vector valued multivariate spectral multipliers, littlewood-paley
TRANSCRIPT
VECTOR VALUED MULTIVARIATE SPECTRAL MULTIPLIERS,LITTLEWOOD-PALEY FUNCTIONS, AND SOBOLEV SPACES
IN THE HERMITE SETTING
Alejandro SanabriaDepartment de Mathematical Analysis
University of La Laguna
Complex Analysis and ApproximationNational University of Ireland, Maynooth
17-19th June 2013
1 IntroductionIntroductionHeat and Poisson semigroupsHermite Littlewood-Paley functionsUMD-Banach spacesγ-radonifying operators.
2 Uniparametric case.Uniparametric Littlewood-Paley function.
3 Multiparametric caseMultiparametric Littlewood-Paley functionsPisier’s (α)-property.Multiparametric Littlewood-Paley functions
4 Multipliers
5 Hermite Sobolev spaces
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 2 / 47
Introduction
1 IntroductionIntroductionHeat and Poisson semigroupsHermite Littlewood-Paley functionsUMD-Banach spacesγ-radonifying operators.
2 Uniparametric case.Uniparametric Littlewood-Paley function.
3 Multiparametric caseMultiparametric Littlewood-Paley functionsPisier’s (α)-property.Multiparametric Littlewood-Paley functions
4 Multipliers
5 Hermite Sobolev spaces
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 3 / 47
Introduction Introduction
♦ jointly with Jorge J. Betancor and Juan C. Fariña (University of La Laguna, Tenerife).
available at arXiv:1304.4018
The Hermite operator (also called harmonic oscillator) H on Rn is defined by
H =−∆ + |x |2,
where ∆ denotes the usual Laplacian operator.
For every m ∈ N we denote by hm the m-th Hermite function given by
hm(u) = (2mm!√
π)−12 Pm(u)e−
u22 , u ∈ R,
where Pm represents the m-th Hermite polynomial.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 4 / 47
Introduction Introduction
If k = (k1, . . . ,kn) ∈ Nn, the k -th Hermite function hk is defined by
hk (x) =n
∏j=1
hkj (xj ), x = (x1, . . . ,xn) ∈ Rn.
We have that, for every k = (k1, . . . ,kn) ∈ Nn,
Hhk = (2|k |+ n)hk ,
where |k |= k1 + · · ·+ kn.
The sequence hkk∈Nn is an orthonormal basis in L2(Rn).
The linear space spanhkk∈Nn generated by hkk∈Nn is dense in Lp(Rn), 1 ≤p < ∞.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 5 / 47
Introduction Introduction
The Hermite operator H is defined by
Hf = ∑k∈Nn
(2|k |+ n)ck (f )hk , f ∈ D(H),
whereD(H) = f ∈ L2(Rn) : ∑
k∈Nn(2|k |+ n)2|ck (f )|2 < ∞
and, for every k ∈ Nn
ck (f ) =∫Rn
hk (x)f (x)dx , f ∈ L2(Rn).
The space C∞c (Rn) of smooth compactly supported functions in Rn is contained in
the domain D(H) of H and Hf = Hf , f ∈ C∞c (Rn)
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 6 / 47
Introduction Introduction
Hermite polynomial setting:
Muckenhoupt (1969): one dimensional setting.
Sjögren (1982), Urbina (1990), Fabes, Gutiérrez and Scotto (1994): harmonic analy-sis operators associated with the Ornstein-Uhlenbech operator in Rn.
In the last decade this topic has been studied by a host of authors: García-Cuerva,Mauceri, Meda, Sjögren, and Torrea (2000), Pérez, and Soria (2000), Harboure,Torrea, and Viviani (2003).
Hermite function setting:
Thangavelu (1993) , K. Stempak and J.L. Torrea (2003): harmonic analysis opera-tors associated with the Hermite operator (Hermite function expansions setting).
Lust Piquard (2006) , Huang (2012) among others.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 7 / 47
Introduction Heat and Poisson semigroups
The heat semigroup W Ht t>0 generated by −H in L2(Rn) is defined by
W Ht (f ) = ∑
k∈Nne−(2|k |+n)t ck (f )hk , f ∈ L2(Rn) and t > 0.
According to Mehler’s formula we can write, for every t > 0 and f ∈ L2(Rn),
W Ht (f )(x) =
∫Rn
W Ht (x ,y)f (y)dy ,
where
W Ht (x ,y) =
1
πn2
(e−2t
1−e−4t
) n2
e− 1
4
(|x−y |2 1+e−2t
1−e−2t +|x+y |2 1−e−2t
1+e−2t
),
where x ,y ∈ Rn and t > 0.
W Ht t>0 is the semigroup of contractions in Lp(Rn), 1≤ p <∞, generated by−H.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 8 / 47
Introduction Heat and Poisson semigroups
The Poisson semigroup PHt t>0 associated with the Hermite operator (generated
by −√
H) in Lp(Rn), 1 ≤ p < ∞, is defined by using the subordination formula asfollows:
For every t > 0,
PHt (f )(x) =
t2√
π
∫∞
0
e−t24u
u32
W Hu (f )(x)du, f ∈ Lp(Rn).
PHt t>0 is a semigroup of contractions in Lp(Rn), 1≤ p < ∞.
The Hermite semigroups W Ht t>0 and PH
t t>0 are not conservative.
W Ht t>0 and PH
t t>0 are not diffusion semigroups (in the sense of Stein).
Hytönen’s results (Revista Iberoaméricana de Matemáticas, ...) about vector valuedLittlewood-Paley functions do not apply in the Hermite setting.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 9 / 47
Introduction Hermite Littlewood-Paley functions
The square function (also called Littlewood-Paley function) gHW ,k associated with
the Hermite semigroup W Ht t>0 is defined by
gHW ,k (f )(x) =
(∫∞
0|tk
∂kt W H
t (f )(x)|2 dtt
) 12
, x ∈ Rn,
for every f ∈ Lp(Rn) and k ∈ N\0.Lp-boundedness properties of the operator gH
W ,k , k ∈ N \ 0, were establishedby Thangavelu (Lectures on Hermite and Laguerre expansions) and Stempak andTorrea (Acta Math. Hungar.).
For every 1 < p < ∞ and k ∈ N\0, there exists C > 0 such that
1C‖f‖Lp(Rn) ≤ ‖gH
W ,k (f )‖Lp(Rn) ≤ C‖f‖Lp(Rn), f ∈ Lp(Rn).
Equivalence above allows to obtain Lp-boundedness for spectral Hermite multi-pliers (Thangavelu - Lectures on Hermite and Laguerre expansions).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 10 / 47
Introduction Hermite Littlewood-Paley functions
The Littlewood-Paley functions associated with the Hermite-Poisson semigroupPH
t t>0 are defined by
gHP,k (f )(x) =
(∫∞
0|tk
∂kt PH
t (f )(x)|2 dtt
) 12
, x ∈ Rn, k ∈ N\0.
For every 1 < p < ∞ and k ∈ N\0, there exists C > 0 such that
1C‖f‖Lp(Rn) ≤ ‖gH
P,k (f )‖Lp(Rn) ≤ C‖f‖Lp(Rn), f ∈ Lp(Rn).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 11 / 47
Introduction Hermite Littlewood-Paley functions
C. Segovia and R. Wheeden (J. Math. Mech.; 19 (1969-70); 247-262) introducedthe notion of fractional derivative:
∂σt f (t,x) =
e−π(m−σ)i
Γ(m−σ)
∫∞
0∂
mt f (t + s,x)sm−σ−1ds, x ∈ Rn and t ∈ (0,∞).
where σ > 0 and m ∈ N\0 such that m−1≤ σ < m.The generalized Littlewood-Paley function associated with the Poisson semigroupfor the Hermite operator is defined by
gHP,σ(f )(x) =
(∫∞
0|tσ
∂σt PH
t (f )(x)|2 dtt
) 12
, σ > 0,
For every 1 < p < ∞ there exists C > 0 such that
1C‖f‖Lp(Rn) ≤ ‖gH
P,σ(f )‖Lp(Rn) ≤ C‖f‖Lp(Rn), f ∈ Lp(Rn), σ > 0.
J.L. Torrea and C. Zhang, Fractional vector-valued Littlewood-Paley-Stein theoryfor semigroups (arXiv: 1105.6022.v3) considered generalized Littlewood-Paley g-functions associated with diffusion semigroups.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 12 / 47
Introduction Hermite Littlewood-Paley functions
Suppose B is a Banach space and σ > 0. A first and natural definition of theLittlewood-Paley function on Lp(Rn,B), 1< p <∞, is the following one. If Lp(Rn,B),1 < p < ∞, we define
GHP,σ,B(f )(x) =
(∫∞
0‖tσ
∂σt PH
t (f )(x)‖2B
dtt
) 12
, σ > 0,
M. Martínez, J.L. Torrea, Q. Xu, C. Zhang, amongst others.
TheoremLet B is a Banach space. The following assertions are equivalent:
i) B is isomorphic to a Hilbert space
ii) For some (equivalently, for any) 1 < p < ∞ and σ > 0, we have that
1C‖f‖Lp(Rn,B) ≤ ‖GH
P,σ,B(f )‖Lp(Rn,B) ≤ C‖f‖Lp(Rn,B), f ∈ Lp(Rn,B).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 13 / 47
Introduction UMD-Banach spaces
Motivated by the ideas in:C. Kaiser, Wavelet transforms for functions with values in Lebesgue spaces, Procee-dings of SPIE Optics and Photonics, Conf. on Mathematical Methods. Wavelets XI5914, 2005.C. Kaiser and L. Weis, Wavelet transform for functions with values in UMD spaces,Studia Math., 186 (2), 101-126, 2008.
Take a different point of view of vector valued Littlewood-Paley functions looking forthe validity of the equivalence
1C‖f‖Lp(Rn,B) ≤ ‖GH
P,σ,B(f )‖Lp(Rn,B) ≤ C‖f‖Lp(Rn,B), f ∈ Lp(Rn,B),
for other Banach spaces that are not Hilbert spaces.
UMD Banach spaces
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 14 / 47
Introduction UMD-Banach spaces
The Hilbert transform of f ∈ Lp(R), 1≤ p < ∞ is defined by
H(f )(x) = limε→0
1π
∫|x−y |>ε
f (y)
y− xdy , a.e. x ∈ R.
TheoremThe operator H is a bounded operator from Lp(R) into itself, for every 1 < p < ∞, andfrom L1(R) into L1,∞(R)
If 1< p <∞ and B is a Banach space, the Hilbert transform is defined on Lp(R)⊗Bin a natural way as the operator H⊗ IdB.
DefinitionA Banach space B is said to be a UMD space when the Hilbert transform H can be ex-tended to the Bochner-Lebesgue space Lp(R,B) as a bounded operator from Lp(R,B)into itself, for some 1 < p < ∞.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 15 / 47
Introduction UMD-Banach spaces
The UMD property does not depend on p.
B is a UMD Banach space if and only if the Hilbert transform can be extended toL1(R,B) as a bounded operator from L1(R,B) into L1,∞(R,B).
There exist a lot of other characterizations for the UMD Banach spaces (Burkholder,Bourgain, Hytönen, Torrea, Xu, Harboure, Viviani, Guerre-Delabrière,...).
ExamplesHilbert spaces,
Lp(Ω;µ;B), 1 < p < ∞ and B is UMD,
Lp1 (Lp2 (...(Lpn )...)) (mixed norm), 1 < pj < ∞, j = 1, . . . ,n,
Lp,q(Ω) (Lorentz spaces), 1 < p,q < ∞,
Cp (Schatten class), 1 < p < ∞.
Hytönen (Revista Iberoaméricana...) used stochastic integration to define Littlewood-Paley functions associated with diffusion semigroups.
Kaiser and Weis studied vector valued square functions defined by convolutions byusing γ-radonifying operators.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 16 / 47
Introduction γ-radonifying operators.
Let gn∞n=1 be a sequence of independent standard Gaussian variables on a pro-
bability space (Ω,P).
TheoremLet H be a separable real Hilbert space with orthonormal basis hn∞
n=1 and let B be areal Banach space. For a bounded linear operator T ∈L(H ,B) the following assertionsare equivalent:
The operator TT ∗ is the covariance of a Gaussian measure m on B;
Almost surely the series ∑∞n=1 gnThn converges in B;
For some 1≤ p < ∞ the series ∑∞n=1 gnThn converges in Lp(Ω,B);
For every 1≤ p < ∞ the series ∑∞n=1 gnThn converges in Lp(Ω,B).
In this situation, for every 1≤ p < ∞, and x = ∑∞n=1 gnThn
∫B‖x‖pdµ(x) = E
∥∥∥∥∥ ∞
∑n=1
gnThn
∥∥∥∥∥p
= supN≥1E
∥∥∥∥∥ N
∑n=1
gnThn
∥∥∥∥∥p
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 17 / 47
Introduction γ-radonifying operators.
DefinitionA bounded operator T ∈ L(H ,B) is called γ-radonifying if it satisfies the equivalentconditions of the theorem. For such an operator we define
‖T‖γ =
(∫B‖x‖2dµ(x)
) 12
=
E
∥∥∥∥∥ ∞
∑n=1
gnThn
∥∥∥∥∥2 1
2
Theorem(γ(H ,B),‖ · ‖γ) is a real Banach space, which is separable provided that B isseparable.
If T ∈ L(H ,B) is γ-radonifying, then T is compact.
If B is a real Hilbert space, then T ∈ L(H ,B) is γ-radonifying if and only if T isHilbert-Schmidt.
γ(H ,C) = H .
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 18 / 47
Introduction γ-radonifying operators.
Suppose thatH = L2(W ,µ) where (W ,B,µ) is a σ-finite measure space with a countably genera-ted σ-algebra B and thatf : W → B is a strongly measurable function such that f is weakly H , that is, for everyS ∈ B∗, the dual space of B, the function S f ∈H .
Then there exists Tf ∈ L(H ,B) satisfying that
〈S,Tf (ϕ)〉B∗,B =∫
W〈S, f (w)〉B∗,Bϕ(w)dµ(w), ϕ ∈H .
where 〈·, ·〉B∗,B denotes the (B∗,B) duality.
We say that f ∈ γ(W ,µ;B) when Tf ∈ γ(H ;B) and then we define
‖f‖γ(W ,µ;B) = ‖Tf‖γ(H ;B)
It is usual to identify f with Tf .
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 19 / 47
Introduction γ-radonifying operators.
C. Kaiser and L. Weis, Suppose that ψ∈ L2(Rn). We consider ψt (x) = 1tn ψ(
xt
), x ∈
Rn and t > 0. The wavelet transform Wψ associated with ψ is defined by
Wψ(f )(x , t) = (f ∗ψt )(x), x ∈ Rn and t > 0,
where f ∈ S(Rn,B), the B-valued Schwartz space.
Kaiser and Weis gave sufficient conditions for ψ in order to
‖Wψf‖E(Rn,γ(H ,B)) ∼ ‖f‖E(Rn;B),
for every f ∈ E(Rn;B), where B is a UMD Banach space and E represents Lp,1 < p < ∞, H1 or BMO.
Taking ψ(x) = ∂t Pt (x)|t=1, x ∈ Rn, we have that
Wψ(f )(x , t) = t∂t Pt (f )(x), x ∈ Rn and t > 0.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 20 / 47
Uniparametric case.
1 IntroductionIntroductionHeat and Poisson semigroupsHermite Littlewood-Paley functionsUMD-Banach spacesγ-radonifying operators.
2 Uniparametric case.Uniparametric Littlewood-Paley function.
3 Multiparametric caseMultiparametric Littlewood-Paley functionsPisier’s (α)-property.Multiparametric Littlewood-Paley functions
4 Multipliers
5 Hermite Sobolev spaces
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 21 / 47
Uniparametric case. Uniparametric Littlewood-Paley function.
J. Betancor, A. Castro, J. Curbelo, J. Fariña, and L. Rodríguez-Mesa. J. Funct.Anal., 263 (12): 3804-3856, 2012.The univariate fractional g-function operator associated with the Hermite-Poissonsemigroup is defined by
GHP,σ;B(f )(t,x) =
∫R
tσ∂
σt PH
t (x ,y)f (y)dy , σ > 0, x ∈ R and t > 0,
for every f ∈ Lp(Rn,B).
TheoremLet H = L2
((0,∞), dt
t
), B be a UMD Banach space and σ > 0. Then, the operator
GHP,σ,B is bounded
from Lp(Rn,B) into Lp(Rn,γ(H ,B)), for every 1 < p < ∞,
from L1(Rn,B) into L1,∞(Rn,γ(H ,B)), and
from H1(Rn,B) into L1(Rn,γ(H ,B)).
Moreover, for every 1 < p < ∞, ‖f‖Lp(Rn,B) ∼ ‖GHP,σ,B(f )‖Lp(Rn,γ(H ,B)), f ∈ Lp(Rn,B).
Note that, since γ(H ,C) = H , if f ∈ Lp(Rn), 1 < p < ∞, then
‖GHP,σ,C(f )‖γ(H ,C) = gH
P,σ(f ).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 22 / 47
Multiparametric case
1 IntroductionIntroductionHeat and Poisson semigroupsHermite Littlewood-Paley functionsUMD-Banach spacesγ-radonifying operators.
2 Uniparametric case.Uniparametric Littlewood-Paley function.
3 Multiparametric caseMultiparametric Littlewood-Paley functionsPisier’s (α)-property.Multiparametric Littlewood-Paley functions
4 Multipliers
5 Hermite Sobolev spaces
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 23 / 47
Multiparametric case Multiparametric Littlewood-Paley functions
Let k = (k1, . . . ,kn) ∈ (N \ 0)n. We define the multivariate square function gHP,k
as follows
gHP,k (f )(x) =
∫(0,∞)n
∣∣∣∣∣∫Rn
n
∏j=1
tkjj ∂
kjtj PH
tj (xj ,yj )f (y1, . . . ,yn)dy
∣∣∣∣∣2
dt1 · · ·dtnt1 · · · tn
12
,
where x ∈ Rn, for every f ∈ Lp(Rn), 1 < p < ∞.
By proceeding as in the proof of Theorem 2.4 by Wróbel. (Monatsh. Math., 168 (1):125-149, 2012) we can show the following property.
TheoremLet 1 < p < ∞ and k = (k1, . . . ,kn) ∈ (N\0)n. Then, there exists C > 0 such that
1C||f ||Lp(Rn) ≤ ||gH
P,k (f )||Lp(Rn) ≤ C||f ||Lp(Rn), f ∈ Lp(Rn).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 24 / 47
Multiparametric case Pisier’s (α)-property.
If εj∞j=1 is a sequence of independent symmetric ±1-valued random variables
(usually called Rademacher variables) on some probability space, we denote byEε the corresponding expectation operator.
Suppose that εj∞j=1 and ηj∞
j=1 are two independent sequences of Rademachervariables. We say that a Banach space B has (Pisier’s) property (α) when thereexists C > 0 such that
EεEη
∥∥∥ N
∑i,j=1
αi,jεiηjxi,j
∥∥∥B≤ EεEη
∥∥∥ N
∑i,j=1
εiηjxi,j
∥∥∥B,
for every αi,j ∈ +1,−1, xi,j ∈ B, i, j = 1, . . . ,N, and N ∈ N\0.UMD and (α) properties of Banach spaces are crucial in order to prove Banachvalued Fourier multipliers of Mikhlin type.
ExamplesHilbert spaces.
Lp(Ω,µ,B) if B has property-(α).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 25 / 47
Multiparametric case Multiparametric Littlewood-Paley functions
Let H n = L2(
(0,∞)n, dt1···dtnt1···tn
). If k = (k1, . . . ,kn) ∈ (N \ 0)n, we consider the
g-function associated with the Hermite operator defined by
GHP,k ;B(f )(t,x) =
∫Rn
n
∏j=1
tkjj ∂
kjtj PH
tj (xj ,yj )f (y1, . . . ,yn)dy1 . . .dyn,
for every f ∈ Lp(Rn,B), 1 < p < ∞, where t = (t1, . . . , tn) ∈ (0,∞)n andx = (x1, . . . ,xn) ∈ Rn.
TheoremLet B be a UMD Banach space with the property (α), k ∈ (N \ 0)n and 1 < p < ∞.Then, there exists C > 0 such that
1C||f ||Lp(Rn,B) ≤ ||GH
P,k ;B(f )||Lp(Rn,γ(H n,B)) ≤ C||f ||Lp(Rn,B), f ∈ Lp(Rn,B).
Note that γ(H n,C) = H n.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 26 / 47
Multiparametric case Multiparametric Littlewood-Paley functions
Proof.
We proceed by induction on n. (We have seen that is true when n = 1)
Assume that f ∈ Lp(R)⊗ Lp(Rn)⊗B, that is f = ∑ki=1 givibi , where gi ∈ Lp(R),
vi ∈ Lp(Rn) and bi ∈ B, i = 1, . . . ,k ∈ N\0.Lp(R)⊗Lp(Rn)⊗B is dense in Lp(Rn+1,B).
GHP,β;B(f )(t,x) =
k
∑i=1
GHP,β1;C(gi )(t1,x1)GH
P,β;C(vi )(t, x)bi ,
where t = (t1, t) = (t1, t2, . . . , tn+1) ∈ (0,∞)n+1, x = (x1, x) = (x1,x2, . . . ,xn+1) ∈Rn+1, and β = (β2, . . . ,βn+1) ∈ (N\0)n.
GHP,β;B(f )(t, ·) is strongly measurable
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 27 / 47
Multiparametric case Multiparametric Littlewood-Paley functions
Since B has the property (α) we have that γ(H n+1,B)' γ(H 1,γ(H n,B)) (γ-Fubiniproperty - Van Neerven, Weis). By using the induction hypothesis and taking intoaccount that γ(H l ,B), l ∈ N\0, is UMD with the property (α) we obtain
‖GHP,β;B(f )‖Lp(Rn+1,γ(H n+1,B))
=
(∫Rn+1‖GH
P,β;B(f )(·,x)‖pγ(H n+1,B)dx
) 1p
=
(∫Rn+1
∥∥∥∥∫R tβ11 ∂
β1t1 PH
t1 (x1,y1)GHP,β;B(f )(t, x)dy1
∥∥∥∥p
γ(H 1,γ(H n,B))dx
) 1p
≤ C
(∫R
∫Rn‖GH
P,β;B(f (x1, y))(t, x)‖pγ(H n,B)dxdx1
) 1p
≤ C‖f‖Lp(Rn+1)
The operator GHP,β;B can be extended to Lp(Rn+1,B) as a bounded operator from
Lp(Rn+1,B) into Lp(Rn+1,γ(H n+1,B)). This extension operator is denoted by GHP,β;B.
We show that, for every f ∈ Lp(Rn+1,B),
GHP,β;B(f )(x) = GH
P,β;B(f )(·,x), a.e. x ∈ Rn+1.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 28 / 47
Multiparametric case Multiparametric Littlewood-Paley functions
The induction is completed and we prove that the operator GHP,β;B is bounded from
Lp(Rn+1,B) into Lp(Rn+1,γ(H n+1,B)), for every β ∈ (N\0)n+1 and n ∈ N.
Our next objective is to see that there exists C > 0 such that
‖f‖Lp(Rn+1,B) ≤ C‖GHP,α;B‖Lp(Rn+1,γ(H n+1,B)), f ∈ Lp(Rn+1,B).
By using standard spectral arguments we can show that, for every f ∈ Lp(Rn+1)⊗B and g ∈ Lp′(Rn+1)⊗B∗, where p′ is the conjugated of p, that is p′ = p
p−1 ,∫Rn+1
∫(0,∞)n+1
〈GHP,α;B∗(g)(t,x),GH
P,α;B(f )(t,x)〉B∗,Bdt1 · · ·dtn+1
t1 · · · tn+1dx
=n+1
∏j=1
Γ(2αj )
22αj
∫Rn+1〈g(x), f (x)〉B∗,Bdx
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 29 / 47
Multiparametric case Multiparametric Littlewood-Paley functions
Let f ∈ Lp(Rn+1)⊗B. We have that
||f ||Lp(Rn+1,B) = supg∈Lp′ (Rn+1)⊗B∗||g||
Lp′ (Rn+1 ,B∗)≤1
∣∣∣∣∫Rn+1〈g(x), f (y)〉B∗,Bdy
∣∣∣∣≤ C sup
g∈Lp′ (Rn+1)⊗B∗||g||
Lp′ (Rn+1 ,B∗)≤1
∣∣∣∣∫Rn+1
∫(0,∞)n+1
〈GHP,α;B∗(g)(t,x),G
HP,α;B(f )(t,x)〉B∗,B
dt1···dtn+1t1···tn+1
dx
∣∣∣∣≤ C sup
g∈Lp′ (Rn+1)⊗B∗||g||
Lp′ (Rn+1 ,B∗)≤1
∫Rn+1‖GH
P,α;B∗(g)(·,x)‖γ(H n+1,B∗)‖GHP,α;B(f )(·,x)‖γ(H n+1,B)dx .
Since B∗ is UMD and it has the property (α), GHP,α;B∗ is bounded from Lp′(Rn+1,B∗)
into Lp′(Rn+1,γ(H n+1,B∗)), and by using Hölder’s inequality we get
‖f‖Lp(Rn+1,B) ≤ C‖GHP,α;B(f )‖Lp(Rn+1,γ(H n+1,B))
Since Lp(Rn+1)⊗B is a dense subspace of Lp(Rn+1,B) and GHP,α;B is a bounded
operator from Lp(Rn+1,B) into Lp(Rn+1,γ(H n+1,B)) we conclude that
‖f‖Lp(Rn+1,B) ≤ C‖GHP,α;B(f )‖Lp(Rn+1,γ(H n+1,B)), f ∈ Lp(Rn+1,B).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 30 / 47
Multipliers
1 IntroductionIntroductionHeat and Poisson semigroupsHermite Littlewood-Paley functionsUMD-Banach spacesγ-radonifying operators.
2 Uniparametric case.Uniparametric Littlewood-Paley function.
3 Multiparametric caseMultiparametric Littlewood-Paley functionsPisier’s (α)-property.Multiparametric Littlewood-Paley functions
4 Multipliers
5 Hermite Sobolev spaces
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 31 / 47
Multipliers
Suppose now that m is a bounded Borel measurable function from (0,∞)n into C.The Hermite multivariate multiplier Tm associated with m is defined by
Tmf = ∑k=(k1,...,kn)∈Nn
m(λk1 , · · · ,λkn )ck (f )hk , f ∈ L2(Rn),
Tm is a bounded operator from L2(Rn) into itself.
Thangavelu (Theorem 4.2.1, Lectures on Hermite and Laguerre operators, Prince-ton Univ. Press, Princeton, N.J., 1993) established Mikhlin-Hörmander type condi-tion on m under that Tm is bounded from Lp(Rn) into itself, for every 1 < p < ∞.
We define the operator Tm⊗ IdB on L2(Rn)⊗B in the usual way.
We are motivated by the results of Meda (Proc. Amer. Math. Soc., 110 (3), (1990),639-647) and Wróbel (Monatsh. Math., 168 (1) (2012), 125-149).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 32 / 47
Multipliers
Notation:
For every α = (α1, . . . ,αn) ∈ Nn, we consider
mα(t,λ) =n
∏j=1
(tjλj )αj e−
tj λj2 M(λ),
where t = (t1, . . . , tn) ∈ (0,∞)n, λ = (λ1, . . . ,λn) ∈ Rn and M(λ) = m(λ21, . . . ,λ
2n),
λ = (λ1, . . . ,λn) ∈ Rn.
We define
Mα(t,u) =∫(0,∞)n
n
∏j=1
λ−iuj−1j mα(t,λ)dλ,
with t = (t1, . . . , tn) ∈ (0,∞)n and u = (u1, . . . ,un) ∈ Rn.
Let β = (β1, . . . ,βn) ∈ Nn. We write Liβ to refer us to the operator Tmβwhere
mβ(λ1, . . . ,λn) = ∏nj=1 λ
iβj , λ = (λ1, . . . ,λn) ∈ (0,∞)n, that is,
Liβf =∞
∑k1,...,kn=0
n
∏j=1
λiβjkj
ck (f )hk , f ∈ L2(Rn).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 33 / 47
Multipliers
The operator Liβ⊗ IdB is defined in the usual way on L2(Rn)⊗B.
Liβ⊗ IdB can be extended to Lp(Rn,B) as a bounded operator from Lp(Rn,B) intoitself, for every 1 < p < ∞, provided that B is a UMD Banach space. (Betancor,Castro, Curbelo, Fariña, and Rodríguez-Mesa, Complex Anal. Op. Th., 2011).
TheoremLet B be a UMD Banach space with the property (α) and 1 < p < ∞. Suppose that m isa bounded Borel measurable function on (0,∞)n, such that for some γ ∈ Nn,∫
Rnsup
t∈(0,∞)n|Mγ(t,u)|‖Li u
2 ‖Lp(Rn,B)→Lp(Rn,B)du < ∞.
Then, the multiplier operator Tm is bounded from Lp(Rn,B) into itself.
We need to use intermediate UMD spaces in the following theorem to get a suitableestimate for the operator norm ‖Liγ‖Lp(Rn,B)→Lp(Rn,B), γ ∈ Nn and 1 < p < ∞.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 34 / 47
Multipliers
We consider a class of UMD Banach spaces, called intermediate UMD spaces,that includes all known examples of UMD spaces.
We say that B is an intermediate UMD space when B is isomorphic to a closedsubquotient of a complex interpolation space [X ,Q ]θ, where θ ∈ (0,1), X is aUMD Banach space and Q is a Hilbert space.
This class of UMD spaces has been used recently by Berkson and Gillespie, Hytö-nen, Hytönen and Lacey, and Taggart.
It is, as far as we know, an open problem if every UMD space is an intermediateUMD space. This question was posed by Rubio de Francia.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 35 / 47
Multipliers
For every ψ ∈ (0,π) we denote by Γψ the Cn-sector
Γψ = (z1, . . . ,zn) ∈ Cn : |Arg zj |< ψ, j = 1, . . . ,n.
TheoremSuppose that B is isomorphic to a closed subquotient of [X ,Q ]θ, where θ ∈ (0,1), X isa UMD Banach space and Q is a Hilbert space and that B has the property (α). If m isa bounded holomorphic function in Γψ for some ψ > π
4 , then the multiplier operator Tmcan be extended to Lp(Rn,B) as a bounded operator from Lp(Rn,B) into itself, provided
that∣∣∣ 2
p −1∣∣∣< θ.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 36 / 47
Multipliers
Proof
Assume that B is isomorphic to a closed subquotient of a complex interpolation
space [H ,X ]θ. If∣∣∣ 2
p −1∣∣∣< θ and 0≤ Ω < π
2 (1−θ), there exists ω < π
2 −Ω suchthat
‖H iα‖Lq(R,B)→Lq(R,B) ≤ Ceω|α|, α ∈ R,Hence, if u = (u1, . . . ,un) ∈ Rn we conclude that
‖Liu‖Lq(R,B) ≤ Ceω∑ni=1 |ui |,
where C does not depend on u.Working coordinate to coordinate we can obtain that
supt∈(0,∞)n
|Mγ(t,u)| ≤ Cn
∏j=1
(1 + |uj |)|Γ(γj − iuj )|, u = (u1, . . . ,un) ∈ Rn.
Then, if∣∣∣ 2
p −1∣∣∣< θ it follows that∫
Rnsup
t∈(0,∞)n|Mγ(t,u)|‖Li u
2 ‖Lp(Rn,B)→Lp(Rn,B)du < ∞.
We deduce that Tm is bounded from Lp(Rn,B) into itself provided that∣∣∣ 2
p −1∣∣∣< θ.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 37 / 47
Hermite Sobolev spaces
1 IntroductionIntroductionHeat and Poisson semigroupsHermite Littlewood-Paley functionsUMD-Banach spacesγ-radonifying operators.
2 Uniparametric case.Uniparametric Littlewood-Paley function.
3 Multiparametric caseMultiparametric Littlewood-Paley functionsPisier’s (α)-property.Multiparametric Littlewood-Paley functions
4 Multipliers
5 Hermite Sobolev spaces
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 38 / 47
Hermite Sobolev spaces
Bongioanni and Torrea (Proc. Indian Acad. Sci. Math. Sci., 116 (3) (2006), 337-360,and Studia Math., 192 (2) (2009), 147-172) studied Sobolev spaces in the Hermitescalar setting.
The Hermite operator H admits the following factorization
H =12
n
∑j=1
(AjA−j + A−jAj ),
where Aj = ddxj
+ xj and A−j =− ddxj
+ xj , j = 1, . . . ,n (creation and the annihilationoperators).
The creation operator and the annihilation operator play an important role in theharmonic analysis for the Hermite operator.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 39 / 47
Hermite Sobolev spaces
Let B be a Banach space, 1 < p < ∞ and ` ∈ N\0.Hermite Sobolev space W p
H,`(Rn,B). Let ji ∈ Z, 1≤ |ji | ≤ n, 1≤ i ≤m ≤ `
W pH,`(R
n,B) = f ∈ Lp(Rn,B) : Aj1 · · ·Ajm f ∈ Lp(Rn,B).
‖f‖W pH,`(Rn,B) = ‖f‖Lp(Rn,B) + ∑
ji∈Z, 1≤|ji |≤ni=1,...,m≤`
‖Aj1 · · ·Ajm f‖Lp(Rn,B).
Hermite Sobolev space W pH,`(R
n,B). Let ji ∈ N, 1≤ ji ≤ n, 1≤ i ≤m ≤ `
W pH,`(R
n,B) = f ∈ Lp(Rn,B) : A−j1 · · ·A−jm f ∈ Lp(Rn,B).
‖f‖W pH,`(Rn,B) = ‖f‖Lp(Rn,B) + ∑
ji∈N, 1≤ji≤ni=1,...,m≤`
‖A−j1 · · ·A−jm f‖Lp(Rn,B).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 40 / 47
Hermite Sobolev spaces
Let β > 0. The −β-power H−β of the Hermite operator is defined by
H−βf = ∑k∈Nn
ck (f )
(2|k |+ n)βhk , f ∈ Lp(Rn,B), 1 < p < ∞.
H−β is bounded, positive and one to one from Lp(Rn,B) into itself, for every 1 <p < ∞.The potential space
LpH,β(Rn,B) = f ∈ Lp(Rn,B) : f = H−βg, for some g ∈ Lp(Rn,B).
‖f‖LpH,β
(Rn,B) = ‖g‖Lp(Rn,B)
TheoremSuppose that B is isomorphic to a closed subquotient of [X ,Q ]θ where θ ∈ (0,1), X isa UMD Banach space, and Q is a Hilbert space, and that B has the property (α). Then,for every ` ∈ N\0,
W pH,`(R
n,B) = W pH,`(R
n,B) = LpH,`(R
n,B), provided that
∣∣∣∣2p −1
∣∣∣∣< θ.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 41 / 47
Hermite Sobolev spaces
Proof
The Hermite Riesz transform Rm,j is defined as follows
Rm,j f = Am11 · · ·A
mjj A
mj+1−(j+1) · · ·A
mn−nH−
|m|2 f , f ∈ FH .
where m = ∑nj=1 mj . FH denotes the linear space spanhkk∈Nn .
PropositionSuppose that B is isomorphic to a closed subquotient of [X ,Q ]θ, where θ ∈ (0,1), X isa UMD Banach space and Q is a Hilbert space and that B has the property (α) and 1 <
p <∞ being∣∣∣ 2
p −1∣∣∣< θ. If m = (m1, . . . ,mn)∈Nn and j = (j1, . . . , jn)∈Zn, being |ji |= i ,
i = 1, . . . ,n, the Hermite Riesz transform R jm defined by R j
mf = ∏ni=1 Ami
ji H−|m|2 f , f ∈
FH , where |m| = ∑ni=1 mi , can be extended to Lp(Rn,B) as a bounded operator from
Lp(Rn,B) into itself.
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 42 / 47
Hermite Sobolev spaces
If j ∈ Z, 1≤ |j| ≤ n, we consider the Hermite Riesz transform R`j by
R`j f = A`
j H− `
2 f , f ∈ FH ,
and we define the operator τ` = ∑nj=1 R`
j R`−j .
τ` is bounded from Lp(Rn,B) into itself.We consider the function
m(z1, . . . ,zn) =(z1 + · · ·+ zn + 2)
`2 (z1 + · · ·+ zn)
`2
∑nj=1 ∏
`m=1(
zj2 + m− 1
2 ), z = (z1, . . . ,zn) ∈ Γ π
2.
The multiplier Tm associated with m is bounded from Lp(Rn,B) into itself.Hence, we obtain
‖g‖Lp(Rn,B) ≤ ‖Tmτ`g‖Lp(Rn,B) ≤ Cn
∑j=1‖A`−jH
− `2 g‖Lp(Rn,B), g ∈ FH .
Then,‖f‖Lp
H,`(Rn,B) ≤ C‖f‖W pH,`(Rn,B), f ∈ FH .
We conclude that W pH,`(R
n,B) is continuously contained in LpH,`(R
n,B).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 43 / 47
Hermite Sobolev spaces
Let B be a Banach space and 1 < p < ∞. Suppose that β > 0 and k ∈N\0. Weconsider the Littlewood-Paley operator GH
P,β,k ;B defined by
GHP,β,k ;B(f )(t,x) = tβ
∂kt PH
t (f )(x , t), x ∈ Rn and t > 0,
for every f ∈ Lp(Rn,B).
F Hβ,k (Rn,B) = f ∈ Lp(Rn,B) : GH
P,β,k ;B(f ) ∈ Lp(Rn,γ(H n,B)).
‖f‖FHβ,k (R
n,B) = ‖f‖Lp(Rn,B) +‖GHP,β,k ;B(f )‖Lp(Rn,γ(H n,B)).
Note that the space F Hβ,k (Rn,B) can be considered as a Triebel-Lizorkin type space
in the Hermite setting.
TheoremLet B be a UMD Banach space and 1 < p < ∞. Suppose that β > 0 and k ∈ N is suchthat k > β. Then, Lp
H,β(Rn,B) = F Hk−β,k (Rn,B).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 44 / 47
Hermite Sobolev spaces
Proof
We have thatGH
P,k−β;B(f ) = GHP,β,k ;B(H−
β
2 f ), f ∈ FH ⊗B.
Then it follows that1C‖f‖Lp
H,β
2
(Rn,B) ≤ ‖GHP,β,k ;B(f )‖Lp(Rn,γ(H 1,B)) ≤ C‖f‖Lp
H,β
2
(Rn,B), f ∈ FH ⊗B.
Suppose now that f ∈ Lp(Rn,B) and GHP,β,k ;B(f )∈ Lp(Rn,γ(H 1,B)). We are going
to see that f ∈ Lp
H, β
2
(Rn,B). Let δ > 0. We define
Fδ = ∑k∈Nn
(2|k |+ n)β
2 e−δ(2|k |+n)12 cH
k (f )hk .
Fδ ∈ Lp(Rn,B). Since H−β
2 Fδ = PHδ
(f ) ∈ Lp(Rn,H ), PHδ
(f ) ∈ Lp
H, β
2
(Rn,B) and
‖PHδ
(f )‖Lp
H,β
2
(Rn,B) = ‖Fδ‖Lp(Rn,B). Hence
‖Fδ‖Lp(Rn,B) ≤ C‖GHP,β,k ;B(PH
δ(f ))‖Lp(Rn,γ(H 1,B)).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 45 / 47
Hermite Sobolev spaces
The set Fδδ>0 is bounded in Lp(Rn,B).
By Banach-Alaoglu’s Theorem there exists a decreasing sequence δkk∈N\0 ⊂(0,∞) and F ∈ Lp(Rn,B) such that δk → 0, as k → ∞,∫
Rn〈g(x),Fδk (x)〉B∗,Bdx →
∫Rn〈g(x),F(x)〉B∗,Bdx , as k → ∞,
for every g ∈ Lp′(Rn,B∗), and ‖F‖Lp(Rn,B) ≤ C‖GHP,β,k ;B(f )‖Lp(Rn,γ(H 1,B)).
Taking into account that H−β
2 is a bounded operator from Lp(Rn,B) into itself, weget, for every g ∈ Lp′(Rn,B∗),∫
Rn〈g(x),PH
δk(f )(x)〉B∗,Bdx →
∫Rn〈g(x),H−
β
2 (F)(x)〉B∗,Bdx , as k → ∞.
Since PHδk
(f )→ f , as k → ∞, in Lp(Rn,B), we conclude that f = H−β
2 (F). Hence,
f ∈ Lp
H, β
2
(Rn,B) and
‖f‖Lp
H,β
2
(Rn,B) ≤ C‖GHP,β,k ;B(f )‖Lp(Rn,γ(H 1,B)).
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 46 / 47
Hermite Sobolev spaces
“Caminante no hay camino"
Caminante, son tus huellasel camino y nada más;
Caminante, no hay camino,se hace camino al andar.
Al andar se hace el camino,y al volver la vista atrás
se ve la senda que nuncase ha de volver a pisar.
Caminante no hay caminosino estelas en la mar.
Antonio Machado (1875 – 1939)
Alejandro Sanabria () Vector valued multivariate spectral multipliers 13CAA Maynooth 47 / 47