤
≤
(sup
r∈[ρη ,+∞[
wλ(r)
)∥f − f ∗ ϕε∥Lp(Rn) for all ε > 0.
Since f ∈ Lp(Rn) (by Lemma 1.9) and p ∈ [1,+∞[ and∫Rn
|ϕ(y)|dy = 1,
standard properties of approximate identities of convolution imply that
limε→0
∥f − f ∗ ϕε∥Lp(Rn) = 0.
Thus, there is exists εη > 0 such that(sup
r∈[ρη ,+∞[
wλ(r)
)∥f − f ∗ ϕε∥Lp(Rn) ≤ η ∀ ε ∈]0, εη].
23
1. Morrey and Sobolev Morrey spaces
So, we obtain
sup(x,r)∈Rn×[ρη ,+∞[
wλ(r)∥f − f ∗ ϕε∥Lp(B(x,r)) ≤ η ∀ ε ∈]0, εη]. (1.7)
By combining inequalities (1.6) and (1.7), we have
sup(x,r)∈Rn×[0,+∞[
wλ(r)∥f − f ∗ ϕε∥Lp(B(x,r)) ≤ η ∀ ε ∈]0, εη].
Hence,
limε→0
f ∗ ϕε = f in Mλp (Rn).
(iv) Clearly, C∞c (Rn) is contained in Mλ,0p (Rn). Since Mλ,0
p (Rn) is closed in
Mλp (Rn) (by Lemma 1.7), we obtain
clMλp (Rn)C
∞c (Rn) ⊂ clMλ
p (Rn)Mλ,0p (Rn) =Mλ,0
p (Rn).
Now let f ∈Mλ,0p (Rn). Then for all k ∈ N
fχB(0,k) ∗ ϕ 1k∈ C∞c (Rn).
Next, using (i) of Theorem 1.22, we obtain
∥fχB(0,k) ∗ ϕ 1k− f∥Mλ
p (Rn) ≤
≤ ∥(fχB(0,k) − f) ∗ ϕ 1k∥Mλ
p (Rn) + ∥f ∗ ϕ 1k− f∥Mλ
p (Rn) ≤
≤ ∥fχB(0,k) − f∥Mλp (Rn)∥ϕ∥L1(Rn) + ∥f ∗ ϕ 1
k− f∥Mλ
p (Rn).
By Lemma 1.21
limk→∞
∥fχB(0,k) − f∥Mλp (Rn) = 0,
and by (iii) of Theorem 1.22
limk→∞
∥f ∗ ϕ 1k− f∥Mλ
p (Rn) = 0.
So, we obtain that
limk→∞
∥fχB(0,k) ∗ ϕ 1k− f∥Mλ
p (Rn) = 0.
Thus, f ∈ clMλp (Rn)C
∞c (Rn) and equality (1.5) holds.
24
1.2 Approximation by C∞ functions in Morrey spaces
Remark 1.23. The limiting relation (1.4) does not hold for all f ∈ Mλp (Rn).
For example,
|x|λ−npχB(0,1) ∗ ϕε 9 |x|λ−
npχB(0,1) in Mλ
p (Rn) (1.8)
as ε→ 0.
Indeed, function |x|λ−npχB(0,1) ∗ ϕε belongs to C∞c (Rn). If
|x|λ−npχB(0,1) ∗ ϕε → |x|λ−
npχB(0,1) in Mλ
p (Rn)
then by (iv) of Theorem 1.22 |x|λ−npχB(0,1) ∈ clMλ
p (Rn)C∞c (Rn) = Mλ,0
p (Rn),
which is not true (see Example 1.6).
Lemma 1.24. Let Ω be an open subset of Rn. Let EΩ be the extension operator
from M(Ω) to M(Rn) defined by
EΩf ≡
f, in Ω,
0, in Rn \ Ω,
Let p ∈ [1,+∞], λ ∈ [0, n/p]. Then for all f ∈Mλp (Ω)
|EΩf |ρ,wλ,p,Rn ≤ 2λ|f |2ρ,wλ,p,Ω (1.9)
for all ρ ∈]0,+∞] and
∥EΩf∥Mλp (Rn) ≤ 2λ∥f∥Mλ
p (Ω). (1.10)
In particular, EΩ maps Mλp (Ω) to M
λp (Rn) and Mλ,0
p (Ω) to Mλ,0p (Rn).
Proof. Let ρ ∈]0,+∞]. Then we have
|EΩf |ρ,wλ,p,Rn = sup(x,r)∈Rn×]0,ρ[
wλ(r)∥EΩf∥Lp(B(x,r)) =
= sup0<r<ρ
supx∈Rn
wλ(r)∥f∥Lp(B(x,r)∩Ω) =
= sup0<r<ρ
supx∈Rn
B(x,r)∩Ω=∅
wλ(r)∥f∥Lp(B(x,r)∩Ω).
If (x, r) ∈ Rn×]0, ρ[ and B(x, r)∩Ω = ∅, then there exists ξ(x) ∈ B(x, r)∩
Ω. By the triangle inequality, we have
B(x, r) ∩ Ω ⊂ B(ξ(x), 2r) ∩ Ω,
25
1. Morrey and Sobolev Morrey spaces
hence,
|EΩf |ρ,wλ,p,Rn ≤ sup0<r<ρ
supx∈Rn
B(x,r)∩Ω=∅
wλ(r)∥f∥Lp(B(ξ(x),2r)∩Ω) ≤
≤ sup0<r<ρ
supη∈Ω
wλ(r)∥f∥Lp(B(η,2r)∩Ω).
We also note that
wλ(ρ) ≤ 2λwλ(2ρ) ∀ ρ ∈ [0,+∞[.
Therefore,
|EΩf |ρ,wλ,p,Rn ≤ 2λ supη∈Ω
sup0<r<ρ
wλ(2r)∥f∥Lp(B(η,2r)∩Ω) = 2λ|f |2ρ,wλ,p,Ω.
Inequality (1.10) follows by inequality (1.9).
Theorem 1.25. Let Ω be an open subset of Rn. Let p ∈ [1,+∞[. Then the
following statements hold.
(i) clMλp (Ω)
(Mλ,0
p (Ω) ∩ C∞(Ω) ∩ C0ub(Ω)
)=
= clMλp (Ω)
(Mλ,0
p (Ω) ∩ C∞(Ω) ∩ C0ub(Ω)
)=Mλ,0
p (Ω).
(ii) If mn(Ω) <∞ and if λ < np, then C0
ub(Ω) ⊆ L∞(Ω) ⊆Mλ,0p (Ω) and
clMλp (Ω)
(C∞(Ω) ∩ C0
ub(Ω))= clMλ
p (Ω)
(C∞(Ω) ∩ C0
ub(Ω))=Mλ,0
p (Ω).
(iii) If Ω is bounded and if λ < np, then
C∞(Ω) ⊆ C∞(Ω) ⊆ C0ub(Ω) ⊆ L∞(Ω) ⊆Mλ,0
p (Ω)
and
clMλp (Ω)C
∞(Ω) = clMλp (Ω)C
∞(Ω) =Mλ,0p (Ω).
Proof. (i) First we observe that the sets Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0
ub(Ω) and
Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0
ub(Ω) are contained in Mλ,0p (Ω).
By Lemma 1.7 Mλ,0p (Ω) is a closed subspace of Mλ
p (Ω).
Then the closure of the set Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0
ub(Ω) is contained in
Mλ,0p (Ω).
26
1.2 Approximation by C∞ functions in Morrey spaces
Similarly, the closure of the set Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0
ub(Ω) is contained
in Mλ,0p (Ω).
Let now f ∈ Mλ,0p (Ω). Then, by Lemma 1.27, there exists a bounded
linear extension operator EΩ :Mλ,0p (Ω) →Mλ,0
p (Rn) such that EΩf∣∣Ω= f
for all f ∈Mλ,0p (Ω).
Thus, we can approximate EΩf by smooth functions, which are, by def-
inition, functions from C∞(Ω).
Therefore, EΩf , as a limit, belongs to
clMλp (Ω)
(Mλ,0
p (Ω) ∩ C∞(Ω) ∩ C0ub(Ω)
)=
= clMλp (Ω)
(Mλ,0
p (Ω) ∩ C∞(Ω) ∩ C0ub(Ω)
).
(ii) Let f ∈ C0ub(Ω), then f is bounded and, thus, it is essentially bounded,
i.e. f ∈ L∞(Ω) and hence
C0ub(Ω) ⊆ L∞(Ω).
Then by Lemma 1.12 (ii) we have L∞(Ω) ⊂Mλ,0p (Ω). Hence,
Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0
ub(Ω) = C∞(Ω) ∩ C0ub(Ω)
and
Mλ,0p (Ω) ∩ C∞(Ω) ∩ C0
ub(Ω) = C∞(Ω) ∩ C0ub(Ω).
Therefore, from (i) we obtain
clMλp (Ω)
(C∞(Ω) ∩ C0
ub(Ω))= clMλ
p (Ω)
(C∞(Ω) ∩ C0
ub(Ω))=Mλ,0
p (Ω).
(iii) Let f ∈ C∞(Ω). By definition there exist an open neighborhood U of Ω
and a function F ∈ C∞(U) such that F |Ω = f . Then f ∈ C∞(Ω).
Now let f ∈ C∞(Ω). Then f is continuous.
Since Ω is bounded and Ω is closed, then, by Cantor’s Theorem, f is
uniformly continuous. In this case f is also bounded. Thus, f ∈ C0ub(Ω).
Hence, we have
C∞(Ω) ∩ C0ub(Ω) = C∞(Ω) and C∞(Ω) ∩ C0
ub(Ω) = C∞(Ω).
27
1. Morrey and Sobolev Morrey spaces
From (ii) we get
clMλp (Ω)C
∞(Ω) = clMλp (Ω)C
∞(Ω) =Mλ,0p (Ω).
1.3 Preliminaries on integral operators
Lemma 1.26. Let f ∈ Lloc1 (Rn). If there exists R ∈]0,+∞[ such that
f∣∣Rn\Bn(0,R)
is essentially bounded, then for each ε > 0 there exists δ > 0
such that ∫E
|f | dmn ≤ ε ∀E ∈ Ln, mn(E) ≤ δ.
Proof. Since f∣∣Rn\Bn(0,R)
is essentially bounded, we have f ∈ L∞(Rn\Bn(0, R))
and f ∈ L1(Bn(0, R)).
We set
E1 = E ∩Bn(0, R), E2 = E \Bn(0, R), for all E ∈ Ln.
Now let ε > 0. Then, by absolute continuity of the Lebesgue integral, there
exists δ1(ε) > 0 such that ∫E1
|f | dmn ≤ ε
2
whenever mn(E1) ≤ δ1(ε).
Next we suppose that
δ2(ε) =ε
2∥f∥L∞(Rn\Bn(0,R)) + 1,
then for E2 satisfying mn(E2) ≤ δ2(ε) we obtain∫E2
|f | dmn ≤ ∥f∥L∞(Rn\Bn(0,R))mn(E2) < ∥f∥L∞(Rn\Bn(0,R))δ2(ε) =
= ∥f∥L∞(Rn\Bn(0,R))ε
2∥f∥L∞(Rn\Bn(0,R))
=ε
2.
Therefore,∫E
|f | dmn =
∫E1
|f | dmn +
∫E2
|f | dmn ≤ ε
2+ε
2= ε.
28
1.3 Preliminaries on integral operators
Lemma 1.27. Let n ∈ N \ 0. Let λ ∈]0, n[. For each ε > 0 there exists
δ > 0 such that ∫E
1
|ξ − η|λdη ≤ ε ∀E ∈ Ln, mn(E) ≤ δ.
for all ξ ∈ Rn.
Proof. Setting ξ − η = z, we have∫E
1
|ξ − η|λdη =
∫ξ−E
dz
|z|λ.
Function 1|x|λ ∈ L1(B(0, 1)) and 1
|x|λ ∈ L∞(Rn \Bn(0, 1)).
Then, by applying the previous lemma, we complete the proof.
Theorem 1.28. Let n ∈ N \ 0. Let α ∈]0, n[.Let Ω be an open subset of Rn
of finite measure. Let Ω1 be an open subset of Rn. Let
DΩ1×Ω ≡ (x, y) ∈ Ω1 × Ω : x = y .
Let k be a function from (Ω1×Ω)\DΩ1×Ω to R such that k(x, ·) is measurable
in Ω \ x for all x ∈ Ω1. Assume that there exists c ∈]0,+∞[ such that
|k(x, y)| ≤ c
|x− y|n−α∀ (x, y) ∈ (Ω1 × Ω) \DΩ1×Ω.
Assume that if x ∈ Ω1, there exists a subset Nx of measure 0 of Ω such that
x /∈ Ω \Nx and such that the function k(·, y) from Ω1 \ y to R is continuous
at x for all y ∈ Ω \Nx.
If f ∈ L∞(Ω), then the function H[f ] from Ω1 to R defined by
H[f ](x) ≡∫Ω
k(x, y)f(y)dy ∀ x ∈ Ω1
is continuous.
Proof. Let x ∈ Ω1. Clearly,
|k(x, y)f(y)| ≤ ∥f∥L∞(Ω)c
|x− y|n−α(1.11)
for almost all y ∈ Ω. Since c|x−y|n−α is integrable in y ∈ Ω, we deduce that
k(x, y)f(y) is integrable in y ∈ Ω.
29
1. Morrey and Sobolev Morrey spaces
Now let x ∈ Ω1. In order to prove the continuity of H[f ] at x, we want to
apply the Vitali Convergence Theorem
By inequality (1.11), we have∫E
|k(x, y)f(y)|dy ≤ c∥f∥L∞(Ω)
∫E
dy
|x− y|n−α≤
≤ c∥f∥L∞(Ω) supx∈Rn
∫E
dy
|x− y|n−α,
for all x ∈ Ω1. Now let ε > 0. Lemma 1.27 implies that there exists δ > 0
such that∫E
dy
|x− y|n−α≤ ε
1 + c∥f∥L∞(Ω)
if E ∈ Ln, mn(E) ≤ δ, x ∈ Rn.
Then we have∫E
|k(x, y)f(y)|dy ≤ ε if E ∈ Ln, mn(E) ≤ δ, x ∈ Ω1. (1.12)
Now let xjj∈N be a sequence in Ω1 \ x which converges to x in Ω1. Let
Nx be a subset of measure 0 of Ω as in assumptions. Then we have
limj→∞
k(xj, y) = k(x, y) ∀ y ∈ Ω \Nx, (1.13)
and, in particular, for almost all y ∈ Ω. Then (1.12) and (1.13) and the Vitali
Convergence Theorem imply that
limj→∞
H[f ](xj) = H[f ](x).
Hence, H[f ] is continuous at x.
Let f ∈ Lloc1 (Rn). Consider the Riesz potential
(Iαf)(x) =
∫Rn
f(y)
|x− y|n−αdy, 0 < α < n.
In [14], in particular, the following statement is proved generalizing the results
of [1], [17], [42], [47], [27] and [16].
Theorem 1.29. Let condition
1 < p ≤ ∞, 0 < q ≤ ∞ and n
(1
p− 1
q
)+
< α < n, (1.14)
30
1.3 Preliminaries on integral operators
or
p = 1, 0 < q <∞ and n
(1− 1
q
)+
< α < n, (1.15)
or
1 < p < q < +∞ and α = n
(1
p− 1
q
)(1.16)
be satisfied. Let also u ∈ Λp,∞, v ∈ Λq,∞ and
I(u, v) =
∥∥∥∥∥∥v(t)tnq∞∫t
sα−np−1
∥u∥L∞(s,∞)
ds
∥∥∥∥∥∥L∞(0,∞)
<∞. (1.17)
Then the operator Iα is bounded from Mu(·)p (Rn) to Mv(·)
q (Rn).
Moreover, if condition (1.15) is satisfied, then condition (1.17) is necessary
and sufficient for the boundedness of Iα from Mu(·)p (Rn) to Mv(·)
q (Rn).
Theorem 1.30. Let n ∈ N \ 0. Let 1 ≤ p ≤ q < +∞. Let 0 ≤ λ ≤ ν < nq.
Let
α ≡(ν − n
q
)−(λ− n
p
). (1.18)
Then the following statements hold.
(i) If λ < ν, then the operator Iα is bounded from Mr−λ
p (Rn) to Mr−ν
q (Rn).
(ii) If λ = ν and if 1 < p < q, then Iα is bounded from Mr−λ
p (Rn) to
Mr−ν
q (Rn).
(iii) If λ = ν and if 1 < p < q, then Iα is bounded from Mλp (Rn) to Mν
q (Rn).
Remark 1.31. If ν = λ = 0, then α = n(
1p− 1
q
), and this is the classical
Hardy-Littlewood-Sobolev theorem.
Proof. (i) We recall that
t+ =
t, if t ≥ 0,
0, if t < 0.
Since p ≤ q, we have n(
1p− 1
q
)+= n
(1p− 1
q
).
Assumptions λ < ν and (1.18) imply α =(ν − n
q
)−(λ− n
p
),
λ < ν,⇒
α = ν − λ+ np− n
q,
ν − λ > 0,⇒ n
(1
p− 1
q
)< α,
31
1. Morrey and Sobolev Morrey spaces
α =(ν − n
q
)−(λ− n
p
),
0 ≤ λ ≤ ν < nq,
⇒
α = ν − λ+ np− n
q,
ν − λ < nq,
⇒ α <n
p≤ n.
Therefore,
n
(1
p− 1
q
)< α < n,
and thus either condition (1.14) or (1.15) satisfied.
Now we want to prove that
I(r−λ, r−ν) = supt>0
r−ν(t)tnq
∞∫t
sα−np−1
r−λ(s)ds <∞.
Indeed,
t−ν+nq
∞∫t
sα+λ−np−1ds = t−ν+
nq
∞∫t
sν−nq−1ds =
=t−ν+
nq+ν−n
q
−ν + nq
=
(−ν + n
q
)−1<∞.
By Theorem 1.29, operator Iα is linear and continuous from Mr−λ
p (Rn)
to Mr−ν
q (Rn).
(ii) If λ = ν, then α = n(
1p− 1
q
)and, hence, condition (1.16) satisfied.
Now we prove that inequality (1.17) holds with u(t) = v(t) = t−λ. In-
deed,
t−λ+nq
∞∫t
sα+λ−np−1ds = t−λ+
nq
∞∫t
sλ−nq−1ds =
=t−λ+
nq+λ−n
q
−λ+ nq
=
(−λ+
n
q
)−1<∞.
Hence, Theorem 1.29 implies that Iα is continuous from Mr−λ
p (Rn) to
Mr−ν
q (Rn).
(iii) If λ = ν, then α = n(
1p− 1
q
)and, therefore, condition (1.16) satisfied.
We want to prove that
I(wλ, wλ) = supt>0
wλ(t)tnq
∞∫t
sα−np−1
wλ(s)ds <∞.
32
1.3 Preliminaries on integral operators
Let first t ∈]0, 1]. Then
t−λ+nq
∞∫t
sα+λ−np−1ds = t−λ+
nq
1∫t
sλ−nq−1ds+ t−λ+
nq
∞∫1
sλ−nq−1ds =
=t−λ+
nq − 1− t−λ+
nq
λ− nq
=
(−λ+
n
q
)−1<∞.
Now let t ≥ 1. Then wλ(t) = 1 and
tnq
∞∫t
sα−np−1ds = t
nq
∞∫t
s−nq−1ds =
q
n<∞.
Thus, (1.17) holds and Theorem 1.29 implies that Iα is continuous from
Mλp (Rn) to Mν
q (Rn).
Lemma 1.32. Let p ∈ [1,+∞[, α ∈]0, n[, λ ∈ [0, n/p]. Let q ∈ [1, p] be such
that (α + λ) > nq. Let
µwλ,q ≡ max
1,
1
(α + λ)− nq
. (1.19)
Then we have∫E∩Bn(x,1)
|f(y)|dy|x− y|n−α
≤
≤ mn(E)1q− 1
pµwλ,q(n+ 2− α)v1− 1
qn ∥f∥Mλ
p (Rn) ∀ f ∈Mλp (Rn), (1.20)
for all measurable subsets E of Rn of finite measure, and for all x ∈ Rn.
Proof. The arguments of this proof are in part based on a development of the
ideas of Campanato [19].
If f ∈ Mλp (Rn), then we know that f
∣∣Bn(x,r)
∈ Lp(Bn(x, r)) ⊆ L1(Bn(x, r))
for all x ∈ Rn and r ∈]0,+∞[. In particular, (χEf)∣∣Bn(x,r)
∈ L1(Bn(x, r)) for
all x ∈ Rn and r ∈]0,+∞[ and for all measurable subsets E of Rn.
Now we fix x ∈ Rn and a measurable subset E of Rn of finite measure.
The almost everywhere defined function from ]0,+∞[ to [0,+∞[ which takes
s ∈]0,+∞[ to∫
∂Bn(x,s)
χE|f |dσ is integrable in ]0, r[ for all r ∈]0,+∞[. Then
33
1. Morrey and Sobolev Morrey spaces
by the Fundamental Theorem of Calculus, the function AE,x from [0,+∞[ to
[0,+∞[ defined by
AE,x(ρ) ≡ρ∫
0
∫∂Bn(x,s)
χE|f |dσ ds ∀ ρ ∈ [0,+∞[,
is locally absolutely continuous and
A′E,x(ρ) ≡∫
∂Bn(x,ρ)
χE|f |dσ,
for almost all ρ ∈ [0,+∞[ (cf. e.g., Folland [25, 3.35]). By the Monotone
Convergence Theorem, we have
∫E∩Bn(x,1)
|f(y)|dy|x− y|n−α
=
=
∫Bn(x,1)
χE(y)|f(y)|dy|x− y|n−α
= limε→0
∫Bn(x,1)\Bn(x,ε)
χE(y)|f(y)|dy|x− y|n−α
. (1.21)
Now let ε ∈]0, 1[. Then we have
∫Bn(x,1)\Bn(x,ε)
χE(y)|f(y)|dy|x− y|n−α
=
=
1∫ε
s−n+α
∫∂Bn(x,s)
χE|f |dσ ds =1∫
ε
s−n+αA′E,x(s)ds. (1.22)
Then by integrating by parts, we obtain
1∫ε
s−n+αA′E,x(s)ds =
=[s−n+αAE,x(s)
]1ε−
1∫ε
(−n+ α)s−n+α−1AE,x(s)ds, (1.23)
(cf. e.g., Folland [25, ex.35 p.108]). Then the Holder inequality and inequality
34
1.3 Preliminaries on integral operators
(1.19) imply that
|AE,x(ρ)| ≤ mn(E ∩ Bn(x, ρ))1− 1
p∥f∥Lp(E∩Bn(x,ρ)) =
= mn(E ∩ Bn(x, ρ))1q− 1
p mn(E ∩ Bn(x, ρ))1− 1
q ∥f∥Lp(E∩Bn(x,ρ)) ≤
≤ mn(E)1q− 1
p v1− 1
qn ρn−
nq ∥f∥Lp(Bn(x,ρ)) ≤
≤ mn(E)1q− 1
p v1− 1
qn ρn−αρ−n+αρn−
nqw−1λ (ρ)wλ(ρ)∥f∥Lp(Bn(x,ρ)) ≤
≤ ρn−αmn(E)1q− 1
p v1− 1
qn ρα−
nqw−1λ (ρ)∥f∥Mλ
p (Rn) ≤
≤ ρn−αmn(E)1q− 1
p v1− 1
qn ρ(α+λ)−n/q∥f∥Mλ
p (Rn) (1.24)
for all ρ ∈]0, 1[. Then by the second last line of inequality (1.24), we have∣∣∣∣∣∣1∫
ε
(−n+ α)s−n+α−1AE,x(s)ds
∣∣∣∣∣∣ ≤≤ mn(E)
1q− 1
p v1− 1
qn ∥f∥Mλ
p (Rn)
∣∣∣∣∣∣1∫
ε
(−n+ α)s−n+α−1sn−αs(α+λ)−nq ds
∣∣∣∣∣∣ ≤≤ mn(E)
1q− 1
p v1− 1
qn ∥f∥Mλ
p (Rn)(n− α)
∣∣∣∣∣∣1∫
ε
s(α+λ)−nq−1ds
∣∣∣∣∣∣ == mn(E)
1q− 1
p v1− 1
qn ∥f∥Mλ
p (Rn)(n− α)1
(α+ λ)− n/q(1− ε) =
= mn(E)1q− 1
p (n− α)v1− 1
qn µwλ,q∥f∥Mλ
p (Rn)(1− ε). (1.25)
Then by combining (1.22)–(1.25), we deduce that∫Bn(x,1)\Bn(x,ε)
χE(y)|f(y)|dy|x− y|n−α
≤
∣∣∣∣∣∣1∫
ε
s−n+αA′E,x(s)ds
∣∣∣∣∣∣ ≤
≤ |AE,x(1)|+ |ε−n+αAE,x(ε)|+
∣∣∣∣∣∣1∫
ε
(−n+ α)s−n+α−1AE,x(s)ds
∣∣∣∣∣∣ ≤≤ 1n−αmn(E)
1q− 1
p v1− 1
qn 1(α+λ)−n/q∥f∥Mλ
p (Rn)+
+ε−n+αεn−αmn(E)1q− 1
p v1− 1
qn ε(α+λ)−n/q∥f∥Mλ
p (Rn)+
+mn(E)1q− 1
p (n− α)v1− 1
qn µwλ,q∥f∥Mλ
p (Rn)(1− ε) ≤
≤ mn(E)1q− 1
pµwλ,qv1− 1
qn ∥f∥Mλ
p (Rn)[1 + 1 + (n− α)(1− ε)].
Then the limiting relation (1.21) immediately implies the validity of inequality
(1.20).
35
1. Morrey and Sobolev Morrey spaces
Corollary 1.33. Let p ∈ [1,+∞[, α ∈]0, n[. Let λ ∈ [0, n/p]. Let α + λ > np.
Let Ω be an open subset of Rn. Then the following statements hold.
If f ∈Mλp (Rn) and if
∫Ω
|f |dx < ∞, then the function from Rn to R which
takes x ∈ Rn to ∫Ω
f(y)dy
|x− y|n−α
is bounded, and satisfies the following inequality
supx∈Rn
∫Ω
|f(y)|dy|x− y|n−α
≤
≤ max1, ((λ+ α)− (n/p))−1(n+ 2− α)v1− 1
pn ∥f∥Mλ
p (Rn) +
∫Ω
|f |dx. (1.26)
If Ω has finite measure, then the map Iα,Ω from Mλp (Ω) to B(Rn) defined
by
Iα,Ωf(x) ≡∫Ω
f(y)dy
|x− y|n−α∀x ∈ Rn,
for all f ∈Mλp (Ω) is linear and continuous.
Proof. By applying Lemma 1.32 with E = Ω, we deduce that∫Ω
|f(y)|dy|x− y|n−α
≤∫
Ω∩Bn(x,1)
|f(y)|dy|x− y|n−α
+
∫Ω\Bn(x,1)
|f(y)|dy|x− y|n−α
≤
≤ mn (Bn(x, 1))1q− 1
p µwλ,q(n+ 2− α)v1− 1
qn ∥f∥Mλ
p (Rn) +
∫Ω\Bn(x,1)
|f |1n−α
dx ≤
≤ µwλ,q(n+ 2− α)v1− 1
qn ∥f∥Mλ
p (Rn) +
∫Ω
|f |dx,
for all x ∈ Rn. Hence, inequality (1.26) follows.
1.4 Sobolev spaces built on Morrey spaces
Definition 1.34. A domain Ω ⊂ Rn is called star-shaped with respect to the
point y ∈ Ω if for all x ∈ Ω the segment [x, y] ⊂ Ω. A domain Ω ⊂ Rn is
called star-shaped with respect to a point if for some y ∈ Ω it is star-shaped
with respect to the point y.
36
1.4 Sobolev spaces built on Morrey spaces
Definition 1.35. A domain Ω ⊂ Rn is called star-shaped with respect to the
ball B ⊂ Ω if for all y ∈ B and for all x ∈ Ω we have [x, y] ⊂ Ω. A domain
Ω ⊂ Rn is called star-shaped with respect to a ball if for some ball B ⊂ Ω it is
star-shaped respect to the ball B.
Definition 1.36. If 0 < d ≤ diamB ≤ diamΩ ≤ D, we set that Ω is star-
shaped with respect to a ball with the parameters d,D.
Definition 1.37. We call the set
Vx ≡ Vx,B =∪y∈B
(x, y)
a conic body with the vertex x constructed on the ball B.
A domain Ω star-shaped with respect to a ball B can be equivalently defined
in the following way: for all x ∈ Ω the conic body Vx ⊂ Ω.
Definition 1.38. If r, h ∈]0,+∞[, then we define the cone K(r, h) as follows
K(r, h) ≡(x′, xn) ∈ Rn : |x′| < rxn
h, xn < h
,
if n > 1, and
K(r, h) ≡]0, h[,
if n = 1.
We denote by O(n) the orthogonal group, i.e., the set of n× n matrices R
with real entries such that RRt = I = RtR.
Definition 1.39. Let Ω be an open subset of Rn.
(i) Let r, h ∈]0,+∞[. We say that Ω satisfies the cone property (or con-
dition) with parameters r, h provided that for each x ∈ Ω there exists
Rx ∈ O(n) such that
x+Rx(K(r, h)) ⊆ Ω.
(ii) We say that Ω satisfies the cone property (or condition), provided that
there exists r, h ∈]0,+∞[ such that Ω satisfies the cone property (or
condition) with parameters r, h.
37
1. Morrey and Sobolev Morrey spaces
Lemma 1.40. 1. A bounded open set Ω ⊂ Rn satisfies the cone condition if,
and only if, there exist s ∈ N and bounded domains Ωk, which are star-shaped
with respect to the balls Bk ⊂ Bk ⊂ Ωk, k = 1, . . . , s, such that Ω =s∪
k=1
Ωk.
2. An unbounded open set Ω ⊂ Rn satisfies the cone condition if, and only
if, there exist bounded domains Ωk, k ∈ N \ 0, which are star-shaped with
respect to the balls Bk ⊂ Bk ⊂ Ωk, k ∈ N \ 0, and are such that
1) Ω =∞∪k=1
Ωk,
2) 0 < infk∈N\0
diamBk ≤ supk∈N\0
diamΩk <∞,
3) the multiplicity of the covering ℵ(Ωk∞k=1) is finite.
(cf. e.g., Burenkov[13, Ch. 3.2])
Lemma 1.41. Let m0 ∈ N \ 0, 1 ≤ p1, . . . , pm0 , q ≤ ∞, λ ∈[0, n
pm
], for all
m = 1, . . . ,m0, 0 ≤ ν ≤ nq. Let Ω =
s∪k=1
Ωk, where s ∈ N\0 and Ωk ⊂ Rn are
bounded open sets. Furthermore, let fm, m = 1, . . . ,m0, and g be measurable
functions on Ω.
Suppose that there exists σm > 0 such that
∥g∥Mρ−νq (Ωk)
≤m0∑m=1
σm∥fm∥Mρ−λpm (Ωk)
, (1.27)
for all m = 1, . . . ,m0.
Then
∥g∥Mρ−νq (Ω)
≤ 2νs1q
m0∑m=1
σm∥fm∥Mρ−λpm (Ω)
. (1.28)
Proof. Let q <∞.
∥g∥qMρ−ν
q (Ω)= sup
x∈Ωρ>0
ρ−νq∥g∥qLq(B(x,ρ)∩Ω) =
= supx∈Ωρ>0
ρ−νq∫
B(x,ρ)∩Ω
|g(y)|qdy ≤ supx∈Ωρ>0
s∑k=1
ρ−νq∫
B(x,ρ)∩Ωk
|g(y)|qdy ≤
≤s∑
k=1
supx∈Ωρ>0
ρ−νq∥g∥qLq(B(x,ρ)∩Ωk)≤
s∑k=1
supx∈Rnρ>0
ρ−νq∥g∥qLq(B(x,ρ)∩Ωk).
If x ∈ Ωk, then
ρ−νq∥g∥qLq(B(x,ρ)∩Ωk)= sup
x∈Ωkρ>0
ρ−νq∥g∥qLq(B(x,ρ)∩Ωk)= ∥g∥q
Mρ−νq (Ωk)
.
38
1.4 Sobolev spaces built on Morrey spaces
Let x ∈ Rn\Ωk. If B(x, ρ) ∩ Ωk = ∅, then ρ−ν∥g∥Lq(B(x,ρ)∩Ωk) = 0. Thus,
we can assume that B(x, ρ) ∩ Ωk = ∅.
Let ξ ∈ B(x, ρ) ∩ Ωk. By the triangle inequality, we have
B(x, ρ) ∩ Ωk ⊂ B(ξ, 2ρ) ∩ Ωk.
Hence,
ρ−νq∥g∥qLq(B(x,ρ)∩Ωk)≤ 2νq sup
ξ∈Ωkρ>0
(2ρ)−νq∥g∥qLq(B(ξ,2ρ)∩Ωk)= 2νq∥g∥q
Mr−νq (Ωk)
.
By (1.27) and the Minkowski inequality it follows, that
∥g∥Mρ−νq (Ω)
≤ 2ν
(s∑
k=1
∥g∥qMρ−ν
q (Ωk)
) 1q
≤ 2ν
(s∑
k=1
(m0∑m=1
σm∥fm∥Mρ−λpm (Ωk)
)q) 1q
≤
≤ 2νm0∑m=1
(s∑
k=1
(σm∥fm∥Mρ−λ
pm (Ωk)
)q) 1q
= 2νm0∑m=1
σm
(s∑
k=1
∥fm∥qMρ−λ
pm (Ωk)
) 1q
.
Then(s∑
k=1
∥fm∥qMρ−λ
pm (Ωk)
) 1q
≤
(s∑
k=1
∥fm∥qMρ−λ
pm (Ω)
) 1q
= s1q ∥fm∥Mρ−λ
pm (Ω)
and inequality (1.28) follows.
The case in which some pm = ∞ is treated in a similar way with suprema
replacing sums.
The case q = ∞ is trivial and the statement holds for Ω =∪i∈I
Ωi, where I
is an arbitrary set of indices:
∥g∥Mν∞(Ω) = sup
i∈Isupx∈Ωρ>0
ρ−ν∥g∥L∞(B(x,ρ)∩Ωi) ≤m0∑m=1
σm∥fm∥Mλpm
(Ω).
Definition 1.42. Let Ω ⊂ Rn be an open set. Let l ∈ N, p ∈ [1,+∞] and
λ ∈[0, n
p
]. Then we define the Sobolev space of order l built on the Morrey
space Mλp (Ω), as the set
W l,λp (Ω) ≡
f ∈Mλ
p (Ω) : Dαwf ∈Mλ
p (Ω) ∀α ∈ Nn, |α| ≤ l,
where Dαwf is the weak derivative of f .
39
1. Morrey and Sobolev Morrey spaces
Then we set
∥f∥W l,λp (Ω) =
∑|α|≤l
∥Dαwf∥Mλ
p (Ω) ∀ f ∈ W l,λp (Ω).
In particular, W 0,λp (Ω) = Mλ
p (Ω) and W l,0p (Ω) = W l
p(Ω), where W lp(Ω)
denotes the classical Sobolev space of exponents l, p in Ω. It is obvious that
W l,λp (Ω) ⊂ W l
p(Ω).
Since Mλp (Ω) is a Banach space, one can exploit standard properties of the
weak derivatives and prove the following.
Theorem 1.43. Let Ω be an open subset of Rn. Let p ∈ [1,+∞] and
λ ∈[0, n
p
]. Then
(W l,λ
p (Ω), ∥ · ∥W l,λp (Ω)
)is a Banach space.
Then we have the following remark, which provides a family of equivalent
norms in W l,λp (Ω).
Remark 1.44. Let Ω be an open subset of Rn. Let l ∈ N, p ∈ [1,+∞] and
λ ∈[0, n
p
]. Let c(l) be the number of multi indexes α ∈ Nn such that |α| ≤ l.
Let q be a norm on Rc(l). Then the map qW l,λp (Ω) from W l,λ
p (Ω) to [0,+∞[
which takes u to qW l,λp (Ω)(u) ≡ q
((∥Dα
wu∥Mλp (Ω))|α|≤l
)is an equivalent norm in
W l,λp (Ω).
In particular,
( ∑|α|≤l
∥Dαwu∥tMλ
p (Ω)
) 1t
for t ∈ [1,+∞[ and max|α|≤l
∥Dαwu∥Mλ
p (Ω)
are all equivalent norms on W l,λp (Ω).
Definition 1.45. Let Ω ⊂ Rn be an open set. Let l ∈ N, p ∈ [1,+∞] and
λ ∈[0, n
p
]. Then we define the Sobolev space of order l built on the little
Morrey space Mλ,0p (Ω), as the set
W l,λ,0p (Ω) ≡
f ∈Mλ,0
p (Ω) : Dαwf ∈Mλ,0
p (Ω) ∀α ∈ Nn, |α| ≤ l.
Since Mλ,0p (Ω) is a closed subspace of Mλ
p (Ω), we can easily deduce the
validity of the following.
Theorem 1.46. Let Ω be an open subset of Rn. Let l ∈ N, p ∈ [1,+∞] and
λ ∈[0, n
p
]. Then W l,λ,0
p (Ω) is a closed proper subspace of W l,λp (Ω).
40
1.4 Sobolev spaces built on Morrey spaces
Proof. Let u ∈ W l,λp (Ω). Let ukk∈N be a sequence in W l,λ,0
p (Ω) which con-
verges to u in W l,λp (Ω). We want to show that u ∈ W l,λ,0
p (Ω).
Since uk → u in W l,λp (Ω) as k → ∞, we have
Dαwuk → Dα
wu ∀ |α| ≤ l in Mλp (Ω)
as k → ∞.
We know that Mλ,0p (Ω) is a closed subspace of Mλ
p (Ω). Therefore,
Dαwu ∈Mλ,0
p (Ω) ∀ |α| ≤ l,
and, thus, u ∈ W l,λ,0p (Ω).
Lemma 1.47. Let l ∈ N \ 0. Let m ∈ N, m < l. Let 1 ≤ p, q ≤ +∞,
0 ≤ λ ≤ np, 0 ≤ ν ≤ n
q. Suppose that for each bounded domain G ⊂ Rn
star-shaped with respect to a ball there exists c1 > 0 such that for each β ∈ Nn
satisfying |β| ≤ m and for all f ∈ W l,λp (G)
∥Dβwf∥Mν
q (G) ≤ c1∥f∥W l,λp (G).
Then for each open bounded set Ω ⊂ Rn satisfying the cone condition there
exists c2 > 0 such that
∥Dβwf∥Mν
q (Ω) ≤ c2∥f∥W l,λp (Ω)
for each β ∈ Nn satisfying |β| ≤ m and for all f ∈ W l,λp (Ω).
Proof. Let Ω satisfy the cone condition with the parameters r, h. By
Lemma 1.40
Ω =s∪
k=1
Ωk,
where s ∈ N and Ωk are bounded domains star-shaped with respect to the
balls Bk ⊂ Bk ⊂ Ωk.
Then
∥Dβwf∥Mν
q (Ωk) ≤ c1(Ωk)∥f∥W l,λp (Ωk)
, k = 1, . . . , s.
Hence, by Lemma 1.41
∥Dβwf∥Mν
q (Ω) ≤ 2νs1q maxk=1,...,s
c1(Ωk)∥f∥W l,λp (Ω).
41
Chapter 2
The embedding and
multiplication operators in
Sobolev Morrey spaces
2.1 The Sobolev Embedding Theorem
First we introduce the following notation.
Definition 2.1. Let p ∈ [1,+∞], l, n ∈ N\0, m ∈ N, m ≤ l,
λ, ν ∈ [0,+∞[. Let l + λ−m− ν = np. Then we set
q∗(l,m, n, p, λ, ν) ≡ n
(n/p)− (l + λ−m− ν).
If λ = ν = 0, then q∗(l,m, n, p, λ, ν) equals the classical Sobolev limiting ex-
ponent. If λ, ν ∈ [0,+∞[, then the exponent q∗(l,m, n, p, λ, ν) can be obtained
from the classical one by replacing l by l + λ and m by m+ ν.
We note that if l + λ − ν = np, then the equality which defines
q∗(l, 0, n, p, λ, ν) is equivalent to the equality
l =
(ν − n
q∗(l, 0, n, p, λ, ν)
)−(λ− n
p
).
We also note that
q∗(l, 0, n, p, λ, ν)
p> 1 whenever
l + λ > ν,
l + λ− ν < np,
43
2. The embedding and multiplication operators in Sobolev Morrey spaces
and
q∗(l −m, 0, n, p, λ, ν) = q∗(l,m, n, p, λ, ν).
Remark 2.2. Before we prove an analogue of the Sobolev Embedding Theorem,
we recall the following:
(i) Mλp (Rn) is continuously embedded into Mr−λ
p (Rn).
(ii) If Ω is a bounded domain, then we have Mr−λ
p (Ω) = Mλp (Ω) with equiv-
alent norms.
We are now ready to prove the following Sobolev Embedding Theorem.
Theorem 2.3. Let p ∈ [1,+∞[, l, n ∈ N\0, m ∈ N, m ≤ l, λ ∈[0, n
p
]. Let
Ω be a bounded open subset of Rn which satisfies the cone property. Then the
following statements hold.
(i) Let l−m+λ < np. Let ν ∈]λ, (l−m)+λ]. Then W l,λ
p (Ω) is continuously
embedded into Wm,νq∗(l,m,n,p,λ,ν)(Ω).
(ii) Let l−m+ λ < np. If p > 1, then W l,λ
p (Ω) is continuously embedded into
Wm,λq∗(l,m,n,p,λ,λ)(Ω).
(iii) Let l−m+λ > np. Then W l,λ
p (Ω) is continuously embedded into Wm∞(Ω).
Proof. (i) First let m = 0.
Let Ω be a bounded domain star-shaped with respect to the ball B =
B(x0, r), B ⊂ Ω. Then by Sobolev’s integral representation there existsM1 > 0
such that
|f(x)| ≤M1
∫B
|f |dy +∑|α|=l
∫Vx
(Dαwf)(y)
|x− y|n−ldy
for almost all x ∈ Ω for each (cf. e.g., Burenkov [13, Ch.3 p.112]).
Hence,
∥f∥Mr−ν
q∗(l,0,n,p,λ,ν)(Ω)
≤M1
∫B
|f |dy · ∥1∥Mr−ν
q∗(l,0,n,p,λ,ν)(Ω)
+
+∑|α|=l
∥∥∥∥∥∥∫Rn
Φα(y)
|x− y|n−ldy
∥∥∥∥∥∥Mr−ν
q∗(l,0,n,p,λ,ν)(Ω)
,
44
2.1 The Sobolev Embedding Theorem
where Φα(y) =
Dαwf(y), if y ∈ Ω;
0, if y /∈ Ω.
Note that
∥1∥Mr−ν
q∗(l,0,n,p,λ,ν)(Ω)
= supx∈Ω
supρ>0
ρ−ν∥1∥Lq∗(l,0,n,p,λ,ν)(B(x,ρ)∩Ω) ≤
≤ supx∈Ω
max
sup
0<ρ≤(diamΩ)
υ1
q∗(l,0,n,p,λ,ν)n ρ−ν+
nq∗(l,0,n,p,λ,ν) , sup
ρ≥(diamΩ)
ρ−νmn(Ω)1
q∗(l,0,n,p,λ,ν)
=
= max
υ
1q∗(l,0,n,p,λ,ν)n mn(Ω)
−ν+ nq∗(l,0,n,p,λ,ν) ,mn(Ω)
−ν+ 1q∗(l,0,n,p,λ,ν)
<∞.
By Theorem 1.30 there exists c > 0 depending only on n, l, p,
q∗(l, 0, n, p, λ, ν) such that
∥∥∥∥∥∥∫Rn
Φα(y)
|x− y|n−ldy
∥∥∥∥∥∥Mr−ν
q∗(l,0,n,p,λ,ν)(Rn)
≤ c∥Φα∥Mr−λp (Rn)
≤
≤ 2λc∥Dαwf∥Mr−λ
p (Ω)≤ 2λc∥Dα
wf∥Mλp (Ω).
By Holder inequality∫B
|f |dy ≤ mn(B)1p′ ∥f∥Lp(Ω).
Therefore, there exist M2 > 0 and M3 > 0 such that
∥f∥Mνq∗(l,0,n,p,λ,ν)
(Ω) ≤ max1, (diamΩ)ν∥f∥Mr−ν
q∗(l,0,n,p,λ,ν)(Ω)
≤
≤ max1, (diamΩ)ν∥f∥Mr−ν
q∗(l,0,n,p,λ,ν)(Rn)
≤
≤M2
∥f∥Lp(Ω) +∑|α|=l
∥∥∥∥∥∥∫Rn
Φα(y)
|x− y|n−ldy
∥∥∥∥∥∥Mr−ν
q∗ (Rn)
≤
≤M3
∥f∥Mλp (Ω) +
∑|α|=l
∥Dαwf∥Mλ
p (Ω)
=M3∥f∥W l,λp (Ω), ∀ f ∈ W l,λ
p (Ω).
Hence, by Lemma 1.47, the statement of Theorem 2.3 follows.
Now let α : |α| = m. Then Dαwf ∈ W
l−|α|,λp (Ω) = W l−m,λ
p (Ω). Hence, there
45
2. The embedding and multiplication operators in Sobolev Morrey spaces
exists a constant c1 > 0 such that
∥f∥Wm,νq∗(l,m,n,p,λ,ν)
(Ω) =∑|α|≤m
∥Dαwf∥Mν
q∗(l,m,n,p,λ,ν)(Ω) =
=∑|α|≤m
∥Dαwf∥Mν
q∗(l−m,0,n,p,λ,ν)(Ω) ≤ c1
∑|α|≤m
∥Dαwf∥W l−m,λ
p (Ω) ≤
≤ c1∑|α|≤m
∑|γ|≤l−m
∥Dγ+αw f∥Mλ
p (Ω) ≤
≤ c1∑|α|≤l
∥Dαwf∥Mλ
p (Ω) = c1∥f∥W l,λp (Ω), ∀ f ∈ W l,λ
p (Ω).
(ii) This case can be analized as case (i) by replacing q∗(l,m, n, p, λ, ν) by
q∗(l,m, n, p, λ, λ).
(iii) Now l − m + λ > np. Let Ω be a bounded domain star-shaped with
respect to the ball B = Bn(ξ, r0), B ⊂ Ω. Then by Sobolev’s integral repre-
sentation there exists c > 0 such that
|f(x)| ≤ c
∫Bn(ξ,r0)
|f |dx+∑|γ|=l
∫Vx
|(Dγwf)(y)|
|x− y|n−ldy
, (2.1)
for almost all x ∈ Ω and for all f ∈ W l,λp (Ω), and where Vx denotes the conical
body based on Bn(ξ, r0) and with vertex x (cf. e.g., Burenkov [13, Ch.3 p.112]).
We first consider case m = 0. So we now assume that l+λ > np. We plan to
estimate the supremum of |f | by exploiting inequality (2.1). Since∫
Bn(ξ,r0)
|f |dx
is a constant, it defines an element of C0b (Ω) ⊆ L∞(Ω). Next we prove that
the sum in the right hand side of (2.1) is bounded if f ∈ W l,λp (Ω).
We plan to treat separately case l < n and case l ≥ n.
Let l < n. Since l+λ > np, we can invoke Corollary 1.33 and conclude that
Il,Ω is linear and continuous from Mλp (Ω) to B(Rn).
Since Vx ⊆ Ω for all x ∈ Ω, we deduce that∣∣∣∣∣∣∫Vx
|h(y)||x− y|n−l
dy
∣∣∣∣∣∣ ≤ Il,Ω(|h|) ∀x ∈ Ω,
for all h ∈ Mλp (Ω). By the continuity of the restriction operator in Morrey
spaces and by the above mentioned continuity of Il,Ω, we deduce that the map
Jl,Ω from Mλp (Ω) to B(Ω) defined by
Jl,Ωh(x) ≡∫Vx
h(y)
|x− y|n−ldy ∀x ∈ Ω,
46
2.1 The Sobolev Embedding Theorem
for all h ∈Mλp (Rn) satisfies the inequality
|J l,Ωh(x)| ≤ |Il,Ω(|h|)(x)| ≤ ∥Il,Ω∥L(Mλp (Ω),B(Rn))∥ |h| ∥Mλ
p (Ω) ≤
≤ ∥Il,Ω∥L(Mλp (Ω),B(Rn))∥h∥Mλ
p (Ω) ∀x ∈ Ω, (2.2)
for all h ∈Mλp (Ω). Then we deduce that
|f(x)| ≤ c
∫Bn(ξ,r0)
|f |dx+∑|γ|=l
|Jl,Ω[Dγwf ](x)|
≤
≤ c
[mn(Bn(ξ, r0))]1− 1
p∥f∥Lp(Ω) + ∥Il,Ω∥L(Mλp (Ω),B(Rn))
∑|γ|=l
∥Dγwf∥Mλ
p (Ω)
≤
≤ c([mn(Bn(ξ, r0))]
1− 1p + ∥Il,Ω∥L(Mλ
p (Ω),B(Rn))
)∥f∥W l,λ
p (Ω), (2.3)
for almost all x ∈ Ω and for all f ∈ W l,λp (Ω).
We now consider case l ≥ n. The embedding of Mλp (Ω) into Lp(Ω) and
inequality (2.1) and the Holder inequality imply that
|f(x)| ≤ c
∫Bn(ξ,r0)
|f |dx+∑|γ|=l
∫Ω
|(Dγwf)(y)|
|x− y|n−ldy
≤ (2.4)
≤ c
[mn(Bn(ξ, r0))]1− 1
p∥f∥Lp(Ω) +∑|γ|=l
∥Dγwf∥L1(Ω)(diamΩ)l−n
≤
≤ c
[mn(Bn(ξ, r0))]1− 1
p∥f∥Lp(Ω) + (diamΩ)l−n[mn(Ω)]1− 1
p
∑|γ|=l
∥Dγwf∥Lp(Ω)
≤
≤ c([mn(Bn(ξ, r0))]
1− 1p + (diamΩ)l−n[mn(Ω)]
1− 1p
)∥f∥W l,λ
p (Ω),
for almost all x ∈ Ω and for all f ∈ W l,λp (Ω). By Lemma 1.47, by the inequality
(2.3) for case l < n and by the inequality (2.4) for case l ≥ n, we deduce the
validity of statement (iii) in case m = 0.
Next we prove the statement (iii) in case m > 0. If f ∈ W l,λp (Ω), then
Dβwf ∈ W l−m,λ
p (Ω) for all |β| ≤ m. Now by assumption, we have (l−m)+λ > np.
Hence, case m = 0 with l replaced by l−m implies that W l−m,λp (Ω) ⊆ L∞(Ω)
and that there exists c1 > 0 such that
∥g∥L∞(Ω) ≤ c1∥g∥W l−m,λp (Ω) ∀ g ∈ W l−m,λ
p (Ω).
47
2. The embedding and multiplication operators in Sobolev Morrey spaces
Hence, Dβwf ∈ L∞(Ω) for all β ∈ Nn such that |β| ≤ m, and
∥f∥Wm∞(Ω) ≤
∑|β|≤m
∥Dβwf∥L∞(Ω) ≤
≤ c1∑|β|≤m
∥Dβwf∥W l−m,λ
p (Ω) ≤ c1∑|β|≤m
∑|γ|≤l−m
∥Dγ+βw f∥Mλ
p (Ω) ≤
≤ c1
∑|β|≤m
∑|γ|≤l−m
1|γ+β|
∥f∥W l,λp (Ω)
for all f ∈ W l,λp (Ω). Hence, the proof of statement (iii) is complete.
2.2 Approximation by C∞ functions in
Sobolev Morrey spaces
First we state a known Leibnitz formula for Sobolev spaces. For the proof one
can see for example [24, 5.2.3]
Theorem 2.4. Let l ∈ N. Let 1 ≤ p < +∞. Let Ω be a bounded open subset
of Rn. Let u ∈ W lp(Ω) and |α| ≤ l. If ζ ∈ C l
c(Ω), then ζu ∈ W lp(Ω) and
Dαw(ζu) =
∑β≤α
α
β
DβζDα−βw u, (2.5)
where
α
β
= α!β!(α−β)!
Proposition 2.5. Let Ω ⊂ Rn be an open set, l ∈ N. Let u, v ∈ Lloc1 (Ω). More-
over, assume that for any β ∈ Nn satisfying |β| ≤ l there exists 1 ≤ pβ < ∞
such that Dβwu ∈ Lloc
pβ(Ω) and Dγ
wv ∈ Llocp′β(Ω) for all γ ∈ Nn : |γ| ≤ l − |β|.
Then for any α ∈ Nn satisfying |α| ≤ l the weak derivative Dαw(uv) exists and
the Leibnitz formula holds:
Dαw(uv) =
∑0≤β≤α
α
β
DβwuD
α−βw v. (2.6)
Proof. Let u ∈ C∞(Ω) and v ∈ Lloc1 (Ω) be such that Dγ
wv ∈ Lloc1 (Ω). Then
Dαw(uv) =
∑0≤β≤α
α
β
DβuDα−βw v.
48
2.2 Approximation by C∞ functions in Sobolev Morrey spaces
Now let u be as in formulation, i.e. Dβwu ∈ Lloc
pβ(Ω). Let also
uk(x) = u(x) ∗ φ 1k(x) ∀ k ∈ N,
where φ 1k(x) as in definition 1.20 with t = 1
k.
Then, by Theorem 2.4, we have
Dαw(ukv) =
∑0≤β≤α
α
β
DβukDα−βw v =
=∑
0≤β≤α
α
β
Dβ[u ∗ φ 1k]Dα−β
w v.
Properties of mollifiers imply that
uk → u in Llocpβ(Ω),
Dβuk → Dβwu in Lloc
pβ(Ω),
as k → ∞. Thus,
Dαw(uv) =
∑0≤β≤α
α
β
DβwuD
α−βw v.
Theorem 2.6. Let l ∈ N. Let 1 ≤ p < +∞, 0 ≤ λ ≤ np. Let Ω be a
bounded open subset of Rn. Then for u ∈ W l,λ,0p (Ω) there exist functions um ∈
C∞(Ω) ∩W l,λp (Ω) such that
um → u in W l,λp (Ω).
Proof. We have Ω =∞∪i=1
Ωi, where
Ωi :=
x ∈ Ω: dist(x, ∂Ω) >
1
i
(i = 1, 2, . . .).
Write Vi := Ωi+3 − Ωi+1.
Choose also any open set V0 ⊂⊂ Ω so that Ω =∞∪i=0
Vi. Now let ζi∞i=1
be a smooth partition of unity subordinate to the open sets Vi∞i=0; that is,
suppose 0 ≤ ζi ≤ 1, ζi ∈ C∞c (Vi)∞∑i=0
ζi = 1 on Ω.
49
2. The embedding and multiplication operators in Sobolev Morrey spaces
Next, choose any function u ∈ W l,λ,0p (Ω). By Theorem 2.4 we know that
Dαw(uζi) =
∑0≤β≤α
α
β
DβwuD
α−βw ζi.
Since Dβwu ∈ Mλ,0
p (Ω), Dα−βw ζi ∈ L∞(Ω), by Theorem 1.18 Dβ
wuDα−βw ζi ∈
Mλ,0p (Ω). Therefore, Dα
w(uζi) ∈ Mλ,0p (Ω) for all |α| ≤ l. Since W l,λ,0
p (Ω) is a
closed subspace of W l,λp (Ω), we have ζiu ∈ W l,λ,0
p (Rn) and supp (ζiu) ⊂ Vi.
Fix δ > 0. Choose then εi > 0 so small that ϕεi ∗ (ζiu) satisfies ∥ϕεi ∗ (ζiu)− ζiu∥W l,λp (Ω) ≤
δ2i+1 , (i = 0, 1, . . .)
supp [ϕεi ∗ (ζiu)] ⊂ Wi (i = 1, . . .),
for Wi := Ωi+4 − Ωi ⊃ Vi (i = 1, . . .).
Write v : =∞∑i=0
ϕεi ∗ (ζiu). This function belongs to C∞(Ω), since for each
open set V ⊂⊂ Ω there are at most finitely many nonzero terms in the sum.
Since u =∞∑i=0
ζiu, we have for each V ⊂⊂ Ω
∥v − u∥W l,λp (V ) ≤
∞∑i=0
∥ϕεi ∗ (ζiu)− ζiu∥W l,λp (Ω) ≤ δ
∞∑i=0
1
2i+1= δ.
Take the supremum over sets V ⊂⊂ Ω , to conclude ∥v − u∥W l,λp (Ω) ≤ δ.
2.3 Multiplication Theorems for Sobolev Mor-
rey spaces
Next we prove the following multiplication result which follows by the Holder
inequality.
Proposition 2.7. Let Ω be a bounded open subset of Rn. Let l ∈ N. Let
p, q, r ∈ [1,+∞] be such that 1r= 1
p+ 1
q. Let also 0 ≤ λ1 ≤ n
p, 0 ≤ λ2 ≤ n
q,
0 ≤ λ ≤ nr, λ ≡ λ1 + λ2. Then if u ∈ W l,λ1
p (Ω) and v ∈ W l,λ2q (Ω), we have
uv ∈ W l,λr (Ω).
Moreover, there exists c > 0 such that
∥uv∥W l,λr (Ω) ≤ c∥u∥
Wl,λ1p (Ω)
∥v∥W
l,λ2q (Ω)
∀ (u, v) ∈ W l,λ1p (Ω)×W l,λ2
q (Ω).
50
2.3 Multiplication Theorems for Sobolev Morrey spaces
Proof. By Proposition 2.5, we know that the Dαw weak derivative of u is de-
livered by the Leibnitz rule. Using Holder’s inequality for Morrey spaces, we
obtain
∥uv∥W l,λr (Ω) =
∑|α|≤l
∥Dαw(uv)∥Mλ
r (Ω) ≤
≤∑|α|≤l
∑|β|≤α
α
β
∥∥Dα−βw uDβ
wv∥∥Mλ
r (Ω)≤
≤∑|α|≤l
∑|β|≤α
α
β
∥∥Dα−βw u
∥∥M
λ1p (Ω)
∥∥Dβwv∥∥M
λ2q (Ω)
≤
≤ c∥u∥W
l,λ1p (Ω)
∥v∥W
l,λ2q (Ω)
.
We now extend a known multiplication result for Sobolev spaces of Zolesio
[61] (see also Valent [58], Runst and Sickel [51]).
Theorem 2.8. Let Ω be a bounded open subset of Rn. Let l ∈ N. Let p, q, r ∈
[1,+∞], and let 0 ≤ λ1 ≤ np, 0 ≤ λ2 ≤ n
q, 0 ≤ λ ≤ n
r. Assume that Ω has the
cone property, and p ≥ r, q ≥ r and
λ1 −n
p≥ λ− n
r, λ2 −
n
q≥ λ− n
r, (2.7)
l + λ1 + λ2 − λ
n>
1
p+
1
q− 1
r, (2.8)
l
n>
1
p+
1
q− 1
r. (2.9)
Then if u ∈ W l,λ1p (Ω) and v ∈ W l,λ2
q (Ω), we have uv ∈ W l,λr (Ω).
Moreover, there exists a positive number c such that
∥uv∥W l,λr (Ω) ≤ c∥u∥
Wl,λ1p (Ω)
∥v∥W
l,λ2q (Ω)
∀ (u, v) ∈ W l,λ1p (Ω)×W l,λ2
q (Ω). (2.10)
Proof. Let α ∈ Nn, |α| ≤ l. Then we set
Qα[u, v] =∑β≤α
α
β
Dα−βw uDβ
wv for all (u, v) ∈ W l,λ1p (Ω)×W l,λ2
q (Ω).
We want to prove that Qα defines a bilinear and continuous map from
W l,λ1p (Ω)×W l,λ2
q (Ω) to Mλr (Ω).
It is sufficies to show that if β ≤ α, |α| ≤ l, then
51
2. The embedding and multiplication operators in Sobolev Morrey spaces
(A) the map from Wl−|α−β|,λ1p (Ω)×W
l−|β|,λ2q (Ω) to Mλ
r (Ω) which takes (u, v)
to uv is bilinear and continuous.
Before we perform the proof in several steps, we remark the following:
If either Wl−|α−β|,λ1p (Ω) or W
l−|β|,λ2q (Ω) is continuously embedded into
L∞(Ω) then (A) is true.
Indeed, if Wl−|α−β|,λ1p (Ω) → L∞(Ω), we know that
W l−|β|,λ2q (Ω) →Mλ2
q (Ω) →Mλr (Ω)
and the product
L∞(Ω)×Mλr (Ω) →Mλ
r (Ω)
is continuous.
Now we split the proof into several steps:
(a) maxl − |α− β|+ λ1 − n
p, l − |β|+ λ2 − n
q
> 0;
(b) maxl − |α− β|+ λ1 − n
p, l − |β|+ λ2 − n
q
< 0;
(c) maxl − |α− β|+ λ1 − n
p, l − |β|+ λ2 − n
q
= 0.
We begin by considering the case (a). Assume that
max
l − |α− β|+ λ1 −
n
p, l − |β|+ λ2 −
n
q
= l − |α− β|+ λ1 −
n
p> 0.
Then Theorem 2.3 (ii) implies that Wl−|α−β|,λ1p (Ω) is continuously embedded
into L∞(Ω). Thus, by remark above (A) follows.
If
max
l − |α− β|+ λ1 −
n
p, l − |β|+ λ2 −
n
q
= l − |β|+ λ2 −
n
q> 0,
then by similar way we see that Wl−|β|,λ2q (Ω) is continuously embedded into
L∞(Ω) and accordingly the truth of (A) follows.
Our next step will be proving of the case (b).
Here we distinguish the following cases (b1) l − |α− β| = 0;
(b2) l − |α− β| > 0.
(b3) l − |β| = 0;
(b4) l − |β| > 0.
52
2.3 Multiplication Theorems for Sobolev Morrey spaces
Before we continue the proof we note that in case (b1)
l = |α− β| ≤ |α| ≤ l ⇔ |α| = l, |β| = 0, (2.11)
and in case (b3)
l = |β| ≤ |α| ≤ l ⇔ α = β.
So we can conclude that (b1) and (b3) cannot be both true.
We start from the case (b1)-(b4). Thus, we have Wl−|α−β|,λ1p (Ω) =
W 0,λ1p (Ω) =Mλ1
p (Ω) and by (2.11) Wl−|β|,λ2q (Ω) = W l,λ2
q (Ω).
We now choose ν ∈]λ2, λ2+l], then we set q1 =nq
n−q(l+λ2−ν) . Since λ2+l <nq,
Theorem 2.3(i) implies the following embedding
W l,λ2q (Ω) →M ν
q1(Ω)
Next we define r1 ∈ [1,+∞] such that 1r1
= 1p+ 1
q1.
Using Holder’s inequality for Morrey spaces, we get that the pointwise
product is bilinear and continuous from Mλ1p (Ω)×Mν
q1(Ω) to Mλ1+ν
r1(Ω).
Now we want to prove that Mλ1+νr1
(Ω) → Mλr (Ω). This embedding holds
provided the following two conditions.
1
r1=
1
p+
1
q1=
1
p+n− q(l + λ2 − ν)
nq=
1
p+
nq− (l + λ2 − ν)
n≤ 1
r,
and
(λ1 + ν)− n
r1≥ λ− n
r.
We can rewrite the second condition as follows
λ1 + ν
n− 1
r1≥ λ
n− 1
r,
λ1 + ν
n− 1
p−
nq− (l + λ2 − ν)
n≥ λ
n− 1
r,
l + λ1 + λ2 − λ
n≥ 1
p+
1
q− 1
r.
So we have the two conditions ln+ λ2−ν
n≥ 1
p+ 1
q− 1
r,
l+λ1+λ2−λn
≥ 1p+ 1
q− 1
r.
By assumptions (2.8), (2.9) we can choose ν sufficiently close to λ2 such
that the above inequalities hold as strict inequalities.
53
2. The embedding and multiplication operators in Sobolev Morrey spaces
Therefore, by Theorem 1.15(iii) Mλ1+νr1
(Ω) is continuously embedded into
Mλr (Ω). Hence, uv ∈Mλ
r (Ω). Moreover, there exists c > 0 such that
∥uv∥Mλr (Ω) ≤ c∥u∥
Mλ1p (Ω)
∥v∥W
l,λ2q (Ω)
∀ (u, v) ∈Mλ1p (Ω)×W l,λ2
q (Ω).
Case (b2)-(b3) can be analyzed as case (b1)-(b4) by switching roles of
Wl−|α−β|,λ1p (Ω) and of W
l−|β|,λ2q (Ω).
Finally, we consider the case (b2)-(b4).
As above we choose γ ∈]λ1, λ1 + l] and µ ∈]λ2, λ2 + l]. Then we have the
following embeddings
W l−|α−β|,λ1p (Ω) →Mγ
p∗(Ω),
and
W l−|β|,λ2q (Ω) →Mµ
q∗(Ω),
where p∗ = npn−p(l−|α−β|+λ1−γ) , q
∗ = nqn−q(l−|β|+λ2−µ) .
Next we define r∗ such as 1r∗
= 1p∗
+ 1q∗.
Holder’s inequality for Morrey spaces implies that the pointwise product is
bilinear and continuous from Mγp∗(Ω)×Mµ
q∗(Ω) to Mγ+µr∗ (Ω).
Now if we recall Theorem 1.15(iii), we see that the embedding
Mγ+µr∗ (Ω) → Mλ
r (Ω) holds provided that
r ≤ r∗, (γ + µ)− n
r∗≥ λ− n
r. (2.12)
As before, we can rewrite this conditions as
1
r∗=
1
p∗+
1
q∗=
=n− p(l − |α− β|+ λ1 − γ)
np+n− q(l − |β|+ λ2 − µ)
nq=
=1
p+
1
q− l
n− l − |α|
n− λ1 − γ
n− λ2 − µ
n≤ 1
r,
l
n+l − |α|n
+λ1 − γ
n+λ2 − µ
n≥ 1
p+
1
q− 1
r,
l
n+l − |α|n
+λ1 − γ
n+λ2 − µ
n≥ l
n+λ1 − γ
n+λ2 − µ
n≥ 1
p+
1
q− 1
r.
54
2.3 Multiplication Theorems for Sobolev Morrey spaces
and
(γ + µ)− n
r∗≥ λ− n
r,
γ + µ
n− 1
r∗≥ λ
n− 1
r,
γ + µ
n− 1
p− 1
q+l
n+l − |α|n
+λ1 − γ
n+λ2 − µ
n≥ λ
n− 1
r,
l + (l − |α|) + λ1 + λ2 − λ
n≥ 1
p+
1
q− 1
r,
l + (l − |α|) + λ1 + λ2 − λ
n≥ l + λ1 + λ2 − λ
n≥ 1
p+
1
q− 1
r.
As before, by assumptions (2.8), (2.9) we can choose γ and µ sufficiently
close to λ1 and λ2 respectively such that the above inequalities hold as strict
inequalities. Thus, we finish the case (b).
Next we will see case (c). We consider the following three separate cases
(c1) l − |α− β|+ λ1 − np< 0, l − |β|+ λ2 − n
q= 0;
(c2) l − |α− β|+ λ1 − np= 0, l − |β|+ λ2 − n
q< 0;
(c3) l − |α− β|+ λ1 − np= 0, l − |β|+ λ2 − n
q= 0.
We start with case (c1).
If l − |β| = 0, then λ2 = nqand W
l−|β|,λ2q (Ω) = L∞(Ω). Therefore, our
remark implies (A).
If l − |β| ≥ 1, we consider separately
(c11) q > 1;
(c12) λ2 > 0;
(c13) λ2 = 0, q = 1.
Let first l − |β| ≥ 1, q > 1. Then we choose q ∈]1, q[ so close to q so that
l + λ1 + λ2 − λ
n>
1
p+
1
q− 1
r,
l
n>
1
p+
1
q− 1
r.
Since q < q we have λ2 + l < nq, if l − |α− β| = 0,
λ2 + l − |β| < nq, if l − |α− β| ≥ 1.
55
2. The embedding and multiplication operators in Sobolev Morrey spaces
But we have already proved such case (see cases (b1)-(b4) and (b2)-(b4)).
Thus (A) is true.
Now we see case (c12): l − |β| ≥ 1, λ2 > 0. Then we choose ν ∈]0, λ2[ so
close to λ2 so thatl + λ1 + ν − λ
n>
1
p+
1
q− 1
r.
Since ν < λ2 we have ν + l < nq, if l − |α− β| = 0,
ν + l − |β| < nq, if l − |α− β| ≥ 1.
But we have already proved such case (see cases (b1)-(b4) and (b2)-(b4)).
Thus (A) is true.
Finally we consider case (c13). Since λ2 = 0, q = 1, we have
W l−|β|,λ2q (Ω) = W
l−|β|1 (Ω).
Condition l − |β| = n implies that
Wl−|β|1 (Ω) → L∞(Ω).
Hence, by our remark (A) follows.
Case (c2) can be analyzed as case (c1) by switching roles of Wl−|α−β|,λ1p (Ω)
and of Wl−|β|,λ2q (Ω).
Now we consider case (c3).
If l − |α − β| = 0 or l − |β| = 0, then λ1 = npor λ2 = n
qrespectively
and Wl−|α−β|,λ1p (Ω) = L∞(Ω) or W
l−|β|,λ2q (Ω) = L∞(Ω). Therefore, our remark
implies (A).
Thus we can assume that
minl − |α− β|, l − |β| ≥ 1.
In such a case, we can split the proof of case (c3) into the following five
cases.
(c31) p > 1, q > 1;
(c32) p > 1, λ2 > 0;
(c33) λ1 > 0, q > 1;
(c34) λ1 > 0, λ2 > 0;
56
2.3 Multiplication Theorems for Sobolev Morrey spaces
(c35) either (p, λ1) = (1, 0) or (q, λ2) = (1, 0).
In case (c31) we choose p ∈]1, p[ so close to p and q ∈]1, q[ so close to q so
thatl + λ1 + λ2 − λ
n>
1
p+
1
q− 1
r,
l
n>
1
p+
1
q− 1
r.
Since p < p, q < q we have λ1 + l − |α− β| < np
λ2 + l − |β| < nq.
But we have already proved such case (see case (b)). Thus (A) is true.
Now we see case (c32): p > 1, λ2 > 0. Then we choose p ∈]1, p[ so close to
p and ν ∈]0, λ2[ so close to λ2 so that
l + λ1 + ν − λ
n>
1
p+
1
q− 1
r,
l
n>
1
p+
1
q− 1
r.
Since p < p, ν < λ2 we have λ1 + l − |α− β| < np,
ν + l − |β| < nq.
But we have already proved such case (see case (b)). Thus (A) is true.
Case (c33): λ1 > 0, q > 1. Then we choose γ ∈]0, λ1[ so close to λ1 and
q ∈]1, q[ so close to q so that
l + γ + λ2 − λ
n>
1
p+
1
q− 1
r,
l
n>
1
p+
1
q− 1
r.
Since q < q, γ < λ1 we have γ + l − |α− β| < np,
λ2 + l − |β| < nq.
But we have already proved such case (see case (b)). Thus (A) is true.
Case (c34): λ1 > 0, λ2 > 0. We choose γ ∈]0, λ1[ so close to λ1 and
ν ∈]0, λ2[ so close to λ2 so that
l + γ + ν − λ
n>
1
p+
1
q− 1
r.
Since γ < λ1, ν < λ2 we have γ + l − |α− β| < np,
ν + l − |β| < nq.
57
2. The embedding and multiplication operators in Sobolev Morrey spaces
But we have already proved such case (see case (b)). Thus (A) is true.
Case (c35): either (p, λ1) = (1, 0) or (q, λ2) = (1, 0). By Sobolev embedding
theorem we have either
W l−|α−β|,λ1p (Ω) = W
l−|α−β|1 (Ω) → L∞(Ω).
or
W l−|β|,λ2q (Ω) = W
l−|β|1 (Ω) → L∞(Ω).
Hence, by our remark (A) follows.
We now show that if α ∈ Nn and |α| ≤ l, then
Dαw(uv) = Qα[u, v] ∀(u, v) ∈ W l,λ1
p (Ω)×W l,λ2q (Ω),
an equality which implies the validity of the Leibnitz rule. To do so, we take
p1 ∈ [1, p], p1 <∞, q1 ∈ [1, q], q1 <∞,
such that
p1 ≥ 1, 0− np1
≥ 0− n1,
q1 ≥ 1, 0− nq1
≥ 0− n1,
andl + 0 + 0− 0
n>
1
p1+
1
q1− 1
1.
Then we note that
W l,λ1p (Ω)×W l,λ2
q (Ω) → W lp(Ω)×W l
q(Ω) →W lp1(Ω)×W l
q1(Ω).
Let now uk and vk be sequences in C∞(Ω)∩W l
p1(Ω) and C∞(Ω)
∩W l
q1(Ω)
respectively such that uk converges to u in W lp1(Ω) and vk converges to v in
W lq1(Ω).
Next we observe that the pointwise product fromW lp1(Ω)×W l
q1(Ω) to L1(Ω)
is continuous.
Indeed, we know that uv = D0w(uv) = Q0[u, v] is continuous from
W lp1(Ω) × W l
q1(Ω) to L1(Ω). Since the pointwise product is bilinear and con-
tinuous from L1(Ω)× L∞(Ω) to L1(Ω) we conclude that∫Ω
ukvkDαφdx→
∫Ω
uvDαφdx.
58
2.3 Multiplication Theorems for Sobolev Morrey spaces
Then ∫Ω
uvDαφdx = limk→∞
∫Ω
ukvkDαwφdx = lim
k→∞
∫Ω
Qα[uk, vk]φdx.
Since Qα is continuous from W lp1(Ω)×W l
q1(Ω) to L1(Ω), we have
limk→∞
∫Ω
Qα[uk, vk]φdx =
∫Ω
Qα[u, v]φdx,
and therefore ∫Ω
uvDαwφdx =
∫Ω
Qα[u, v]φdx.
Corollary 2.9. Let Ω be a bounded open subset of Rn which satisfies the cone
property. Let p, q ∈ [1,+∞]. Let l ∈ N\0. Let λ1 ∈ [0, n/p], λ2 ∈ [0, n/q],
p ≥ q, λ1 −n
p≥ λ2 −
n
q, (2.13)
and
l >n
p. (2.14)
Then the pointwise product from W l,λ1p (Ω) ×W l,λ2
q (Ω) to W l,λ2q (Ω) is bilinear
and continuous and the Leibnitz rule (2.6) holds.
Proof. It suffices to choose r = q, λ = λ2 in Theorem 2.8. We note that for
such a specific choice of the exponents, condition (2.9) implies the validity of
condition (2.8).
Then we have the following immediate consequence of the previous corol-
lary.
Corollary 2.10. Let Ω be a bounded open subset of Rn which satisfies the cone
property. Let p ∈ [1,+∞]. Let l ∈ N\0. Let λ1 ∈ [0, n/p],
l >n
p. (2.15)
Then the pointwise product from W l,λ1p (Ω) ×W l,λ1
p (Ω) to W l,λ1p (Ω) is bilinear
and continuous and the Leibnitz rule (2.6) holds.
Proof. It suffices to choose p = q, λ2 = λ1 in the previous corollary.
59
2. The embedding and multiplication operators in Sobolev Morrey spaces
We now prove in case l = 1 the following stronger result.
Proposition 2.11. Let Ω be a bounded open subset of Rn which satisfies the
cone property. Let p ∈ [1,+∞]. Let λ ∈ [0, n/p],
1 + λ >n
p. (2.16)
Then the pointwise product from W 1,λp (Ω)×W 1,λ
p (Ω) to W 1,λp (Ω) is bilinear and
continuous and the Leibnitz rule (2.6) holds.
Proof. We want to prove that if u, v ∈ W 1,λp (Ω), then uv ∈ W 1,λ
p (Ω).
To do so, we observe that ∀ j ∈ 1, . . . , n
(uv)xj= uxj
v + uvxj,
uxj∈Mλ
p (Ω), vxj∈Mλ
p (Ω).
Since 1 + λ > np, Theorem 2.3 (iii) implies that W 1,λ
p (Ω) is continuously
embedded into L∞(Ω). Then by Theorem 1.18 the pointwise product is bilinear
and continuous from Mλp (Ω)×L∞(Ω) to M
λp (Ω) and from L∞(Ω)×Mλ
p (Ω) to
Mλp (Ω). Thus,
(uxjv) ∈Mλ
p (Ω), (uvxj) ∈Mλ
p (Ω),
and, therefore, (uv)xj∈Mλ
p (Ω) for all j ∈ 1, . . . , n.
60
Chapter 3
The composition operator in
Sobolev Morrey spaces
3.1 Composition operator in Morrey spaces
Lemma 3.1. Let Ω be an open subset of Rn. Let Ω1 be a subset of R. Let
y ∈ Ω1. Let f be a Lipschitz continuous map from Ω1 to R. Let g ∈ M(Ω) be
such that g(x) ∈ Ω1 for almost all x ∈ Ω. Then
|f(g(x))| ≤ Lip(f)|g(x)|+ Lip(f)|y|+ |f(y)| (3.1)
for almost all x ∈ Ω.
Proof.
|f(g(x))| ≤ |f(g(x))− f(y)|+ |f(y)| ≤ Lip(f)|g(x)− y|+ |f(y)| ≤
≤ Lip(f)|g(x)|+ Lip(f)|y|+ |f(y)|
for almost all x ∈ Ω.
Lemma 3.2. Let Ω be a bounded open subset of Rn. Let p ∈ [1,+∞]. Let
λ ∈[0, n
p
]. Then 1 ∈Mλ
p (Ω).
Proof. If p = +∞, then we have λ = 0 and accordingly Mλp (Ω) = Lp(Ω) =
L∞(Ω) and 1 ∈ L∞(Ω) =Mλp (Ω).
Now let p ∈ [1,+∞[. Then we have
wλ(r)∥1∥Lp(Bn(x,r)∩Ω) ≤ wλ(r) (mn(Bn(x, r) ∩ Ω))1p
61
3. The composition operator in Sobolev Morrey spaces
for all x ∈ Ω and r ∈]0,+∞[. Hence,
wλ(r)∥1∥Lp(Bn(x,r)∩Ω) ≤ v1pn r
np−λ ≤ v
1pn
for all x ∈ Ω and r ∈]0, 1] and
wλ(r)∥1∥Lp(Bn(x,r)∩Ω) ≤ (mn(Ω))1p
for all x ∈ Ω and r ∈]1,+∞[ and accordingly 1 ∈Mλp (Ω) and
∥1∥Mλp (Ω) ≤ sup
v
1pn , (mn(Ω))
1p
.
Now we consider the case of Morrey spaces and we introduce the following
sufficient condition.
Proposition 3.3. Let Ω be a bounded open subset of Rn. Let Ω1 be a subset
of R. Let y ∈ Ω1. Let p ∈ [1,+∞]. Let λ ∈[0, n
p
]. Let g ∈ Mλ
p (Ω) be such
that g(x) ∈ Ω1 for almost all x ∈ Ω. Let f be a measurable function from Ω1
to R. Assume that there exists a, b > 0 such that
|f(y)| ≤ a|y|+ b, y ∈ Ω1. (3.2)
Then f g ∈Mλp (Ω) and for any y ∈ Ω1
∥f g∥Mλp (Ω) ≤ a∥g∥Mλ
p (Ω) + b∥1∥Mλp (Ω). (3.3)
Proof.
∥f g∥Mλp (Ω) ≤ ∥a|g|+ b∥Mλ
p (Ω) ≤ a∥g∥Mλp (Ω) + b∥1∥Mλ
p (Ω).
Remark 3.4. If f is a Lipschitz continuous function on Ω1, then Lemma 3.1
implies that condition (3.2) is satisfied with a = Lip(f), b = Lip(f)|y|+ |f(y)|.
Hence, by Proposition 3.3 for any y ∈ Ω1
∥f g∥Mλp (Ω) ≤ Lip(f)∥g∥Mλ
p (Ω) + ∥1∥Mλp (Ω)(Lip(f)|y|+ |f(y)). (3.4)
62
3.2 Composition operator in Sobolev Morrey spaces
Corollary 3.5. Let conditions of Proposition 3.3 are satisfied. If also 0 ∈ Ω1
and f is a Lipschitz continuous function on Ω1, then
∥f g∥Mλp (Ω) ≤ Lip(f)∥g∥Mλ
p (Ω) + |f(0)| ·∥1∥Mλp (Ω).
Corollary 3.6. Let conditions of Proposition 3.3 are satisfied. If also 0 ∈ Ω1,
f(0) = 0 and f is a Lipschitz continuous function on Ω1, then
∥f g∥Mλp (Ω) ≤ Lip(f)∥g∥Mλ
p (Ω).
Corollary 3.7. Let Ω be a bounded open subset of Rn. Let p ∈ [1,+∞]. Let
λ ∈[0, n
p
]. Let f be a locally Lipschitz continuous function from R to itself.
Then
Tf [Mλp (Ω) ∩ L∞(Ω)] ⊆Mλ
p (Ω) ∩ L∞(Ω).
(Note that in general Mλp (Ω) * L∞(Ω)).
Proof. Let g ∈Mλp (Ω)∩L∞(Ω). We set Ω1 =
[−∥g∥L∞(Ω), ∥g∥L∞(Ω)
]. Since Ω1
is a finite segment, f is Lipschitz continuous on Ω1. Hence, by Corollary 3.5
∥f g∥Mλp (Ω) < +∞.
We also have
∥f g∥L∞(Ω) ≤ ∥f∥L∞(Ω1) < +∞.
So, f g ∈Mλp (Ω) ∩ L∞(Ω).
3.2 Composition operator in Sobolev Morrey
spaces
Next we try to understand whether the Lipschitz continuity of a function f of
a real variable is enough to ensure that Tf [W1,λp (Ω)] ⊆ W 1,λ
p (Ω) under suitable
conditions on the exponents. To do so, we face the problem of taking the
distributional derivatives of the composite function f g, and we expect to
prove that
Dj(f g) = (f ′ g)Djg ∀ j ∈ 1, . . . , n.
63
3. The composition operator in Sobolev Morrey spaces
However, it is not clear what f ′g should mean. Indeed, f ′ is defined only up to
a set of measure zero Nf and g←(Nf ) may have a positive measure, and even fill
the whole of Ω, and (f ′ g)(x) makes no sense when x ∈ g←(Nf ). Classically,
one circumvents such a difficulty by introducing a result of de la Vallee-Poussin
which states that both Dj(f g) and Djg vanish on g←(Nf ). Accordingly, it
suffices to define (f ′ g)(x) when x ∈ Ω\ g←(Nf ), and to replace (f ′ g)(x) by
0 in g←(Nf ). We find convenient to introduce a symbol for the function which
equals (f ′ g)(x) when x ∈ Ω \ g←(Nf ) and 0 elsewhere. Then we introduce
the following.
Definition 3.8. Let Ω be an open subset of Rn. Let Ω1 be a measurable
subset of R. Let g be a measurable function from Ω to R. Let the set Ng ≡
x ∈ Ω: g(x) /∈ Ω1 have measure zero.
Let H be a Borel subset of Ω1 of measure zero. Let h be a Borel measurable
function from Ω \H to R. Let hg be the function from Ω to R defined by
hg ≡
0, if x ∈ g←(H) ∪Ng,
(h g)(x), if x ∈ Ω \ (g←(H) ∪Ng).(3.5)
By definition, the function hg is measurable. Next we note that the fol-
lowing holds.
Lemma 3.9. Let Ω, Ω1, h, H be as in Definition 3.8. Let g, g1 be a measurable
functions from Ω to R such that g(x), g1(x) ∈ Ω1 for almost all x ∈ Ω. If
g(x) = g1(x) for almost all x ∈ Ω, then (hg)(x) = (hg1)(x) for almost all
x ∈ Ω.
Proof. Let N be a measurable subset of measure zero of Ω such that
g(x) = g1(x) and g(x), g1(x) ∈ Ω1 for all x ∈ Ω \ N . Since N has mea-
sure zero, it suffices to show that (hg)(x) = (hg1)(x) for all x ∈ Ω \N .
If x ∈ (Ω\N)∩g←(H), then g1(x) = g(x) ∈ H and x ∈ (Ω\N)∩g←(H), and
accordingly (hg1)(x) = 0 = (hg)(x). If instead x ∈ (Ω \N) ∩ (Ω \ g←(H)),
then g1(x) = g(x) /∈ H and accordingly x ∈ (Ω \ N) ∩ (Ω \ g←(H)) and
(hg1)(x) = (h g1)(x) = (h g)(x) = (hg)(x). Hence, (hg1)(x) = (hg)(x)
for all x ∈ Ω \N .
By the previous Lemma, it makes sense to introduce the following.
64
3.2 Composition operator in Sobolev Morrey spaces
Definition 3.10. Let Ω, Ω1, h, H be as in Definition 3.8. If G is an equiv-
alence class of measurable functions g from Ω to R such that g(x) ∈ Ω1 for
almost all x ∈ Ω, then we define hG to be the equivalence class of measurable
functions from Ω to R which are equal to hg almost everywhere for at least a
g ∈ G.
If Ω be an open subset of Rn. We say that g ∈ Lloc1 (Ω) is zero on a subset A
of Ω provided that g(x) = 0 for almost all x ∈ A, for at least a representative
g of g (and thus for all representatives of g).
Remark 3.11. Let g1, g2 ∈ Lloc1 (Ω). Let g1 = g2 almost everywhere in Ω. If
A is a subset of R, then the symmetric difference g←1 (A)g←2 (A) has measure
zero. Indeed, g←1 (A)g←2 (A) ⊆ x ∈ Ω: g1(x) = g2(x).
Then we have the following n dimensional form of a result of de la Vallee-
Poussin [21]. For a proof, we refer to Marcus and Mizel [34].
Theorem 3.12. Let Ω be an open subset of Rn. Let g ∈ W 1,loc1 (Ω) and if N
is a subset of R of measure zero, then (D1g, . . . , Dng) = 0 on g←(N) for any
representative g of g.
Then we introduce the following form of the chain rule (see Marcus and
Mizel [34]).
Proposition 3.13. Let Ω be an open subset of Rn. Let Ω1 be an interval of
R. Let f be Lipschitz continuous function from Ω1 to R. Let
W 1,loc1 (Ω,Ω1) ≡ g ∈ W 1,loc
1 (Ω): g(x) ∈ Ω1 for almost all x ∈ Ω
for all representatives g of g.
Let Nf be the subset of Ω1 such that Ω1 \ Nf is the set of points of Ω1 where
f is differentiable. Let g ∈ W 1,loc1 (Ω,Ω1). Let f ′g be defined as in Definition
3.10 (with h = f ′, H = Nf). Then the chain rule
Dj(f g) = (f ′g)Djg, (3.6)
holds in the sense of distributions in Ω for all j ∈ 1, . . . , n.
65
3. The composition operator in Sobolev Morrey spaces
Then we introduce the following sufficient condition for Sobolev Morrey
spaces of order one.
Proposition 3.14. Let Ω be a bounded open subset of Rn which satisfies the
cone property. Let Ω1 be an interval of R. Let p ∈ [1,+∞]. Let λ ∈[0, n
p
].
Let f be Lipschitz continuous function from Ω1 to R. Let
W 1,λp (Ω,Ω1) ≡ g ∈ W 1,λ
p (Ω) : g(x) ∈ Ω1 for almost all x ∈ Ω
for all representatives g of g.
Then
Tf [W1,λp (Ω,Ω1)] ⊆ W 1,λ
p (Ω).
Let Nf be the subset of Ω1 such that Ω1 \Nf is the set of points of Ω1 where
f is differentiable. Let g ∈ W 1,λp (Ω,Ω1). Let f ′g be defined as in Definition
3.10 (with h = f ′, H = Nf). Then f ′g ∈ L∞(Ω) and the chain rule formula
(3.6) holds in the sense of distributions in Ω for all j ∈ 1, . . . , n. Moreover,
∥f g∥W 1,λp (Ω) ≤
≤ (Lip(f)|y|+ |f(y)|) + Lip(f)(∥g∥W 1,λp (Ω) + ∥1∥Mλ
p (Ω)), (3.7)
for all g ∈ W 1,λp (Ω,Ω1) and for all y ∈ Ω1.
Proof. By Remark 3.4, we know that inequality (3.4) holds for all
g ∈ W 1,λp (Ω,Ω1) ⊆ Mλ
p (Ω) and for all y ∈ Ω1.
Now let g ∈ W 1,λp (Ω,Ω1). Let g be a representative of g. Let Ng be a subset
of measure 0 of Ω such that
g ∈ Ω1 ∀ x ∈ Ω \Ng.
If x ∈ Ω \ (Ng ∪ g←(Nf )), then
|f ′(g(x))| =∣∣∣∣ limη→g(x)
f(g(x))− f(η)
g(x)− η
∣∣∣∣ ≤ Lip(f).
Since f ′g = 0 for all x ∈ g←(Nf ), we conclude that
|(f ′g)(x)| ≤ Lip(f) a.e. in Ω.
66
3.2 Composition operator in Sobolev Morrey spaces
and accordingly that f ′g ∈ L∞(Ω) and that ∥f ′g∥L∞(Ω) ≤ Lip(f) < +∞.
Then by the multiplication Theorem 1.18 and by the membership of Djg in
Mλp (Ω), we have (f ′g)Djg ∈Mλ
p (Ω) and
∥(f ′ g)Djg∥Mλp (Ω) ≤ Lip(f)∥Djg∥Mλ
p (Ω) (3.8)
for all j ∈ 1, . . . , n. Thus the right hand side of equality (3.6) belongs to
Mλp (Ω) for all g ∈ W 1,λ
p (Ω).
By the formula (3.6) for the chain rule, the inequalities (3.3), (3.8) imply
that
∥f g∥W 1,λp (Ω) = ∥f g∥Mλ
p (Ω) +n∑
j=1
∥(f ′g)Djg∥Mλp (Ω) ≤
≤ Lip(f)∥g∥Mλp (Ω) + ∥1∥Mλ
p (Ω)(Lip(f)|y|+ |f(y)|) +n∑
j=1
Lip(f)∥Djg∥Mλp (Ω) ≤
≤ (Lip(f)|y|+ |f(y)|) + Lip(f)(∥g∥W 1,λp (Ω) + ∥1∥Mλ
p (Ω)), (3.9)
for all g ∈ W 1,λp (Ω) and y ∈ Ω1, and thus inequality (3.7) holds true.
Corollary 3.15. Let Ω be a bounded open subset of Rn which satisfies the cone
property. Let p ∈ [1,+∞]. Let λ ∈[0, n
p
]. Let f be a function from R to itself.
Then the following statements hold.
(i) If (1+λ) > npand if f is locally Lipschitz continuous, then Tf [W
1,λp (Ω)] ⊆
W 1,λp (Ω).
(ii) If (1 + λ) ≤ npand if f is Lipschitz continuous, then Tf [W
1,λp (Ω)] ⊆
W 1,λp (Ω).
Proof. We first consider statement (i). The Sobolev Embedding Theorem im-
plies thatW 1,λp (Ω) is continuously embedded into L∞(Ω). Thus if g ∈ W 1,λ
p (Ω),
there exists a bounded subset Ω1 of R such that g(x) ∈ Ω1 for almost all
x ∈ Ω. Since f∣∣Ω1
is Lipschitz continuous, Proposition 3.14 implies that
f g ∈ W 1,λp (Ω).
Statement (ii) is an immediate consequence of Proposition 3.14 with
Ω1 = R.
67
3. The composition operator in Sobolev Morrey spaces
We summarize in the following statement some facts we need in the sequel
in case (1 + λ) > npand which are immediate consequence of Proposition 2.11
and Proposition 3.14.
Corollary 3.16. Let p ∈ [1,+∞], λ ∈[0, n
p
], (1+λ) > n
p. Let Ω be a bounded
open subset of Rn which satisfies the cone property. Let Ω1 be a bounded open
interval of R. Then the following statements hold.
(i) W 1,λp (Ω) is a Banach algebra.
(ii) If (f, g) ∈ C0,1(Ω1)×W 1,λp (Ω,Ω1), then f g ∈ W 1,λ
p (Ω). Moreover, there
exists an increasing function ψ from [0,+∞[ to itself such that
∥f g∥W 1,λp (Ω) ≤ ∥f∥C0,1(Ω1)ψ(∥g∥W 1,λ
p (Ω)) (3.10)
for all (f, g) ∈ C0,1(Ω1)×W 1,λp (Ω,Ω1).
3.3 Continuity of the composition operator in
Sobolev Morrey spaces
Corollary 3.16 shows that if (1 + λ) > npthe composition T maps C0,1(Ω1) ×
W 1,λp (Ω,Ω1) to W
1,λp (Ω). Now we want to understand for which f ’s the com-
position operator Tf is continuous in W 1,λp (Ω,Ω1). By following [31],[30], the
idea is that if f is a polynomial, then Tf is continuous in W 1,λp (Ω). Indeed,
for (1 + λ) > np, the space W 1,λ
p (Ω) is a Banach algebra. Then we exploit
inequality (3.10) to show that if f is a limit of polynomials, then Tf is contin-
uous. Actually, such a scheme can be applied in a somewhat abstract general
setting, which we now introduce. Let X be a Banach algebra with unity. Let
m ∈ N\0. In the applications of the present notes we are interested in the
specific case m = 1, but here we present a more general case, which can be
applied to analyze vector valued functions of Sobolev Morrey.
We first note that if p belongs to the space P(Rm) of polynomials in m
real variables with real coefficients, then it makes perfectly sense to compose
68
3.3 Continuity of the composition operator in Sobolev Morrey spaces
p with some x ≡ (x1, . . . , xm) ∈ Xm. Namely, if p is defined by the equality
p(ξ1, . . . , ξm) ≡∑|η|≤deg p, η∈Nm
aηξη11 . . . ξηmm , with aη ∈ R, (ξ1, . . . , ξm) ∈ Rm, (3.11)
then we set
τp[x] ≡∑
|η|≤deg p, η∈Nm
aηxη11 . . . xηmm , ∀x ≡ (x1, . . . , xm) ∈ Xm, (3.12)
where the product between the xj’s is that of X , and where we understand
that x0 is the unit element of X , for all x ∈ X . Next we state the following
result of [30, Thm. 3.1].
Theorem 3.17. Let ∥ · ∥Y be a norm on P(Rm). Let Y be the completion of
P(Rm) with respect to the norm ∥ · ∥Y . Let X be a real commutative Banach
algebra with unity. Let X be a real Banach space. Assume that there exists a
linear continuous and injective map J of X into X . Let A be a subset of Xm.
Assume that there exists an increasing function ψ of [0,+∞) to itself such that
∥J [p(x1, . . . , xm)]∥X ≤ ∥p∥Y ψ (∥(x1, . . . , xm)∥Xm) , (3.13)
for all (p, (x1, . . . , xm)) ∈ P(Rm) × A. Then there exists a unique map A of
Y ×A to X such that the following two conditions hold
(i) A[p, x] = J [p(x)], for all (p, x) ∈ P(Rm)×A.
(ii) For all fixed x ≡ (x1, . . . , xm) ∈ A, the map y 7−→ A[y, x] is continuous
from Y to X .
Furthermore, the map A[·, x] of (ii) is linear, and A is continuous from Y×
A to X , and if y ∈ Y, y = limj→∞ pj in Y, pj ∈ P(Rm), x ≡ (x1, . . . , xm) ∈ A,
then
(iii) A[y, x] = limj→∞ J [pj(x)] in X ;
(iv) ∥A[y, x]∥X ≤ ∥y∥Y ψ (∥x∥Xm).
We shall call A[y, x] the ’abstract’ composition of y and x.
69
3. The composition operator in Sobolev Morrey spaces
We now turn to apply the above theorem to the case of Sobolev Morrey
spaces. To do so, we need the following.
Proposition 3.18. Let Ω1 be a nonempty bounded open interval of R. Then
C1(Ω1) is a completion of the space (P(R), ∥ · ∥C0,1(Ω1)).
Proof. We first note that
supΩ1
|f ′| = Lip(f) ∀ f ∈ C1(Ω1).
Then we have
∥f∥C1(Ω1) = ∥f∥C0(Ω1) + ∥f ′∥C0(Ω1) = ∥f∥C0(Ω1) + Lip(f) = ∥f∥C0,1(Ω1)
for all f ∈ C1(Ω1). Hence,
∥f∥C0,1(Ω1) = ∥f∥C1(Ω1) ∀ p ∈ P(R).
Since C1(Ω1) is a Banach space and the restriction map from P(R) to C1(Ω1)
which takes p ∈ P(R) which takes p to p∣∣Ω1
is a linear isometry from (P(R), ∥ ·
∥C1(Ω1)) to(p∣∣Ω1
: p ∈ P(R), ∥ · ∥C1(Ω1)
)it suffices to show that p
∣∣Ω1
: p ∈
P(R) is dense in C1(Ω1). Let f ∈ C1(Ω1). Let x0 ∈ Ω1. Then
f(x) = f(x0) +
x∫x0
f ′(t)dt ∀x ∈ Ω1.
Now by the Weierstrass approximation Theorem, there exists a sequence
qjj∈N in P(R) such that
limj→N
qj = f ′ uniformly in Ω1.
Then if we set
pj(x) ≡ f(x0) +
x∫x0
qj(t)dt ∀x ∈ R,
for all j ∈ N, we have
∥f − pj∥C1(Ω1) ≤ supx∈Ω1
∣∣∣∣∣∣x∫
x0
(f ′ − qj)dt
∣∣∣∣∣∣+ ∥f ′ − qj∥C0(Ω1) ≤
≤ m1(Ω1)∥f ′ − qj∥C0(Ω1) + ∥f ′ − qj∥C0(Ω1) ≤
≤ (1 +m1(Ω1))∥f ′ − qj∥C0(Ω1) ∀ j ∈ N,
and accordingly limj→∞
∥f − pj∥C1(Ω1) = 0.
70
3.4 Lipschitz continuity of the composition operator in Sobolev Morrey spaces
Then by applying Theorem 3.17, we obtain the following.
Theorem 3.19. Let p ∈ [1,+∞], λ ∈[0, n
p
], (1 + λ) > n
p. Let Ω be a
bounded open subset of Rn which satisfies the cone property. Let Ω1 be a
bounded open interval of R. Then the composition operator T is continuous
from C1(Ω1)×W 1,λp (Ω,Ω1) to W
1,λp (Ω).
Proof. We set ∥ · ∥Y = ∥ · ∥C0,1(Ω1), X = X = W 1,λp (Ω), A = W l,λ
p (Ω,Ω1), J
equal to the identity map, m = 1. As we have shown, C1(Ω1) is a completion
of (P(R), ∥ · ∥C0,1(Ω1)). By Corollary 3.16, W 1,λp (Ω) is a Banach algebra and
there exists a function ψ as in (3.10). Then by Theorem 3.17, there exists a
unique map A from C1(Ω1)×W 1,λp (Ω,Ω1) to W
1,λp (Ω) such that the following
two conditions hold
(i) A[p, g] = τp[g] for all (p, g) ∈ P(R)×A.
(ii) For each fixed g ∈ W 1,λp (Ω,Ω1), the map from C1(Ω1) to W
1,λp (Ω) which
takes f to f g is continuous.
Moreover, A is continuous. Clearly, T satisfies (i), and inequality (3.10) implies
that T satisfies (ii). Hence, we must necessarily have
A[f, g] = T [f, g] ∀(f, g) ∈ C1(Ω1)×W 1,λp (Ω,Ω1).
As a consequence, T is continuous on C1(Ω1)×W 1,λp (Ω,Ω1).
3.4 Lipschitz continuity of the composition
operator in Sobolev Morrey spaces
Next we prove a Lipschitz continuity statement for the composition opera-
tor. For related results in Besov spaces, we refer to Bourdaud and Lanza de
Cristoforis [9].
Theorem 3.20. Let p ∈ [1,+∞], λ ∈[0, n
p
], (1+λ) > n
p. Let Ω be a bounded
open subset of Rn which satisfies the cone property. If f ∈ C1,1loc (R), then Tf
maps W 1,λp (Ω) to itself and Lipschitz continuous on the bounded subsets of
W 1,λp (Ω).
71
3. The composition operator in Sobolev Morrey spaces
Proof. Let B be a bounded subset of W 1,λp (Ω). Since W 1,λ
p (Ω) is continuously
embedded into L∞(Ω), the set B is a bounded subset of L∞(Ω) and there exists
a closed interval B of R such that
[−∥g∥L∞(Ω), ∥g∥L∞(Ω)
]⊆ B ∀ g ∈ B.
Now let g1, g2 ∈ B. Since f is continuously differentiable, we can write
(f g2)(x)− (f g1)(x) =
=
1∫0
f ′[g1(x) + t(g2(x)− g1(x))](g2(x)− g1(x))dt ∀ x ∈ Ω.
Next we fix x ∈ Ω, r ∈]0,+∞[. By the Minkowski inequality for integrals, we
have
wλ(r)∥f g2 − f g1∥Lp(Ω∩Bn(x,r)) ≤
≤1∫
0
wλ(r)∥f ′[g1(·) + t(g2(·)− g1(·))](g2(·)− g1(·))∥Lp(Ω∩Bn(x,r)) ≤
≤1∫
0
wλ(r) supB
|f ′| ∥g2 − g1∥Lp(Ω∩Bn(x,r)) ≤
≤ supB
|f ′| ∥g2 − g1∥Mλp (Ω).
Hence,
∥f g2 − f g1∥Mλp (Ω) ≤ sup
B|f ′| ∥g2 − g1∥Mλ
p (Ω). (3.14)
Next we fix j ∈ 1, . . . , n and we try to estimate
∥(Dj)w f g2 − f g1∥Mλp (Ω) = (3.15)
= ∥f ′(g2)(Dj)w g2 − f ′(g1)(Dj)w g1∥Mλp (Ω) ≤
≤ ∥f ′ g2 − f ′ g1∥L∞(Ω) ∥(Dj)w g2∥Mλp (Ω)+
+∥f ′ g1∥L∞(Ω) ∥(Dj)w g2 − (Dj)w g1∥Mλp (Ω) ≤
≤ Lip(f ′∣∣B)∥g2 − g1∥L∞(Ω) sup
g∈B∥g∥W 1,λ
p (Ω) + supB
|f ′| ∥g2 − g1∥W 1,λp (Ω) ≤
≤Lip(f ′
∣∣B)∥I∥L(W 1,λ
p (Ω),L∞(Ω)) supg∈B
∥g∥W 1,λp (Ω) + sup
B|f ′|∥g2 − g1∥W 1,λ
p (Ω).
72
3.5 Differentiability properties of the composition operator in Sobolev Morreyspaces
By inequalities (3.14) and (3.15), we conclude that
∥f g2 − f g1∥W 1,λp (Ω) ≤
(1 + n) sup
B|f ′|+
+nLip(f ′∣∣B)∥I∥L(W 1,λ
p (Ω),L∞(Ω)) supg∈B
∥g∥W 1,λp (Ω)
∥g2 − g1∥W 1,λ
p (Ω).
3.5 Differentiability properties of the compo-
sition operator in Sobolev Morrey spaces
Next we turn to the question of differentiability, and by following [30], we note
that the following holds.
Lemma 3.21. Let X be a commutative real Banach algebra with unity 1X .
Let P(Rm) be the set of real polynomials in m real variables. Let p ∈ P(Rm)
be defined by
p(η) ≡∑
|γ|≤deg p
aγxγ11 . . . xγmm , ∀ η ≡ (η1, . . . , ηm) ∈ Rm.
The map τp of Xm to X defined by setting
τp[x1, . . . , xm] ≡∑
|γ|≤deg p
aγxη11 . . . xηmm , ∀ (x1, . . . , xm) ∈ Xm,
with the understanding that x0 ≡ 1X , for all x ∈ X , is of class Cr(Xm,X ), for
all r ∈ N ∪ ∞. Furthermore, the differential of τp [·] at x♯ ≡ (x♯1, . . . , x♯m) is
delivered by the map
Xm ∋ (h1, . . . , hm) 7→m∑i=1
τ ∂p∂xi
[x♯] ∗ hi ∈ X .
Once more, we plan to proceed by approximation and show that Tf is of
class Cr if f is a limit of polynomials with an appropriate norm. As we shall
see, it turns out that a right choice for the norm is the following
∥p∥Yr =∑
|γ|≤r, γ∈Nm
∥Dγp∥Y , ∀p ∈ P(Rm). (3.16)
Then we define Yr to be the completion of the space (P(Rm), ∥·∥Yr). As is well
known, Yr is unique up to a linear isometry, and we always choose Yr ⊆ Y .
Then we have the following obvious
73
3. The composition operator in Sobolev Morrey spaces
Remark 3.22. If r, s ∈ N, s ≤ r, then
Yr ⊆ Ys, ∥y∥Ys ≤ ∥y∥Yr , ∀ y ∈ Yr.
Now we have the following (cf. Lanza de Cristoforis [30, Thm. 2.4]).
Theorem 3.23. Let r, s ∈ N, γ ∈ Nm, r − |γ| = s. Let ∥ · ∥Y be a norm
on P(Rm), and let ∥ · ∥Yr be the norm defined in (3.16), and let Yr be the
completion of (P(Rm), ∥ · ∥Yr). Then there exists one and only one linear
and continuous operator of Yr to Ys which coincides with the ordinary partial
derivation of multi index γ on the elements of P(Rm). By abuse of notation,
we shall denote such operator by Dγ, just as the usual partial derivative of
multi index γ. We have
Dγy = limj→∞
Dγpj in Ys, whenever limj→∞
pj = y in Yr, (3.17)
and
∥y∥Yr =∑
|γ|≤r, γ∈Nm
∥Dγp∥Y , ∀ y ∈ Yr.
With analogy with the usual derivations, Dy denotes the matrix
(D1y, . . . , Dmy).
Then we state the following result of Lanza de Cristoforis [30, Thm. 4.1].
Theorem 3.24. Let r ∈ N\0. Let ∥ · ∥Y be a norm on P(Rm). Let Yr be
the completion of P(Rm) with respect to the norm ∥ · ∥Yr defined in (3.16). Let
X be a real commutative Banach algebra with unity. Let X be a real Banach
space. Assume that there exists a linear continuous and injective map J of X
into X . Let (·) ∗ (·) be a continuous and bilinear map of X × X to X . Let ‘∗’
satisfy the following condition:
J [x1] ∗ x2 = J [x1, x2], ∀x1, x2 ∈ X . (3.18)
Let A be an open subset of Xm. Assume that there exists an increasing function
ψ of [0,+∞) to itself satisfying condition (3.13), for all (p, x) ∈ P(Rm)×A.
Then the restriction of the map A of Theorem 3.17 to Yr × A is of class Cr
from Yr ×A to X . (Note that Yr ⊆ Y0, and that Y0 equals the space Y defined
74
3.5 Differentiability properties of the composition operator in Sobolev Morreyspaces
in Theorem 3.17.) Furthermore, the differential of A at each (y#, x#) ∈ Yr×A
is given by
(u, v) 7−→ A[v, x#] +m∑l=1
A[Dly#, x#] ∗ wl,
for all (u, v) ≡ (u, (w1, . . . , wm)) ∈ Yr × Xm. (For the definition of Dly#, see
Theorem 3.23)
Now that we have introduced the above result on the r times differentiabil-
ity of A, we introduce a formula for the differentials dsA of order s = 1, . . . , r
of A of [30, p. 932]
Proposition 3.25. Let all the assumptions of Theorem 3.24 hold. Let r, s ∈ N,
1 ≤ s ≤ r. The differential of order s of A at (y#, x#) ∈ Yr × A, which can
be identified with an element of L(s)(Yr ×Xm, X ), is delivered by the formula
dsA[y#, x#]((v[1], w[1]), . . . (v[s], w[s])) =
=s∑
j=1
m∑l1,...,lj ,...,ls=1
A[Dls · · · Dlj · · ·Dl1v[j], x
#]∗(ws,ls · · · wj,lj · · ·w1,l1
)+
+m∑
l1,...,ls=1
A[Dls · · ·Dl1y
#, x#]∗ (ws,ls · · ·w1,l1) , (3.19)
for all (v[j], w[j] ≡ (wj,1, . . . , wj,n)) ∈ Yr × Xm, j = 1, . . . , s. In (3.19), the
symbols l1, . . . , ls denote summation indexes ranging from 1 to m.
Next we return to the applications to Sobolev Morrey spaces, and we prove
the following.
Proposition 3.26. Let r ∈ N\0. Let Ω1 be a nonempty bounded interval of
R. Then Cr+1(Ω1) is a completion of the space (P(R, ∥ · ∥Yr), where
∥p∥Yr ≡r∑
l=0
∥∥∥∥ dldtl p∥∥∥∥C0,1(Ω1)
∀ p ∈ P(R).
If f ∈ Cr+1(Ω1) and if pjj∈N is a sequence of P(R) which converges to f in
the ∥ · ∥Yr-norm and if l ∈ 0, . . . , r, then
dl
dtlf = lim
j→∞
dl
dtlpj, (3.20)
in Cr−l+1(Ω1).
75
3. The composition operator in Sobolev Morrey spaces
Proof. As we have already pointed out
∥f∥C0,1(Ω1)= ∥f∥C1(Ω1)
∀ f ∈ C1(Ω1).
Hence,
∥p∥Yr =r∑
l=0
(∥∥∥∥ dldtl p∥∥∥∥C0(Ω1)
+
∥∥∥∥ dl+1
dtl+1p
∥∥∥∥C0(Ω1)
)∀ p ∈ P(R),
and
∥p∥Cr+1(Ω1)≤ ∥p∥Yr ≤ 2∥p∥Cr+1(Ω1)
∀ p ∈ P(R).
Hence, the norm ∥ · ∥Yr is equivalent to the norm ∥ · ∥Cr+1(Ω1)on P(R).
Since Cr+1(Ω1) is a Banach space and the restriction map which takes
p in P(R) to p∣∣Ω1
in Cr+1(Ω1) is linear isometry of (P(R), ∥ · ∥Cr+1(Ω1))
onto(p∣∣Ω1
: p ∈ P(R), ∥ · ∥Cr+1(Ω1)
), it suffices to show that for each f ∈
Cr+1(Ω1), there exists a sequence of polynomials pjj∈N in P(R) such that
f = limj→∞
pj∣∣Ω1
in Cr+1(Ω1),
i.e., p∣∣Ω1
: p ∈ P(R) is dense in Cr+1(Ω1). We already know that such a
statement is true for r = 0. We now assume that the statement holds for r
and we prove it for r + 1.
By inductive assumption, there exists a sequence of polynomials qjj∈Nsuch that
limj→∞
qj∣∣Ω1= f ′ in Cr(Ω1).
Now let x0 ∈ Ω1. Then
f(x) = f(x0) +
x∫x0
f ′(t)dt ∀x ∈ Ω1.
Then we set
pj ≡ f(x0) +
x∫x0
qj(t)dt ∀x ∈ Ω1.
Clearly, pj ∈ P(R) for all j ∈ N. Since limj→∞
qj∣∣Ω1= f ′ uniformly in Ω1, the
inequality
|f(x)− pj(x)| ≤ m1(Ω1)∥f ′ − qj∥C0(Ω1)≤
≤ m1(Ω1)∥f ′ − qj∥Cr(Ω1)∀x ∈ Ω1,
76
3.5 Differentiability properties of the composition operator in Sobolev Morreyspaces
shows that limj→∞
∥f ′ − pj∥C0(Ω1)= 0. Hence,
limj→∞
∥f ′ − pj∥C0(Ω1)+
r∑l=0
∥∥∥∥ dldtl f ′ − dl
dtlqj
∥∥∥∥C0(Ω1)
= 0
and accordingly
limj→∞
∥f ′ − pj∥Cr+1(Ω1)= 0
Equality (3.20) is a well-known corollary of the theorem on passing to the limit
under the differentiation sign.
Remark 3.27. Under the assumptions of the previous theorem, the oper-
ator Dγ defined by (3.17) coincides with the ordinary Dγ-differentiation in
Cr+1(Ω1).
Theorem 3.28. Let p ∈ [1,+∞], λ ∈[0, n
p
], (1+λ) > n
p. Let Ω be a bounded
open subset of Rn which satisfies the cone property. Let Ω1 be a bounded open
interval of R. Then the composition operator T from Cr+1(Ω1)×W 1,λp (Ω,Ω1)
to W 1,λp (Ω) defined by
T [f, g] ≡ f g ∀ (f, g) ∈ Cr+1(Ω1)×W 1,λp (Ω,Ω1)
is of class Cr. If (f0, g0) ∈ Cr+1(Ω1)×W 1,λp (Ω,Ω1), then the first order differ-
ential of T at (f0, g0) is given by the formula
dT [f0, g0](v, w) = v g0 + f ′(g0)w
for all (v, w) ∈ Cr+1(Ω1)×W 1,λp (Ω,Ω1).
If s ∈ 1, . . . , r, then the s-th order differential of T at (f0, g0) is given by
the formula
dsT [f0, g0][(v[1], w[1]), . . . , (v[s], w[s])] =
=s∑
j=1
ds−1v[j]dts−1
g0w[1] . . . w[j] . . . w[s] +dsf0dts
g0w[1] . . . w[s],
for all (v[1], w[1]), . . . , (v[s], w[s]) ∈ Cr+1(Ω1)×W 1,λp (Ω,Ω1).
Proof. We set ∥ · ∥Yr = ∥ · ∥C0,1(Ω1), X = X = W 1,λp (Ω), A = W l,λ
p (Ω,Ω1), J
equal to the identity map, m = 1. As we have shown, Cr+1(Ω1) is a completion
of (P(R), ∥ · ∥Yr). By Corollary 3.16, W 1,λp (Ω) is a Banach algebra and there
77
3. The composition operator in Sobolev Morrey spaces
exists a function ψ as in (3.10). As we have already proved in the proof of
Theorem 3.19, the abstract composition A of Theorem 3.17 coincides with T .
Then we can invoke Theorem 3.24 and Proposition 3.25 conclude that T is
of class Cr from Yr × A = Cr+1(Ω1) ×W 1,λp (Ω,Ω1) to W
1,λp (Ω) and that the
formulas for the differentials hold.
78
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