interpolation of weighted spaces of differentiable ... · interpolation of weighted spaces of...

interpolation of Weighted Spaces of Differentiable Functions on R ~ (%

5. L S ~ s ~ 6 ~ (G6teborg, Svezia)

S u m m a r y . - Let W~(w) be the Sobolev space of functions f such that D~f ~ L~(w) for [~[ <~ m. The paper charaeteristizes the interpolation spaces between Sobolev spaces by means of the rear and complex interpolation methods. The interpolation spaces are characterized as potential spaces and Besov spaces on Lr(w).

O. - I n t r o d u c t i o n .

This paper is concerned with the characterization and interpolation of spaces

of differentiable functions on ~d. As an example of the kind of spaces we study,

we mention here the Sobolev space W ~ ( w ) defined by the norm

I I w ' D = / [ b .

Here w is a weight function of a certain class, including for instance

w(x) ---- (1 -F lx1]~)"~/2"... �9 (1 -F [xkle) "~/2 , sx, ..., s~ rea l ,

where x ~ J ' ~-- = (six1, ..., s~xd) , e~-- 0 or 1. The aim of the paper is to characterize the interpolation spaces between two Sobolev spaces or between two of their gener- alizations. I n particular we s tudy the spaces

(w,;:(%), w;':(w,))o.o and (W2(wo),

We are following the presentation given in [1] for the case Wo ~ wl = 1 very

closely. Our main results are given in sections 4-7, while sections 1-3 contains some basic results on translation invariant operators on weighted L~-spaces.

Several authors have been concerned with weighted Sobolev and their gener-

alizations: Besov spaces and potential spaces. However there are not so m a n y

papers deMing with interpolation theory of such spaces. Our main source of inspira-

(*) Entrata in Redazione il 5 ~prile 1982.

13 - AnnaI i di Matemat ica

~90 J . L6~sTaS~I: Interpolation o] weighted spaces, etc.

t ion has been the book of TnlEBE~ [5] where the th i rd chapter is devoted to the subject. We also refer the reader to GRISVARD [2] and in par t icular TI~IEBEL [6], where similar results are obta ined b y different methods.

1. - W e i g h t e d L e b e s g u e s p a c e s .

When we come to the in terpolat ion of spaces of differentiable functions (Sobolev

and Besov spaces) we shall only deal with weight functions which are powers of polynomials . However , to begin wi th no difficulties will arise if we consider any

non-negative~ locally integrable weight funct ion w on R ~. Then the weighted Lebesgue space L~(w) is defined for l < p < o o b y means of the norm

tta

A basic fact on weighted Lebesgue spaces in the Stein-Weiss interpolat ion

theorem.

Tm~Ol~E~[. - Le t /T be a linear mapp ing defined on a linear space containing

L~o(Wo ) and L~l(w~), where we and w~ are given weight functions and l < p o < O O , l < p ~ < o o . Assume t h a t the restr ict ions of T to JL~o(W0) and to L~(w~) are bounded

with norms Me and M~ respectively. Le t 0 < 0 < 1 and pu t

% = w~-~ 1/p = ( 1 - 0)/po + 0/pl.

Then T is defined on L~(w) and its restr ict ion to L~(w) is bounded with norm Me, where

~< M o M1 �9 [] M- ~ 1-o o

Since we are going to s tudy convolut ion operators on L~(w), i t will be essential

for us to know if the t ranslat ion operator rh~ defined by (~h])(x) = ](x-- h), is a bounded operator on L~(w). Assuming t h a t w(x) > 0 almost everywhere, i t is easily

seen t ha t the est imate. []w~YII~<w*(h)]Iw/l[~ holds for all ] ~ L,(w), if and only if w(x + h)<w*(h)w(x). Thus all the t rans la t ion operators r~, h ~ R ~, are bounded if

and only if

(1.1) w(x + y)<~w*(y)w(x), (for a lmost all x, y).

W e shall say t h a t the weight funct ion w is regular on R d if w(x) > 0 everywhere and if there is a weight funct ion w* wuch tha t (1) holds. I f w is a regular weight funct ion then the t ransla t ion operators z~ are bounded on L~(w) for a lmost all h

3. LSFSTRS~: Interpolation o/ weighted spaees, etc. 191

with norm <w*(h). As a consequence we see t ha t if w*g is a bounded measure wi th to ta l mass II[w*glll~ then

(1.2) lily" (g*?)]l~ < ill*v* gill Ilw/l]~.

I f w is a regular weight function~ then there is an equivalent~ infinitely differen-

t iable regular weight function w~. I n order to prove this~ let h belong to the class C~ r of infinitely differentiable functions with compact suppor t . Assume also t h a t h is

non-negat ive and h ( 0 ) > 0. Then define w~ by the formula

w~(x) = (w , h)(x) = f w ( ~ - t)h(t)dt =fw( t )h(x- t)dt.

Clearly w~ is infinitely differentiable and non-negative. B y (1) we see t ha t w*(y) ~ 0 everywhere. Wri t ing w*(y) = l /w*(- - y) we get f rom (1.1) t ha t w*(y)w(x)<w(x

y)<w*(y)w(x) for a lmost ~11 x, y. Thus

w(~)~w,(- t)h(t) dt<~,,(x)<w(x)f~*(- t)h(t) dr,

for a lmost all x. This proves t ha t wl is equivalent to w. I n order to prove t h a t wl is regular we have only to observe t ha t

w~(x + y)<w(x)fw*(y- t)h(t) dt = w(x)fw*(t)h(y- t)dr.

I f w and wl are equivalent weight functions then L~(w) ---- L~(wl), with equivalent

norms. Thus it is no restr ict ion to assume tha t a regular weight funct ion w is infinitely differentiable. We can also normalize w so t ha t for instance w(0)--~ 1.

Final ly we n o t e t ha t if wl and w2 are regular weight functions~ then so are w~'w2, w~ (s real) and ~ , defined by ~ ( x ) = w ( - - x ) .

As examples of regular weight functions we note

(1.3) (1 + Ixl) 8

(1.4) (1 -N Ix11)" ' . . . ' (1 + I~1) '~ ,

or more generally

(~.5) (1 + Ix~l),~.....(1 + ix~l) ~ ,

�9 ~ ~ = 0 or 1, and where s, sl, s~, ... are real numbers . where x ~ = (s~ xl, ..., s~ x~), ~,

192 J. L @ s ~ 6 ~ : Interpolation o/ weighted spaces, etv.

Other examples are

(1.6)

(1.7)

(1.s)

exp (s~x~ + ,.. + s~x~) ,

exp(slx?),

2. - Trans la t ion i n v a r i a n t operators.

Le t w be a weight funct ion on R g. A bounded linear mupping T on L~(w) is said

to be translation invariant if T commutes wi th all t ranslat ions ie if rh(T]) = T(7:h]) for all ] e L~(w) and all h e / ~ . We shall see t h a t if w is a regular weight funct ion

then a t ransla t ion invar ian t opera tor :T can be represented uniquely as a convolu-

t ion b y a distr ibution k, ie we can write

T ] = k . ] for all ] E C ~ ~

_Note t ha t in the case w = 1, k is always a t empered distribution, whereas in the general case, k is jus t a distribution. The proof is a lmost word b y word the same as the proof in the case w = 1 given in t tS rmanders paper on t ransla t ion invar ian t

operators. (See [4], theorem 1.1.) Fo r conveniance we sketch the proof here.

F i rs t we prove t h a t D~(T/) = T(D~/) (in the distr ibution sense) for all ] c C~ ~

I t is sufficient to give the proof for first order derivatives. Le t us restr ict ourselves to the case D ~ = D~ = (5/(~x~.

P u t ]~(x) = ] ( x ~ h, x~, ..., x~). Then h-~(]~-- ]) tends to D~j in L~(w) (for every

] c C~). I n fact , write A~ = h-~(]a - / ) - - D1]. Then

h

-= h - l i ( h - t)D~f(x~-F t, x~, ..., xa) Ah dt . 0

Since w is assumed to be regular we have t ha t w(xl, x2, ..., xd) < w*(-- t, O, ..., .., 0)w(xl@ t, x2, ..., x~). Assuming for simplici ty t ha t h > 0, we get

h

IIwA ]i < h-'f(h-- t ) w , ( - t, o, . . . , o) dtllwD r . 0

Since w* is locally integrable, the r ight hand side tends to zero as h - * 0. I t

follows that Ah-+ 0 in L~(w). :Now the t ransla t ion invar iance of T implies t ha t if g = T], then h-l(gh--g) =

= T(h-l(f~ - ])). Since the r ight hand side converges in L~(w) to T(D1/) we conclude

t h a t Dig ~-L~(w) and D i g = T(Dt]). More generally D~g = D~(T]) = T(D~]). Thus all distr ibution der ivat ives of g are in L~(w).

J. LSFST~5~: Interpolation o] weighted spaces, etc. 193

Since w is regular, w -z is locally bounded. I f K is a compact set in R ~ we therefore have t ha t

f \z/~

K

Therefore all derivat ives of g = T] are locally in L~. B y a special ease of

Sobolev 's l e m m a (see HSn~A~Dn~ [4], l emma 1.1) we conclude t ha t g, af ter cor- rection on ~ set of measure zero, is continuous and tha t

e Z 12: �9 Ig(o),<. f lD~ ]lwZ) glI, I~{<:

Since D~g = T(D~]) and T is bounded on L~(w), with norm say M, we conclude t ha t

ITy(O)]< O1M ~ [[wD~]]I, , f ~ ~ .

Writ ing i ( x ) = ] ( - - x ) we see t ha t the mapp ing i - + T](O) is a distr ibution k,

such t ha t T](0) = k(]) = k , ] ( 0 ) . . N o w the mapp ing ] --> k , ] is t ranslat ion invariant. Thus we arr ive f rom the relat ion T](O) ~ k , ](0), to the relat ion T/(x) ----- -= k , ](x) for all x c R ~. Clearly k is uniquely determined b y T.

Thus, assuming tha t w is a regular weight function, we have showed tha t a bounded t ransla t ion invar iant operator T can be identified with a distr ibution k via the formula T] -~ k �9 f, ] ~ C~. The space of all t ranslat ion invar iant operators

on L~(w) will be denoted b y T~(w). This space will be normed b y the operator norm

T,(w; z~) = sup Ilw(k � 9 ]lw~ll~ = z

I t is easy to see t ha t T~(w) is complete. :Note also t ha t if T1]= k l , ] and

T2f ---- k~* / are two t ransla t ion invar iant operators on L~(w), then T~T2 also belong

to T~(w). Consequently (T1T~)] = k �9 ]. We shall write k = kl* ks. Then clearly

T~(w; k~, kD<T~(w; /~)T~(w; kD.

Thus T~(w) is a Banach algebra under convolution. F r o m (1.2) we see t ha t C~ is a subspace of T~(w).

We shah now derive a few fundamenta l propert ies of t ransla t ion invar ian t operators on L~(w). All the t ime we shall assume tha t w is a regular weight func-

tion. As we explained in section 1 it is no restr ict ion to assume tha t w is infinitely differentiable and tha t w(0)----1.

194 J. LhFSmRh~[: Interpolation el weighted spaces~ etc.

First we shall prove tha t

(2.1) T~,(w) : T~(ff; -~) with equal norms ,

if @(x) ---- w(-- x) and 1/pr= 1- - 1/p~ (l<<p<oo). This result is proved by means of a dual i ty argument. First we note tha t ][w(k */)i]~' is the supremum of l ( ( k , / ) , �9 (@g))(0)] where ][glib---- 1 and /, g e C?. l~ow ( k , / ) , (@g)= k * (]* (@g)) = = ( k , (~g)) * / . l~ote tha t v~g e C~. Thus

( (k , n , �9 �9 (wn)(o).

The absolute value of the right hand side can be estimated by

since the norm of @g in L,(~-~) is ~:]gll~ = 1 . w e conclude tha t

I',w(k �9 < T~(~,-~; k)Hws%,, / z C?'.

Thus

T~,(w; k) < 2~(e-I; k).

l~eplacing p by pr and w by ~-i we get the converse inequality. This proves (2.1).

Note that (2.1) gives T1(w) = Too(e-l). If T e Too(~ -~) then the functional / --> -+ l(/)= T(~/)(O) is bounded in the supremum norm. In fact, if M is the norm

of m in T ~ ( e - : ) we h a v e 7(/)1 = [ e ( 0 ) - : m ( e . / ) ( o ) l < M I / / H ~ , if w(0) ---- 1. Thus l(/) = -= # */(0) where / t is a bounded measure such tha t I[]/~[tl =fld~l<<ig. Consequently T(ff~/) = # , / ( 0 ) = ( ~ - ~ # ) , (~/)(0) ie T / - ~ (w-~#) , / (0) . Hence T / : k , / where k ---- w-1~ and lllw.klll<M.

Thus~ if w(0)= 1

(2.2) '~llwktl I < T~(w; k) = T~o(e-1; k) .

If we have w*(y)< Cw(y) i.e. if

(2.3) w(x 4- Y) < Cw(x)w(y)

then (1.2) implies the converse inequality

m~(w; k ) < Clllwl~lll �9

Thus if (2.3) holds, then T~(w) = T~(fT ~) is the space of all distributions k such tha t wk is a bounded measure,

J. L6]~s~1%6~: Interpolation o/ weighted spaces, etc. 195

Another impor tan t inequali ty is given b y the Stein-Weiss interpolat ion theorem. If we assume 0 < 0 < 1 and let wo and w~ be two regular weight functions then

1 - - 0 _ 0 W 0 = W 0 'W 1

is a new regular weight function. Thus we have

(2.4) T~(Wo; k)< T~(Wo;/~)l-~ k) ~

if 1 /p = ( 1 - o)/po + o/pl , o<0<1 . Combining (2.4) and (2.1) we get some interesting results provided tha t w is

symmetric i.e. ~ = w. In fact, use (2.4) with Wo = w and w~---- w - ~ = ~-~. Since T~(w; k) -= T~,(w-~; k) in this case, we conclude tha t

/'2(1; k )< T~(w; k)~/2T~,(w-~; k)~/~= Y~(w; k)

consequently T~(w)c T~(1) for all symmetric regular weight functions. Bu t now T~(1) = 5~-~L~, the space of all tempered distributions k whose Fourier t ransform belongs to L~. Thus we have proved tha t

(2.5) T~(w) r Y-IL~ (w is symmetr ic ) .

Using the same idea we also get

T~(w~; k)<T~(w; k)l-~ k)~ TAw; ~)

if w is symmetr ic and

Q = ~ - 2o , ~/q = ( ~ - o)/p + o/p' .

Thus we get the following norm-increasing inclusion

(2.6) T~(w) c T~(we), (w is symmetr ic ) ,

where p =/= 2 and

= ( 1 - 2 / q ) . ( 1 - 2//)) -I , l < p < q < 2 or 2<q<.p<.oo.

3. - Estimates for T~(w; k) for special weight functions.

In this section we shall consider arbi t rary powers of a polynomial wo. can write

(3.1) Wo(X § y) = Z a~(x)y ~, ve]

Then we

196 J. L6FSmR6z: Interpolation o] weighted spaces~ etc.

where G

tha t is

are polynomials and 5 is a finite set of multi-indices. Hence we have t ha t

(3.2) wo(I~ * i) ---- ~ I~ , (a~i) where k (y ) ---- y~k(y) .

Now let M~ denote the space of Fourier t ransforms of the distributions in Tall) .

This means t ha t M~ is the famil iar space of Fourier multipliers. (See HSz~A~-

DE~ [4].) The norm on M~ is given b y I[~l!~ = T~(1; k)~ where/c denotes the Fou-

rier t r ans fo rm of k. Thus

, ~ , ^

I lk, t11~< llill~ilkll~,, 1 ~ c ~ .

Since ]~ = D~]~ we conclude f rom (3.2) t h a t

(3.3)

I n order to get an es t imate for T~(Wo; k) we want to replace I]G]][~ b y [iwo/lI~. Thus we shall assume tha t IG(x) i<A~wo(x) , i.e.

(3.~) [D~ wo(X) l < v ! A~wo(x) �9

(Note t ha t this means t ha t Wo is regular and strict ly positive.) Now we arr ive at

the basic es t imate

(3.5) ~, A~j!D kltM~ s ~l ~ , .

~ e x t we replace wo by ~w o' (r = 1, 2, 3, ...). Since

Wo(X § y)r= ~ a~(1)(x).....a~(~)(x).y~(~)+..+~(~) v(1) . . . . . v ( r )e :J

we conclude t ha t

(3.6) 17(~0 ; ~ ) < ~ ~ -~ , A , ,llD~(1)+'"+v(r)]~l[~1 v(1) ..... v(r)ea

Now let Wo be a given polynomial such t ha t (3.4) holds. Then w~ is a regular weight funct ion for any real s. B y (2.1) we have T~(wo ~) : T~,(WSo). Moreover Wo satisfies the same assumptions as wo~ I n order to es t imate T~(wSo; k) we can there-

fore restr ict ourselves to the case s > 0,

J. LSFSTR5~: Interpolation o] weighted spaces, etc. 197

For non-integral s we can write s ~ r~-O, where 0~<0~<1 and r = 0 , 1 , 2 , .... Then Stein-Weiss interpolat ion theorem implies

(3.7) r~(w~; k)< T~(w~; ~)~-~ k) ~

Combining this es t imate with (3.6) we are able to get est imates for T~(w~; k) for s > 0 .

~ e x t let w we a p roduc t of a rb i t ra ry powers of weight functions like w0. Then Stein-Weiss can again be used to get est imates for T~(w; k).

I n fact , we have

(3.8) , ,wl . . . . . , , k) . . . . . r A w , , k ) ~

if O<Oj<~l,j = 1, ..., n and 0 ~ - ... ~- 0 . = 1.

Note also t ha t

(3.9) T~(w, + w~; k)<2(T~(w~; k) + T~(w~; k)),

since lt(wl -}- w~)]]l~< Iiw~]]]~ + I[w2]H~<21I(Wl + w2)]I[~ �9 Using (3.5)-(3.9) we are now able to get est imates for T~(w; k) for all weight

functions w, belonging to the class defined in the following definition.

DEFIS!ITIOST. - A weight funct ion w is said to be polynomial ly regular if w is equivalent to a finite sum of products of finitely m a n y polynomials wl, ..., w,, sat-

isfying (3.4). Thus w ~ ~ w 1 . . . . ... w~J. I f the degree of w, is m, we define the degree

of w to be the largest of all numbers m~[s~[-~ ... ~-m~Isj] occuring in the sum representing w.

~ o t e t ha t a polynomial ly regular weight funct ion is str ict ly posit ive and regular. For examples see formulas (1.3)-(1.5).

We shall now derive a few l emmata which will be used in our s tudy of weighted Sobolev and Besov spaces.

IJEM25A 1. -- Le t w be a polynomial ly regular weight function. Assume tha t )~ is infinitely differentiab]e and tha t there is a number b > 0, such t ha t

for all a and ~. Then k e T~(w) for 1 < p < c~. The norm T~(w; k) can be es t imated

b y a constant t imes the m a x i m u m of finitely m a n y Ca. The constant and the number of Ca :s needed depend on w and the dimension.

PnooP. - Using (3.5)-(3.9) we see tha t if suffices to prove tha t D~l~e M~ for all v. Moreover we can assume tha t ]c(~) = 0 for [$J < 2, since any infinitely differentiable funct ion with compact suppor t belong to M s.

]98 J. L6FS~R6~: Interpolation o/ weighted spaces, etc.

We shall use the well known estimate

(3.10) c [l o',ll s u p / i D ~ ] l ~ if 0 = d/2L, I~i=L

where L > d/2. (See for instance [2], 1emma 6.1.5.) This est imate will be applied to the functions ]~z(~) = ?(2-~)/~(~), where 90 is infinitely differentiable with support on 2-1~<i~1<2 and ~ o ( 2 - ~ ) = 1 for ~v~ 0. Then the assumptions on ]c implies

19%(~) l << r 2-'(b+I~'l),

and thus

Then (3.10) implies tha t

ilD@~il2 < C'~,d2-'(b§

liD"k,/!2 < Q,d 2-'(b+I=i) �9

Consequently ~ D"]~ converges in M~ and hence D')~ belongs to M~. This proves the 1emma.

LEM~rA 2. - Suppose tha t w is a polynomially regular weight function of degree at most m, where m is an integer. Assume tha t D ' f c e M ~ for all tv]<m. P u t

]~t(x) = tdk(tx) for t > 0 ,

that is

Then ] ~ T~(w) for all t > 0 and

~r~(w; ~:t)< C max (1, t -<~) .

The kons tant depends on k, w and the dimension.

P~ooF. - We can write w as a finite sum of products of the form w]'.....w~S~, where wl, ..., wj are polynomials of degree say ml, ..., mj: Then m'---- ml]sl] ~- . . . -~ mjlsj] <~m and by (3.8) we have tha t

.T~(w~'. ., ~. k~ )<T ,,,~-'</ ..... ~t) ~ ~S/,,~. ... Wj , -~w~l , " . . . 'T~(wj , kt) ~

J." L6FSTR6~: Interpolation o] weighted spaces, etc. 199

where e~ = sign s~ and 0~-~ mds,J/m'. Since M~, = M, with equal norms, we get f rom (2.1); (3.6) and (3.7) t ha t

Observing tha t

l"l<m

ILD~fc,]FM~ = t-I~liD~fCIJM~ < c max (1, t - " ) ,

we get the conclusion of the lemma.

LE~I~A 3. - Suppose tha t w is a polynomially regular weight function of degree at most n b where m is an integer. Moreover let 1 be a distribution on R such tha t D~l belong to M~ (on R) for all j<m. Then we pu t

where a e R d. Then k e Y~(w) and T~(w; k(~)) is uniformly bounded on ]a[ = 1.

P~ooF. - :Note tha t /)~]C(a)(~) = a~( Dl~i l)((a, ~}) and tha t the M~-norm of (D P ll)"

�9 ((a, ~}) is equal to the M~-norm, in one dimension, of DNl. Therefore D~(~)e M~ for all I~[~<m. This implies the result just as in the proof of lemma 2.

4. - Weighted Besov spaces and generalized Sobolev spaces.

Let k be a positive integer. Then the Sobolev space of all ] e Z~ such tha t D~/eL~ for t~]~<k will be denonted by W~. The norm of W~ is

i~l<k

Given a weight function w we shall simply replace the L~-norms on the right

hand side by norms in L~(w). In this way we define the weighted Sobolev spaces W~(w), by means of the norms

(4.1) W~(w; /) = ~ llwD~/ll~ .

This is the simplest estension of the ordinary Sobolev spaces. A more com-

plicated extension could involve several weight functions w~, one for each derivative D~/, with some natural connection between the different w~ :s. The reader is refered to TICIEBELS book [5]~ chapter 3.

200 J. L 6 F s ~ 6 ~ : Interpolation o] weighted spaces, etc.

The spaces W~ (with weight funct ion 1) can also be defined as potent ia l spaces. Le t J be the potent ial operator defined by (J])^(~) = (1 + [~]~)~/2]^(~). Then the

norm on W~ is equivalent to UJ~]li]~ for 1 < p < co. (See [1], Theorem 6.2.3.) l~low the potent ia l space P~ (or generalized Sobolev space) is defined to be the space of all / e L~ such tha t J~] ~ L~. The norm on 2~ is

P~( I ) = ]l(Jst)ll~ �9

Similarly, given a weight funct ion w, we define the weighted potent ia l space P~(w) (or generalized Sobolev spaces) by means of the norms

(~.2) P~(w~/) = ]lw(JV)i[~, s rea l , l < p < o o , / e L ~ ( w ) .

We shall prove later on tha t W~(w) : P~(w) if 1 < p < 0% provided tha t w is a polynomial ly regular weight function.

In a similar way we shall ex tend the definition of the Besov spaces. Follow- ing [1] (section 6.2) we define the weighted Besov spaces B~'q(w) in the following way. Le t ~ be an infinitely differentiable function with support on the annulus 2 -~<[~I<2 and assume tha t ~ 9 ( 2 - z ~ ) : ! for ~ # 0 . Then define % and %0 by the formulas

}(~) = i- Z ~(2-~). l = l

Then the spuces B~'q(w) are defined by means of the norms

(4.3) B;'q(w; ]) -: ([[w'(~, * ])il~ n- ~ (2~Zi]w'(% * f)l]~)q) 1/q l = 1

for ! 4 p 4 o % I < q < o % s real. Although this definition looks ra ther innocent it can hide some complications. For instance one would expect tha t any funct ion ] of the form qJj. g, g ~ L~(w)

~s,~,o.~ Bu t if this were the case then belongs to ~ ~ j .

2"[]w. (Tj, ~j , g)L< cliwg][~.

This is however not t rue for all weight functions. (See [7].) Consequently some restrict ion has to be imposed on w. In this paper we shall

assume tha t w is a polynomially regular weight function. (See the definition of sec- t ion 3.) Then the lemnlata of the preceeding section makes it possible to take over almost word by word the proofs given in the case w = I in [1], chapter 6.

Firs t we note tha t if w is polynomiMly regular then

(4.4) T~(w; J~T~)< C2 ~ if l > 0 .

~~ 3. L6~s~r~6~: interpolation o] weighted spaces, etc. 20J_

In fact~ the Fourier transform of J~% is of the form 2 ' ~ ( 2 - ~ ) where ]c~(~)= = ( 2 - ~ ISp)~/~(~). Clearly )~ is i~dinitely differentiablc with compact support und all derivatives of ~ are uniformly bounded in l ~ 0. Thus lemma 1 and 2 imply (4.4).

From (4A) follows a series of simple results. :For instance we have (with con-

tinuous injections)

(4.5)

(4.6)

B~,l(w) c ~ ( w ) c B~'~(w),

P~(w) c P~(w) if s~ < s~.

~-ote also tha t

(4.7)

(4.8)

: "B~'r c B~'~(w) if s 1 > s~,

8~q 1 8~q~ B~ (w) cB~ (w) if q~Kq~.

T h e last two inclusions are immediate consequences of the definition of Besov spaces. Clearly (4.6) follows from (4.5), (4.7) and (4.8). Therefore it is sufficient to prove (4.5). How the norm of a given ] in P~(w) can be estimated by l [w ' ( J~ ') * �9 ]][s-~ ~ ] [ w ' ( J ~ ) * ]l]~. BY lemma 1 and (4.4) this sum is clearly bounded by

l>/1

constant times the norm of ] in B~'~(w). Thus the left hand inclusion of (4.5) follows. The right hand side is a consequence of (4.4) and the identi ty % . ] ~ (J-~,) * (J']).

Note glso tha t Jr is an isomorfism from B~'q(w) onto B~=r and from P~(w) onto _P~-~(w). Using this fact one easily finds tha t the dual of P~(w) is P y ( w -~) if 1 < p < c ~ . Jus t observe tha t the dual of L,(w) is Z,,(w-~). Similarly the du~l of B~'~(~z) is B~S'~'(w-1) if 1 < p < c ~ and 1 < q < c~, since it is easily seen tha t the dual of B~'q(w) can be identified with B~

We shall now give an explicite characterization of the Besov and Sobolev spaces by means of derivatives and difference quotients.

Tm~0RE~ 1. - Let w be a polynomially regular weight function. Assume tha t s > 0 and let M and • be integers such tha t M -k ~Y > s > M~>0. Then the norm

of B~'~(w) is equivalent to

1

0

where

G q w ; t, g) = sup/ Iw. ( ( ~ - 1)~g)lI~ �9 lhl<t

Here ra is the translation operators.

202 J. L6~s~Oz~: Interpolation o] weighted spaces, etc.

The norm on P~(w) for 1 )~.

Let m be the degree of w. Then lemma 1-3 imply tha t T.(w; o~h) and T.(w; 8h) can be estimated by constants times max (1, Ibis). Thus the L.(w)-norm of

can be estimated by a constant times

max (1, Iht "~) min (1, Iht~2 ~) ~'~llw" (~ , !)ll~ �9

Thus it follows tha t

(4.9) e~('--') o~;~(w; 2 - ' , ~ - ' ! / ~ ) < C.max (1, 2-"~) �9 (]lw" (~, * 1)[I,'2~('-~'-~*~) +

-~- ~ 2 (~-O(8-x) rain (1, 2 -(~-~)z~) s~[lw" (q~b * !)i]~) . I = 1

Here the infinite sum on the right hand side is the convolution of the/x-sequence (2 ~(s-~') rain (1, 2-JM))_~ and the sequence (2~llw.(~j. ])1]~)~, which is a / - sequence

Thus

or equivalently 1

( f , - (4.1o) ~ w ~l ~ (t '~-'o)~(w; t, ]l @ ,~=1 8~]/dx;Y)) ~dt/t) 0

< CB~'~(w; 1).

In order to prove the converse inequality we put

~(~) z~(~)m(~)'(e p (i;~) ~ ) - ' ~ 7 ~

J. LS~'s~rlr Interpolation of weighted spaces, etc. 2o$

d where Zr is infinitely differcntiable such that ~ ~(~)~0(~)= ~(~) and with support

5=1 in {$: t~r 0}. (For the existence of such a function see lemma 6.2.6. in [1].) Then D~k~ M~ for l < p < o o and all v.

Writing

e;,~(~) = (exp ( i2-'$;) - l ) ~

and

z~,~( ) = z~(2 ~) , ,~,~g:) = ~ ( 2 - ~ ) ,

we now have that

d d

J = l i = 1

Thus lemma 2 implies that

d (4.11) I[w.(%, l)[I,<emax(1,2-~ml Lf 2-~%~(w;2-~,~l/~C).

J = l

Consequently

%q I B~ (w;])<C IIw]lI,+

J

\i[q\

which implies the converse of (4.10).

In order to prove the second part of the theorem we rewrite ~:r in the form k , J ~ / where ~(~) = C(1 + [~1~) - ~ .

Iqow

sup s u p l ~ p ~ l D ~ D ~ ( ~ ) l < ~ , ( L > d l 2 ) .

Thus Mihlin's multiplier (see for instance [1], theorem 6.1.6), implies that D ~ e M ~ for a l ly if l < p < c ~ . Thus

Conversely we rewrite J~'] as the sum of / and lj, 5~]/~x~' where j = 1, ..., N and i j ( ~ ) : ~(~)~(~) with

: (1 + + i = l

,~ ~-~ ~j(~)=~(~j) ~ ~ �9

204 J. L6FST~6~r: Interpolation of weighted spaces, etc.

Here 2 is a non-neggtiv% infinitely differentiable function such tha t 2 ( ~ ) = 0

for i~[ < 1 and 2(~) = I for 1~1 > 2. Then Mihlin's multiplier theorem implies tha t v ~ D ~ and D vj belong to M~ for all v and l < p < c ~ . Thus we conclude t h a t

d

llw.gffl!,<C(Ilwlll:+ ~, llw.,~ff/ax~ll~), : [ < p < oo.

The proof is complete. Next we come to our first interpolat ion result.

re~der to [1]. For the nota t ion we refer the

Tm~olcE~ 2. - Le t w be a polynomiMly regular weight function. 0 < 0 < 1 and pu t

s = ( 1 - O)so + Os~, 1/q = (1 - O)/qo + O/q~.

Then we have

Assume tha t

2(,-'0)'j(2(,o-~% ~ , , 1; P~o(w), P~,(w)) < c2"liw(%, 1)I1~ .

Thus

(4.12) (p~o(w), ivY'(w))0,, = B~.'(w), 1 < p , r<oo, 8o~ s~,

~~ l < p , %, ql, r<oo, So:/: s 1 , (4.13) (B~ (w), B~l'ql(w))o,,= B~"(w) ,

(4.14) ~,,,~o,~,,, B~,~,(w))0, = B~,~(w), t~,~ ~ j , l <p, %, ql<c~, qova ql

t)~ooF. - I t suffices to prove (4.12) since the other two formulas follow from the rei terat ion theorem (see theorem 3.5.3 and 3.5.4 in [1]).

In order to prove (4.12) we note thg~t (4.4) implies tha t ] lw(~. ])II~ < c2-~~ Hence

if / ~ 1o + i~. Tuking the infimum of the right hand side we then get

i[w" (v, , t)11~> < 02-"K(2"~ t; -PD~ S>$,(w)).

l~rom this we easily get the inclusion

91 BV(w) ~ (P~0(w), P~ (w))0,, if So~ s~

In order to get the converse inclusion we note tha t

i[w'(d~ */)1[~< C2-~[iw(% * ])i]~, (again by (4.4)) .

3. LSFSTI~6Z~: Interpolation o/ weighted spaces, etc. 265

This easily implies

(P~o(w), P~'(w))0,~ BV(w), i~ Sor s~.

(For fur ther details see [1], pug. 143.) F r o m the proof of theorem 2 one easily gets

COROLLARY ]. - - Under the assumptions of theorem 2 we have

$1 (4.~5) (w~o(w), w~ (w))0,, = B~(w),

if s o and s~ are different non-negat ive integers.

P~ooF. - Fi rs t note thut W~J(w)c 2~J(w) and hence

s~r $1 $i B~ (w) = (P~(w), p; (W))o,~D (w~o(w), w~ (w))0,~.

Now lemma 2 implies

IIwD~(% */) L < ~21~1~[[w(%* /)][~,

and hence

l > ~ t ,

2(~-~0)~J(2 (~o~'~)~, % �9 w~.(w), w~(w)) < c2~*l[w(%. I)[1~.

This implies the converse inclusion.

Rn~A~=. - Le t w be an infinitely differentiable and polynomial ly regular weight function. Then it is eusily seen t ha t the norm on W~(w) is equivalent to

Thus the mapp ing co: ] -+ w] is a bounded linear mapp ing f rom W~(w) onto W~ with u bounded invers. Using theorem 1 und 2 we conclude tha t we huve the fol-

lowing equivalences:

P~(w; ] ) ~ []JS(w/)[]~, r

8q ~ ( B~ (w,/) ~ liV * (w])lll + Z (2s~]1%* (w/)lly) TM

1=1

where s > 0, 1 < p < c~. This r emark leads to our second corollary.

14 - A n n a l t d i M a t e m a l i c a

06 J . L 6 F s ~ 5 ] [ : Interpolation o] weighted spaceS~ etc.

C0n0LLA~Y 2. - Under the assumptions of theorem 2 we have

(4.16) ~~ 1 0 . (p~ (w), P~'(w))~0~: _p~(w),

81 $ P ~ o o F . - I t is known tha t (p~o, P~)[o]-:-P~ if 1 /%'(w)

j : O , 1

id

which implies the corollary.

5. - S o m e interpolat ion results:

Besov spaces are obviously closely connected with the vector-valued sequence

spaces l~(A) defined by the norm

: , a : (a , )~ . Z = 0

B y the definition of the Besov spaces we s imply have

BV(w; / ) = l~(LAw); a)

where a t : ~ * f for l > 0 and a o : ~ , ] . l~ext let us recall the definition of retract. A normed space B is a re t rac t of A

if there are bounded linear mappings 3: B - + A and if: A - > B such t h a t if3 is the

ident i ty on B. (Cf. the proof of corollary 2 section 4.)

Tt{E0~E~[ 3. -- Le t w be a polynomial ly regular weight function. Then B~(w) is a re t rac t of l~(L~(w)) for l <p, q<~oo, s ~R.

3. LGFSTR5~: interpotat~on o/ weighted spaces, etc. 207

I f 1 0 , ( 3 / ) 0 = ~ v * ] , o o

/ = 0

where @o= ~ § q~, q~= ~ § q ~ § q~ and q~= ~ _ ~ § ~ § q~+~ for l>2 . Then ~3 is the ident i ty operator o n B~q(w) and

l~(L,~(w); 3]) -~ B~'q(w; ]).

Since T~(w; @~) is uniformly bounded in l~ (by (4.4)) we also see tha t

c ~ 8 , , ) <

Thus B~'q(w) is a retract of l~(L~(w)). I t is slightly harder to prove the second part of the theorem. Again we can,

however~ immitate the proof given in [1], theorem 6.4.3, for the case w = 1. First we note tha t the Fourier transform of 3f is a sequence defined by multip-

lication of (J~/)^ with the sequence Z = (%~)o~ ~o, where

io(~) = (i + 1~19-'"~(~) �9

Then

and consequently

) since the sum contains at most three terms for each ~. Thus the vector-valued version of Mihlin's multiplier theorem implies t h a t HDVf~llMv<C~ for all v and for 1 < p < c~. Hence 3 is a continuous mapping from P~(w) into L~(w; l~) for i < p < o o .

In order to show tha t ~" is a continuous mapping from L~(w; l~) into P~(w) for

208 J . L S ~ s ~ 5 ~ : Interpolation o] weighted spaces, etc.

1 L~(w). ~ o w

r

/ = 0

where ~(~) is the m app i ng f rom [~ into C defined b y

oo

~(~)@)o Z S ~ ( ~ + ~ ~ l = 0

The norm of D~+~]c(~) is

Again the sum o~ the r ight hand side contains a t mos t three terms. Thus easily

F r o m Mihlin~s mult ipl ier theorem it follows tha t ff maps L~(w, l~) into L~(w) continuously if 1 ~ p < co. This concludes the proof of the theorem.

Tm~o~E)~ 4. - Le t w be a polynomial ly regular weight function. Assume tha t

0 < 0 < 1 and pu t

S ---- ( ! - - O)So -~- OSl

~nd

Then

(5.~)

(5.2)

(5.3)

:L/q = (~ - o)/qo + o/q~ , ~ /p = (1 - o)/po + o/p~

1 - - 0 . 0 W ~ W 0 'tO 1 .

$o~qo 3lJql - - (B~ o (Wo),B~ (wJ)o,o-B;'~(w)

So~qo ~ sl~ql (B~. (~0),B~ (w~,)~0~= B;'~

(P;:(wo), P;:(w~)):0~ = ~%(w),

if p ~ q~po~Pl~ qo~ q l < C ~

if p ~ co~ q ~ co ,

i f l ~ P o ~ P~ ~ co"

P~ooF. - The first two formulas follow f rom theorem 3 and the fact t h a t

81 _ A 1 ) o , ~ ) ~ i f qo ql < co (l;:(Ao), ~o1(A1))o,o-- ~;((Ao,

(l~:(Ao), Z;:(A1))E0~ = ~;((Ao, AJEo~) , if ~ < co

ft. LSFST~6~: Interpolation o] weighted spaces, etc. 209

and that

(Go(~O), G~(w3)o,~ = G ( ~ ) if po, p~ < ~ ,

(Go(~%), G~(w~)):0~ = G(w) if p < co .

In order to prove (5.3), let L~(w, A) denote the space of all A-v~lued functions f such tha t I[](x)]]~eL~(w). Then we have

(5.4) (L,.(wo; Ao) , L~(w~; A~))io~ = L~(w; (Ao, At)co]) , if p < c~.

i n fact, consider the mapping ] -+ ], defined by

f ( z , x) = ~o(X)~-~w~(x)# ( z , x) .

This is an isometry between 5(L~o(Wo, Ao), L~,(w~, A~)) and 5"(L~o(Ao), I%(A,)). Thus (5.4) follows f rom the fact tha t (L~o(Ao) , L,,(A~))coj~- Z~((A0, A~)coj) if p < oo. (See [1], theorem 5.1.2. For the notation see also [1], chapter 4.)

Now we combine (5.4) with theorem 3 to get (5.3). [] From theorem 3 we also get the following result.

COI~OLLAIr165 - Let w be a polynomially reg~llar weight function. Then

B~w) cP;(w)cB~2(w) if 1 < p < 2 ,

For the proof we refer the reader to [1], theorem 6.4.4.

6 . - The homogeneous spaces.

The homogeneous ]~esov spaces /~q(w) are defined by means of the seminorms

c o

�9 s q B~ (w; !) = ~ (2~*ilw(%* IG)~ TM �9 - - c o

Similarly the homogeneous potential spaces w~(w) are defined by

The reason for introducing these spaces in the case w = I is mostly technical. The philosophy then is tha t anything tha t is true for the noR-homogeneous spaces also holds for the homogeneous spaces (and coaversly) but it is often somewhat

210 J. L6Fsm~5~: Interpolation o] weighted spaces, etc.

easier to write down the proofs in the homogeneous case. However in the situation considered here this does not seem to be true. Many of the statements we have proved in this paper for the non-homogeneous spaces can not be proved (by our method at least) for the homogeneous spaces. The reason for this is that the basic estimate (4.4) will be replaced by

(6.1) T~(w; I*q~) < C2" max (1, 2 -z,~) ,

where m is the degree of w. (See lemma 2.) The nice characterization of the Besov spaces in theorem 1 must be replaced by two inequalities, namely

co

~l(f ) (rain (1, t-'~_)t ~'-~ as~(w; t, ($~]l~xf)) q dtlt \~lq <-< CB;q(w; ]) J

0

and c>c*

(f B: (w, !)< c ~:~" (max (1, t")~-"~(w; t, a~Ila~f)) ~ dtit) '~' 0

These estimates easily follow from the proof of theorem 1, which also shows t h a t the norm of /5~'(w) is equivalent to

j = l

The interpolation results given in the previous two sections can not be proved on the basis of (6.1).

Finally we note that one can not get rid of the factor 2 -'~ in (6.1). In fact, put w ( x ) = ( 1 ~ Ixf2) ' ( m = 2 r ) and p = l . Then (by section 2)

~(w; ~,)=fw(x)l~,(~)l dx =fw(2 ,x)l~(x)l dx. R~ R~

Thus

7. - E m b e d d i n g and trace theorems .

The familar Sobolev embedding theorem can be extended to Besov spaces and potential spaces in the weighted case considered here.

Tn:EORE)I 5. - Assume that s~-~ s - - d . ( 1 / p - l/p1) where l~<p < p 1 < c ~ . Then

J. L @ S ~ 5 ~ : Interpolation o/ weighted spaces~ etc. 211

Moreover~ if 1 < p < pl < (x) then

P~(w) c _p~i(w).

Here w is any polynomial ly regular weight function.

PROOF. - I n the case w - ~ 1 the proof depends on Young~s inequal i ty:

(7.x) Ilk �9 i ] l~ ,< Ilkll~lllll~

where 1/pl = Zip -- 1/~', 1 < p < ~'. :Now assume tha t wo is a given weight function such t ha t

wo(x § y) = y_, %(x)y~ , N<~m

where [a~(x)[<A~wo(x). Then (7.1) implies t ha t

Ity kll~'liwj[]~, r = x, 2, . . . .

:Now let N(w) = N(w; k) denote the norm of the mapp ing L~(w) ~ ] ---> k �9 ] ~ L~(w) .

Then we have proved tha t

This es t imate will be used when k = ~ where ~ is defined in the proof of theo-

rem 3. Then the L -norm of y~k can be es t imated by a constant t imes 2(-Ivl+ d/~')t.

Thus

.N(Wo)<~ C2 a~/e' , (for all r = O, 1, 2, . . .) .

B y Stein-Weiss interpolat ion theorem we see t ha t the same es t imate holds for

all non-negat ive real powers of w0. The same est imate is however t rue also for

negat ive powers. I n fact~ we have t ha t

P

k * / ( x ) = | w o ( x - y ) k ( y ) ' w o ( x - y ) - l l ( x - y) dy =

-~ ~, a (x){( - - y)Vk(y)wo(x -- y)- l / (x - - y) dy . J

Thus

[lwo ~. ( k , i ) l l~< c ~ Ily~kIl~llw~-~]ll~.

212 J. L6~smz6)~: Interpolation o/ weighted spaces, etc.

Similarly we deal wi th o ther powers Wo e, r --: 1, 2, ... and then we apply Stien- Weiss interpolat ion theorem. Thus we have t ha t N(w~)~ C2 d~tr for any real s and

for any admi t t ed po lynomia l w. A final in te rpola t ion a rgumen t shows t h a t

N (w ; q~z) < r ,

for any polynomial ly regular weight funct ion w. Thus we have t ha t

where 1/ez = 1 / p ~ - l ip. Similarly ][w(~ �9 ]) l]~< CIIw(YJ * ])][~.

Thus we conclude

r

r I " q\l/q B~,~t~,, �9 < o I]w(~ */) r]~ + ~ (~("~+~~ ~[]w. (v~*,)l~) )

Since s - - s ~ - ~ - d / p ' this proves the first pa r t of the theorem.

The second p a r t is p roved exact ly as in [1], theorem 6.5.1.

Nex t we shall p rove a t race theorem. Le t ws denote a polynomial ly regular

weight funct ion on /7 s. Then we pu t

w~_~(x') = w~(x', 0 ) , x ~ = (Xl~ ..., x~_~).

Clearly w~_~ ~:s polynomial ly regular. Similarly we define the t race operator Tr

b y the formula

( m r / ) ( x ' ) ~ ,' = ~ ( x , 0 ) .

I n the following theorem we let B~q(w~) j = d, d - 1 s tand for weighted Besov spaces on t7 d (with weight wa) and on R a-1 (with weight wa_1) and similarly for

P~(wj). Then we have the following t race theorem.

TttEORE~I 6. - Suppos t t ha t w~ is a polynomial ly regular weight funct ion on R d

~nd t h a t there is a constant e such t h a t

w,,_l(oo') = w<~(oo', O) < cw,~(x', x<,)

for all .X~o Assmne tha t 1 l ip. Then the t race oper- a tor can be extended as a bounded linear mapp ing f rom B~q(wd) into ~.~(~-~i~)q(~,,~d_~,

B(S-~l~)~w ~ tha t is and f rom P~(w~) into , ~ ~_~,

Tr: s<~ B(s- ~l~)<~w B ~ ( w ~ ) ~ ~ ~ ~-1~,

J . LSFST~5~: Interpolation o/ weighted spaces, etc. 213

PROOF. - We just [sketch the proof following [1] ( theorem 6.6.1). For an other

me thod of proof in the case w = I see JAtt-WERTH [3], who also proves t ha t there is gn inverse mapping.

I n order to get the results for s > ! it is sufficient to prove tha t

7:~(m- 1/~)~ [a~ Tr: P~(wa) --> ~ ~'~-11 , m = 1, 2, . . . .

(Just use theorem 2.) Now

= ~ o = ~ / m p 7Q(m--1 /2) )~[~ , , ~ ( 3 0 ( W d _ l ) ' L ~ ( W d _ I ) ) O ~ ' . u ~ ~ d - - 1 ! , "

Using the (( espaces de traces ~) characterizution of interpolat ion spaces one eusily finds t ha t

co

0 t~ e-~ d co

\ ~-~' , l ~ - r ~/" 0 R cz-1

Since w~_~(x')<cwd(x', xd) we conclude t ha t

B(• - l l r ) • r . . . . Tr] ) < CP'~(wa; ]) k~Jd - 1 ,

I t remains to prove the result for l i p < s < l . Le t us write

~(t ; ](x', .)) = sup ll](x', x~ + t~) + ](x', x ~ - h) - - 2/(x')H~. I h l ~ t

Then theorem 5 and 1 give co 1

- -co 0

Since Wd_l(X')<cw~(x', xa) we get 1

0

i.e.

Tr: B~ll~)~(wd) -~ L~(wa_l) .

This implies

n(~-l/~q/~,, ~ for ] > ~ s > l / p .

The second pa r t follows just as in [1], theorem 6.6,1.

214 J . L6FST~6~: Interpolation o] weighted spaces, etc.

REFERENCES

[1] BERGtt- L~FSTR(~M, Interpolation spaces, an introduction, Springer, 1976. [2] P. GRISV~D, Espaces intermediaires entre espaces de Sobolev avec poids, Ann. Scuola Norm.

Sup. Pisa, 52 (1963), pp. 255-296. [3] B. JAWE~TH, The trace of Sobolev and Besov spaces i / 0 < p < 1, Studia Math., 62 (1978),

pp. 65-71. [4] L. H61~AND]~R, Estimabes /or translations, invariant operators in L~ spaces, Acta Math.,

104 (1960), pp. 93-140. [5] H. TRIEBEL, Interpolation theory, Hunction spaces, Di]]erential operators, North-Holland

Publishing Comp., 1978 . [6] H. T~I]~B]~L, Spaces o] distributions with weights. Multipliers in L~-spaces with weights,

Math. Naehr., 78 (1977), pp. 339-355. [7] YOUNG - Wo-SA~G, Weighted norm inequalities ]or multipliers. Par t I : Harmonic analysis

in Euclidian spaces, Proc. Syrup. Pure Math., 50 (1979), pp. 133-139.

interpolation of weighted spaces of differentiable ... · interpolation of weighted spaces of...

Documents