Vector valued multivariate spectral multipliers, Littlewood-Paley

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




4 Multipliers



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.



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 < ∞.



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)



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.


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.


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.


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 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).



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 0 such that


P,k (f )‖Lp(Rn) ≤ C‖f‖Lp(Rn), f ∈ Lp(Rn).



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 0 such that


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.



Suppose B is a Banach space and σ > 0. A first and natural definition of theLittlewood-Paley function on Lp(Rn,B), 1 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 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).


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



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 < ∞.



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.


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



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 .



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 .



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 0.


Uniparametric case.




4 Multipliers



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 ).


Multiparametric case




4 Multipliers



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 0 such that

1C||f ||Lp(Rn) ≤ ||gH

P,k (f )||Lp(Rn) ≤ C||f ||Lp(Rn), f ∈ Lp(Rn).


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).



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 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.



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



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.



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



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).


Multipliers




4 Multipliers



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).


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).


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 < ∞.


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.


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.


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.


Hermite Sobolev spaces




4 Multipliers




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.



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).



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.



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.



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).



Let B be a Banach space and 1 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 0 and k ∈ N is suchthat k > β. Then, Lp

H,β(Rn,B) = F Hk−β,k (Rn,B).



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)).



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)).



“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)

Vector valued multivariate spectral multipliers, Littlewood-Paley

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