Properties of the Linear multiplier operator for the Weinstein transform ...

Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 124, pp. 1–18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

PROPERTIES OF THE LINEAR MULTIPLIER OPERATOR FOR THE WEINSTEIN TRANSFORM AND APPLICATIONS ABDESSALEM GASMI, ANIS EL GARNA Communicated by Vicentiu Radulescu

Abstract. In this article, we use the theory of reproducing kernels to study the Weinstein multiplier operators on Sobolev type spaces. Some applications are given and an associated H¨ ormander type theorem on Lp -boundedness is established.

1. Introduction d+1 = In this article, we consider the Weinstein operator ∆α,d W defined on R+ d R ×]0, +∞[, by

∆α,d W =

d+1 X ∂2 2α + 1 ∂ + = ∆d + Lα , 2 ∂xi xd+1 ∂xd+1 i=1

1 α>− , 2

(1.1)

where ∆d is the Laplacian for the d first variables and Lα is the Bessel operator for the last variable defined on ]0, +∞[ by Lα u =

∂2u ∂  2α+1 ∂u  2α + 1 ∂u 1 x . + = 2α+1 2 ∂xd+1 xd+1 ∂xd+1 xd+1 ∂xd+1 d+1 ∂xd+1

The Weinstein operator ∆α,d W , mostly referred to as the Laplace-Bessel differential operator is now known as an important operator in analysis, because of its applications in pure and applied Mathematics, especially in Fluid Mechanics [10]. The relevant harmonic analysis associated with the Bessel differential operator Lα goes back to Bochner, Delsarte, Levitan and has been studied by many other authors such as L¨ ofstr¨ om and Peetre [19], Kipriyanov [17], Stempak [28], Trim`eche [29], Aliev and Rubin [1]. The Weinstein transform generalizing the usual Fourier transform, is given for d+1 f ∈ L1α (Rd+1 + ) and λ ∈ R+ , by Z α,d FW (f )(λ) = f (x)Λα,d (x, λ)dµα,d (x), Rd+1 +

2010 Mathematics Subject Classification. 11F60, 42B25, 35S30, 42A38. Key words and phrases. Linear multiplier operator; Weinstein operator; extremal function; H¨ ormander multiplier theorem. c

2017 Texas State University. Submitted January 10, 2017. Published May 5, 2017. 1

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where dµβ,d (x) =

x2β+1 d+1 dx and Λα,d is given later by (2.2). (2π)d/2 2β Γ(β + 1)

In this article, we deal with the theory of multiplier operators in the Weinstein s settings. For s ∈ R, we consider the Sobolev type spaces Hα,β , when α ≥ β > −1/2, 0 d+1 consisting of all f ∈ S∗ (R ), (the space of tempered distributions, even with α,d respect to the last variable), such that FW (f ) is a function and α,d (1 + |z|2 )s/2 FW (f ) ∈ L2 (dµβ,d ). s The space Hα,β is an Hilbert space when endowed with the inner product Z α,d α,d s hf, giHα,β = FW (f )(z)FW (g)(z)dµsβ,d (z), Rd+1 +

and it is continuously embedded in L2 (dµα,d ), when s ≥ α − β, where dµsβ,d (z) = (1 + |z|2 )s dµβ,d (z). For m ∈ L∞ (dµβ,d ), we define the Weinstein multiplier operators Tα,β,m , for s f ∈ Hα,β , by β,d −1 α,d Tα,β,m f (x) = (FW ) (mFW (f ))(x),

x ∈ Rd+1 + .

These operators are a generalization of the usual linear multiplier operator Tm associated with a bounded function m and given by Tm (f ) = F −1 (mF(f )), where F(f ) denotes the ordinary Fourier transform on Rn . These operators gained the interest of many Mathematicians and they were generalized in many settings, (see for instance [3, 9, 15, 14, 24]). For m ∈ L∞ (dµβ,d ) the operator Tα,β,m is shown to be a bounded operator from 0 0 Hα,β onto L2 (dµβ,d ), and for f ∈ Hα,β we have kTα,β,m f kL2 (dµβ,d ) ≤ kmkL∞ (dµβ,d ) kf kH0α,β . α,d Furthermore, if f is -concentrated on E and FW (f ) is ν-concentrated on S, where d+1 E and S are two measurable subsets on R+ . Using Donoho-Stark uncertainty principle for the Weinstein transform, we obtain the following estimation cα,d (µα,d (E))1/2 (µβ,d (S))1/2 ≥ (1 − ν − ), cβ,d

where cα,d is the constant given by cα,d =

1 . (2π)d/2 2α Γ(α + 1)

(1.2)

As in [20, 26, 30], the theory of reproducing kernels is used to give the best s approximation of the operator Tα,β,m on the Sobolev-Weinstein spaces Hα,β . More 2 precisely, for all η > 0 and g ∈ L (dµβ,d ), we show that there exists a unique ∗ function fηg , where the infimum inf {ηkf k2Hsα,β + kg − Tα,β,m f k2L2 (µβ,d ) }

f ∈Hsα,β

is attained. ∗ The function fηg is called the extremal function and it is given by ∗ fηg = hg, Tα,β,m (Ks (·, y))iL2 (dµβ,d ) ,

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3

s where Ks is the reproducing kernel of the space (Hα,β , h·, ·iη,Hsα,β ). When α = β, as in [2], we develop the original H¨ormander’s technique to establish an analogous of the well-known H¨ormander theorem (see [16]), which gives a sufficient condition on m guaranteeing the boundedness of Tm on Lp (Rn ), for 1 < p < ∞. This paper is organized as follows. In section 2, we recall some basic Harmonic Analysis results related with the Weinstein operator developed in [4, 5] and [6]. In the third section, the Weinstein multiplier operators are studied on the space s Hα,β , for α ≥ β > −1/2. In the fourth section, the extremal function associated with the Weinstein operators is given using the theory of the reproducing kernel and we list some of its properties in Corollary 4.4 and Corollary 4.5. In the last section, we prove the H¨ ormander multiplier theorem for the operators Tα,β,m , when α = β > −1/2.

2. Harmonic analysis and the Weinstein-Laplace operator In this section, we shall collect some results and definitions from the theory of d+1 the harmonic analysis associated with the Weinstein operator ∆α,d W defined on R+ by the relation (1.1). Main references are [4, 5, 6, 7, 12, 13, 21, 22]. Let us begin by the following result, which gives the eigenfunction Ψα,d of the λ . Weinstein operator ∆α,d W Proposition 2.1. For all λ = (λ1 , λ2 , . . . , λd+1 ) ∈ Rd+1 + , the system ∂2u (x) = −λ2j u(x), ∂x2j

if 1 ≤ j ≤ d

Lα u(x) = −λ2d+1 u(x), ∂u (0) = 0, ∂xd+1

u(0) = 1,

(2.1)

∂u (0) = −iλj , ∂xj

if 1 ≤ j ≤ d

has a unique solution Ψα,d given by λ 0

0

−ihz ,λ i Ψα,d jα (λd+1 zd+1 ), λ (z) = e

∀z ∈ Cd+1 ,

(2.2)

where z = (z 0 , zd+1 ), z 0 = (z1 , z2 , . . . , zd ) and jα is the normalized Bessel function of index α, defined by jα (ξ) = Γ(α + 1)

∞ X

ξ (−1)n ( )2n n!Γ(n + α + 1) 2 n=0

∀ξ ∈ C.

Remark 2.2. The Weinstein kernel Λα,d : (λ, z) 7→ Ψα,d λ (z) has a unique extension to Cd+1 × Cd+1 and can be written in the form Z 1 1 −ihx0 ,y 0 i Λα,d (x, y) = aα e (1 − t2 )α− 2 cos(txd+1 yd+1 )dt ∀x, y ∈ Cd+1 , (2.3) 0 0

0

where x = (x , xd+1 ), x = (x1 , x2 , . . . , xd ) and aα is the constant given by 2Γ(α + 1) aα = √ . πΓ(α + 12 ) The following result summarizes some of the Weinstein kernel’s properties.

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Proposition 2.3. (i) For all λ, z ∈ Cd+1 and t ∈ R, we have Λα,d (λ, 0) = 1, Λα,d (λ, z) = Λα,d (z, λ),

Λα,d (λ, tz) = Λα,d (tλ, z).

(ii) For all ν ∈ Nd+1 , x ∈ Rd+1 and z ∈ Cd+1 , we have + |Dzν Λα,d (x, z)| ≤ kxk|ν| exp(kxkk Im zk), where Dzν =

(2.4)

∂ν νd+1 ∂z1ν1 . . . ∂zd+1

and |ν| = ν1 + · · · + νd+1 . In particular |Λα,d (x, y)| ≤ 1, ∀x, y ∈ Rd+1 + .

(2.5)

In this article, we use the following notation: • C∗ (Rd+1 ), the space of continuous functions on Rd+1 , even with respect to the last variable. • C∗,c (Rd+1 ), the space of continuous functions on Rd+1 with compact support, even with respect to the last variable. • C∗p (Rd+1 ), the space of functions of class C p on Rd+1 , even with respect to the last variable. • E∗ (Rd+1 ), the space of C ∞ -functions on Rd+1 , even with respect to the last variable. • S∗ (Rd+1 ), the Schwartz space of rapidly decreasing functions on Rd+1 , even with respect to the last variable. • D∗ (Rd+1 ), the space of C ∞ -functions on Rd+1 which are of compact support, even with respect to the last variable. • S 0 ∗ (Rd+1 ), the space of temperate distributions on Rd+1 , even with respect to the last variable. It is the topological dual of S∗ (Rd+1 ). • Lp (dµα,d ), 1 ≤ p ≤ +∞, the space of measurable functions on Rd+1 such that + kf kL∞ (dµα,d ) = ess supx∈Rd+1 |f (x)| < +∞, + i1/p hZ p |f (x)| dµα,d (x) < +∞, if 1 ≤ p < +∞, kf kLp (dµα,d ) = Rd+1 +

where µα,d is the measure on Rd+1 given by + dµα,d (x) = cα,d x2α+1 d+1 dx,

(2.6)

dx is the Lebesgue measure on Rd+1 , and cα,d is the constant given by relation (1.2) • H∗ (Cd+1 ), the space of entire functions on Cd+1 , even with respect to the last variable, rapidly decreasing and of exponential type. Definition 2.4. The Weinstein transform for f ∈ L1α (Rd+1 + ) is Z α,d FW (f )(λ) = f (x)Λα,d (x, λ)dµα,d (x), ∀λ ∈ Rd+1 + ,

(2.7)

Rd+1 +

where µα,d is the measure on Rd+1 given by relation (2.6). + α,d Some basic properties of the transform FW are summarized in the following results. For the proofs, we refer to [6, 7, 8].

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Proposition 2.5. (i) For all f ∈ L1 (dµα,d ), we have α,d kFW (f )kL∞ (dµα,d ) ≤ kf kL1 (dµα,d ) .

(ii) For m ∈ N and f ∈ S∗ (Rd+1 ), we have  α,d  m m 2m α,d FW ∆α,d FW (f )(y), W ) f (y) = (−1) |y| (iii) For all f in S∗ (Rd+1 ) and m ∈ N, we have
m  α,d  α,d ∆α,d FW (f ) (λ) = FW (Pm f )(λ), W

(2.8)

∀y ∈ Rd+1 + .

∀λ ∈ Rd+1 + ,

(2.9)

(2.10)

where Pm (λ) = (−1)m kλk2m . α,d Theorem 2.6. (i) The Weinstein transform FW is a topological isomorphism from S∗ (Rd+1 ) onto itself, from D∗ (Rd+1 ) onto H∗ (Cd+1 ) and from S 0 ∗ (Rd+1 ) onto itself. α,d −1 (ii) Let f ∈ S∗ (Rd+1 ). The inverse transform (FW ) is given by
α,d −1 α,d FW (f )(x) = FW (f )(−x), ∀x ∈ Rd+1 (2.11) + ,. α,d (iii) Let f ∈ L1 (dµα,d ). If FW (f ) ∈ L1 (dµα,d ), then we have Z α,d f (x) = FW (f )(y)Λα,d (−x, y)dµα,d (y), a.e. x ∈ Rd+1 + .

(2.12)

Rd+1 +

Theorem 2.7. (i) For all f, g ∈ S∗ (Rd+1 ), we have the Parseval formula Z Z α,d α,d f (x)g(x)dµα,d (x) = FW (f )(λ)FW (g)(λ)dµα,d (λ).

(2.13)

(ii) (Plancherel formula) For all f ∈ S∗ (Rd+1 ), we have Z Z α,d 2 (f )(λ)|2 dµα,d (λ). |f (x)| dµα,d (x) = |FW

(2.14)

Rd+1 +

Rd+1 +

Rd+1 +

Rd+1 +

α,d extends uniquely to an isometric (iii) (Plancherel theorem) The transform FW 2 isomorphism on L (dµα,d ).

Definition 2.8. The translation operator τxα , x ∈ Rd+1 + , associated with the Wed+1 instein operator ∆α,d is defined on C (R ), for all y ∈ Rd+1 ∗ + , by W Z π q aα 2 τxα f (y) = f (x0 + y 0 , x2d+1 + yd+1 + 2xd+1 yd+1 cos θ)(sin θ)2α dθ, 2 0 Z (2.15) = f (x0 + y 0 , u)qα (xd+1 , yd+1 , u)u2α+1 du, R+ 0

0

where x + y = (x1 + y1 , . . . , xd + yd ), and qα (v, w, u) = 22α−1 √

Γ(α + 1) Υ(v, w, u)2α−1 1[|v−w|,v+w] (z), (uvw)2α πΓ(α + 1/2)

with Υ(v, w, u) =

1p (v + w + u)(v + w − u)(v − w + u)(w + u − v). 4

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We should note that for all v, w > 0, we have Z qα (v, w, u)u2α+1 du = 1.

(2.16)

R+

The following proposition summarizes some properties of the Weinstein translation operator. Proposition 2.9. (i) For f ∈ C∗ (Rd+1 ), we have τxα f (y) = τyα f (x) and τ0α f = f,

∀x, y ∈ Rd+1 + .

α (ii) For all f ∈ E∗ (Rd+1 ) and y ∈ Rd+1 + , the function x 7→ τx f (y) belongs to d+1 E∗ (R ). (iii) We have α,d α α ∆α,d ∀x ∈ Rd+1 + . W ◦ τx = τx ◦ ∆W , α (iv) Let f ∈ Lp (dµα,d ), 1 ≤ p ≤ +∞ and x ∈ Rd+1 + . Then τx f belongs to p L (dµα,d ) and we have

kτxα f kLp (dµα,d ) ≤ kf kLp (dµα,d ) . satisfies the product formula (v) The function Λα,d (·, λ), λ ∈ Cd+1 , on Rd+1 + Λα,d (x, λ)Λα,d (y, λ) = τxα [Λα,d (., λ)](y),

∀y ∈ Rd+1 + .

(2.17)

(vi) Let f ∈ Lp (dµα,d ), p = 1 or 2 and x ∈ Rd+1 + , we have α,d α α,d FW (τx f )(y) = Λα,d (x, y)FW (f )(y),

∀y ∈ Rd+1 + .

(2.18)

(vii) The space S∗ (Rd+1 ) is invariant under the operators τxα , with x ∈ Rd+1 + . Definition 2.10. The Weinstein convolution product of f, g ∈ C∗ (Rd+1 ) is given by: Z (2.19) f ∗W g(x) = τxα f (−y)g(y)dµα,d (y), ∀x ∈ Rd+1 + . Rd+1 +

Proposition 2.11. (i) Let p, q, r ∈ [1, +∞] be such that p1 + 1q − 1r = 1. Then for all f ∈ Lp (dµα,d ) and g ∈ Lq (dµα,d ), the function f ∗W g belongs to Lr (dµα,d ) and kf ∗W gkLr (dµα,d ) ≤ kf kLp (dµα,d ) kgkLq (dµα,d ) .

(2.20)

(ii) For all f, g ∈ L1 (dµα,d ), (resp. S∗ (Rd+1 )), f ∗W g ∈ L1 (dµα,d ) (resp. S∗ (Rd+1 )) and α,d α,d α,d FW (f ∗W g) = FW (f )FW (g). (2.21) 3. Weinstein multiplier operators on a Sobolev type space Throughout this section α and β denote two real numbers satisfying α ≥ β > − 12 . s Let s ∈ R, we define the Sobolev-Weinstein type space of order s, denoted by Hα,β , α,d 0 d+1 as the set of all f ∈ S∗ (R ) such that FW (f ) is a function and s

α,d (1 + |z|2 ) 2 FW (f ) ∈ L2 (dµα,d ). s The space Hα,β is endowed with the inner product Z α,d α,d hf, giHsα,β = FW (f )(z)FW (g)(z)dµsβ,d (z) Rd+1 +

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and the norm kf k2Hsα,β

Z = Rd+1 +

7

α,d |FW (f )(z)|2 dµsβ,d (z),

where dµsβ,d (z) = (1 + |z|2 )s dµβ,d (z). s Lemma 3.1. Let s ∈ R, the space Hα,β is an Hilbert space. α,d s Proof. Let (fn )n be a Cauchy sequence on Hα,β . It is easy to see that (FW (fn ))n 2 s is a Cauchy sequence of L (µβ,d ), which is a complete space. Therefore there exists a function g ∈ L2 (µsβ,d ) satisfying α,d lim kFW (fn ) − gkL2 (µsβ,d ) = 0.

n→+∞

α,d −1 Then g ∈ S 0 ∗ (Rd+1 ) and if we denote f the distribution given by f = (FW ) (g), α,d according to Theorem2.6, we deduce that f ∈ S 0 ∗ (Rd+1 ) and F (f ) = g ∈ + W s . Furthermore, L2 (µsβ,d ), which proves that f ∈ Hα,β α,d lim kfn − f kHsα,β = lim kFW (fn ) − gkL2 (µsβ,d ) = 0.

n→+∞

This proves that

n→+∞

s Hα,β

is a complete space.



s Remark 3.2. (i) For s ≥ α − β and f ∈ Hα,β , the Plancherel Theorem associated with the Weinstein transform leads to Z α,d (f )(z)|2 2(α−β) s |FW cα,d kf k2L2 (µα,d ) ≤ z dµβ,d (z). d+1 cβ,d R+ (1 + |z|2 )s d+1

Also for s ≥ α − β, we have 2(α−β)

zd+1 |z|2(α−β) ≤ ≤ 1. (1 + |z|2 )s (1 + |z|2 )s So, we deduce that

cα,d 1/2 ) kf kHsα,β . cβ,d s Consequently, the space Hα,β is continuously contained in L2 (dµα,d ). α,d d s , then FW (f ) ∈ L1 (dµα,d ) and (ii) Let s > 2α − β + 2 + 1. If f ∈ Hα,β kf kL2 (µα,d ) ≤ (

(3.1)

α,d kFW (f )kL1 (µα,d ) ≤ Cα,β kf kHsα,β ,

where Cα,β = (

cα,d cβ,d

Z Rd+1 +

2(α−β)

zd+1

1/2 dµ−s . α,d (z))

α,d Using (3.1), we deduce that the function FW (f ) ∈ L1 (dµα,d ) ∩ L2 (dµα,d ) and Z α,d f (x) = FW (f )(z)Λα,d (−x, z)dµα,d (z); a.e. x ∈ Rd+1 + . Rd+1 +

Definition 3.3. Let m be a function in L∞ (dµβ,d ), we define the Weinstein muls tiplier operator Tα,β,m on Hα,β , by β,d −1 α,d Tα,β,m f (x) = (FW ) (mFW (f ))(x),

x ∈ Rd+1 + .

Using Plancherel Theorem associated with the Weinstein transform, we get the following result.

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0 Theorem 3.4. Let m ∈ L∞ (dµβ,d ) and f ∈ Hα,β , then we have

kTα,β,m f kL2 (dµβ,d ) ≤ kmkL∞ (dµβ,d ) kf kH0α,β . Definition 3.5. (i) Let E be a measurable subset of Rd+1 + , we say that the function 0 is -concentrated on E, if f ∈ Hα,β kf − χE f kH0α,β ≤ kf kH0α,β , where χE is the indicator function of the set E. α,d 0 (ii) Let S be a measurable subset of Rd+1 and let f ∈ Hα,β . We say that FW (f ) + is ν-concentrated on S, if α,d α,d kFW (f ) − χS FW (f )kL2 (dµβ,d ) ≤ νkf kH0α,β .

The following theorem can be obtained from Donoho-Stark Uncertainty Principle for the Weinstein transform (see [23]). α,d 0 (f ) is νTheorem 3.6. Let f ∈ Hα,β , if f is -concentrated on E and FW concentrated on S, then cα,d (µα,d (E))1/2 (µβ,d (S))1/2 ≥ (1 − ν − ). cβ,d

4. Extremal functions for the operator Tα,β,m The theory of extremal functions and reproducing kernels on Hilbert spaces, is an important tool in this section concerning the study of the extremal function associated with the Weinstein multiplier operators Tα,β,m . In this section, s denotes a real number satisfying s > 2α − β + d2 + 1. s Let η > 0. We denote by h·, ·iη,Hsα,β the inner product on the space Hα,β by hf, giη,Hsα,β = ηhf, giHsα,β + hTα,β,m f, Tα,β,m giL2 (dµβ,d ) , and k · kη,Hsα,β the associated norm. We remark that the two norms k · kη,Hsα,β and k · kHsα,β are equivalent, therefore s the pair (Hα,β , h·, ·iη,Hsα,β ) is an Hilbert space with a reproducing kernel given in the following theorem. s Theorem 4.1. Let η > 0 and m ∈ L∞ (dµβ,d ). The space (Hα,β , h·, ·iη,Hsα,β ) has the reproducing kernel Z Λα,d (−x, z)Λα,d (y, z) 2(α−β) cα,d z dµα,d (z); Ks (x, y) = 2 + η(1 + |z|2 )s d+1 cβ,d Rd+1 |m(z)| +

that is s (i) For all y ∈ Rd+1 + , the function Ks (·, y) belongs to Hα,β . s (ii) The reproducing property: For all f ∈ Hα,β and y ∈ Rd+1 + , hf, Ks (·, y)iη,Hsα,β = f (y). Proof. (i) Let y ∈ Rd+1 + . Using the relation (2.5), one can see that the function φy : z 7→

Λα,d (y, z) cα,d 2(α−β) z cβ,d |m(z)|2 + η(1 + |z|2 )s d+1

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belongs to L1 (dµα,d ) ∩ L2 (dµα,d ). Then the function Ks is well defined and α,d −1 Ks (x, y) = (FW ) (φy )(x), x ∈ Rd+1 + .

(4.1)

Plancherel Theorem for the Weinstein transform and relation (2.5) give 4(α−β)

α,d (1 + |z|2 )s |FW (Ks (·, y))(z)|2 ≤ (

cα,d 2 zd+1 , ) cβ,d η 2 (1 + |z|2 )s

So kKs (·, y)kHsα,β ≤ ≤

cα,d  cβ,d

4(α−β) 1/2 zd+1 dµ (z) β,d η 2 (1 + |z|2 )s

Z Rd+1 +

1 cα,β < ∞. η

s This proves that for all y ∈ Rd+1 the function Ks (·, y) belongs to Hα,β . + d+1 s (ii) Let f ∈ Hα,β and y ∈ R+ , using relation (4.1), we have

hf, Ks (·, y)iη,Hsα,β α,d −1 α,d −1 = ηhf, (FW ) (φy )iHsα,β + hTα,β,m f, Tα,β,m (FW ) (φy )iL2 (dµβ,d ) .

Using Parseval formula for the Weinstein transform, hf, Ks (·, y)iη,Hsα,β Z Z α,d FW (f )(z)φy (z)dµsβ,d (z) + =η

Rd+1 +

Rd+1 +

Z = Rd+1 +

Z = Rd+1 +

α,d FW (f )(z)

α,d |m(z)|2 FW (f )(z)φy (z)dµβ,d (z)

cα,d 2(α−β) Λα,d (y, z)zd+1 dµβ,d (z) cβ,d

α,d FW (f )(z)Λα,d (y, z)dµα,d (z) = f (y).

Hence Ks (·, y) is a reproducing kernel.



s Remark 4.2. The space Hα,β has the reproducing kernel Z cα,d 2(α−β) Ks (x, y) = Λα,d (−x, z)Λα,d (y, z)zd+1 dµ−s α,d (z). cβ,d Rd+1 +

Theorem 4.3. Let m ∈ L∞ (dµα,d ), for any function g ∈ L2 (dµβ,d ) and for any ∗ , where the infimum η > 0, there exists a unique function fη,g inf {ηkf k2Hsα,β + kg − Tα,β,m f k2L2 (µβ,d ) }

f ∈Hsα,β

∗ is attained. Moreover, the extremal function fη,g is given by Z ∗ fη,g (y) = g(x)Qα,β (x, y)dµβ,d (x), Rd+1 +

where Z Qα,β (x, y) =

Rd+1 +

m(z)Λα,d (x, z)Λα,d (−y, z) dµα,d (z). |m(z)|2 + η(1 + |z|2 )s

(4.2)

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∗ Proof. The existence and the uniqueness of the extremal function fη,g satisfying (4.2) is obtained in [11, 20, 25]. Especially, using the reproducing kernel of the s ∗ space (Hα,β , k · kη,Hsα,β ), fη,g is given by ∗ fη,g = hg, Tα,β,m (Ks (·, y))iL2 (dµβ,d ) .

(4.3)

The relation (4.1) leads to Z Tα,β,m (Ks (·, y))(x) =

Rd+1 +

α,d m(z)FW (Ks (·, y))(z)Λβ,d (−x, z)dµβ,d (z)

Z =

m(z) Rd+1 +

Λα,d (−x, z)Λα,d (y, z) dµα,d (z). |m(z)|2 + η(1 + |z|2 )s

Which gives the result.

 ∗ fη,g

2

Corollary 4.4. Let g ∈ L (dµβ,d ) and η > 0. The extremal function cα,d ∗ |fη,g | ≤ √ kgkL2 (dµβ,d ) , 2 η Z  1/2 |x|2 cα,α ∗ kfη,g kL2 (dµα,d ) ≤ √ |g(x)|2 e 2 dµβ,d (x) ; 2 η Rd+1 + Z Λα,d (−y, z)m(z) β,d ∗ FW (g)(z)dµα,d (z). fη,g (y) = d+1 |m(z)|2 + η(1 + |z|2 )s R+

satisfies:

Proof. To prove (4.4), have ∗ fη,g (y) = hg, Tα,β,m (Ks (·, y))iL2 (dµβ,d ) .

This leads to α,d ∗ |fη,g (y)| ≤ kgkL2 (dµβ,d ) kmFW (Ks (·, y))kL2 (dµβ,d ) Z  1/2 |m(z)|2 |φy (z)|2 dµβ,d (z) ≤ kgkL2 (dµβ,d ) Rd+1 +

≤ kgkL2 (dµβ,d )

Z Rd+1 +

2(α−β) 1/2 zd+1 |m(z)|2 cα,d dµα,d (z) . 2 2 s 2 cβ,d [|m(z)| + η(1 + |z| ) ]

Using the inequality  2 |m(z)|2 + η(1 + |z|2 )s ≥ 4η(1 + |z|2 )s |m(z)|2 , we obtain ∗ |fη,g (y)|

≤ kgkL2 (dµβ,d )

c

2(α−β)

Z

α,d

cβ,d

Rd+1 +

1/2 zd+1 dµ (z) α,d 4η(1 + |z|2 )s

cα,β ≤ √ kgkL2 (dµβ,d ) . 2 η To prove (4.5) we write Z ∗ fη,g (y) =

e−

|x|2 4

Rd+1 +

e

|x|2 4

g(x)Qα,β (x, y)dµβ,d (x).

H¨ older inequality gives ∗ |fη,g (y)|2 ≤

Z e Rd+1 +

|x|2 2

|g(x)|2 |Qα,β (x, y)|2 dµβ,d (x).

(4.4) (4.5) (4.6)

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WEINSTEIN TRANSFORM

11

Now, applying Fubini-Tonelli’s Theorem we obtain Z |x|2 ∗ 2 kfη,g (y)kL2 (dµα,d ) ≤ e 2 |g(x)|2 kQα,β (x, ·)k2L2 (dµα,d ) dµβ,d (x). Rd+1 +

Let ψx be the function defined by ψx (z) =

m(z)Λα,d (x, z) . [|m(z)|2 + η(1 + |z|2 )s ]2

α,d −1 Since s > 2α−β+ d2 +1, ψx ∈ L1 (dµα,d )∩L2 (dµα,d ) and Qα,β (x, y) = (FW ) (ψx )(y), we have α,d (Qα,β (x cdot))k2L2 (dµα,d ) kQα,β (x, ·)k2L2 (dµα,d ) = kFW Z |m(z)|2 dµα,d (z) ≤ [|m(z)|2 + η(1 + |z|2 )s ]2 Rd+1 + Z dµα,d (z) ≤ d+1 4η(1 + |z|2 )s R+ cα,α ≤ √ . 2 η

Equality (4.6) follows from relation (4.3), Plancherel Theorem for the Weinstein transform and (4.1).  s Corollary 4.5. For every f ∈ Hα,β and g = Tα,β,m f , we have

(i) |m(z)|2 F α,d (f )(z); m(z)2 + η(1 + |z|2 )s W

α,d ∗ FW (fη,g )(z) = kmkL∞ (dµ

)

∗ √ β,d kf kHs ; (ii) kfη,g kHsα,β ≤ 2 η α,β ∗ − f kHsα,β = 0; (iii) limη→0+ kfη,g ∗ − f kL∞ (dµα,d ) = 0. (iv) limη→0+ kfη,g s Proof. (i) For every f ∈ Hα,β and g = Tα,β,m f , we have α,d ∗ FW (fη,g )(z) Z α,d = g(x)FW (Qα,β (x cdot))(z)dµβ,d (x) Rd+1 +

Z = Rd+1 +

= =

Tα,β,m f (x)ψx (z)dµβ,d (x)

m(z) 2 m(z) + η(1 + |z|2 )s

Z Rd+1 +

α,d −1 α,d (FW ) (mFW (f ))(x)Λα,d (x, z)dµβ,d (x)

|m(z)|2 F α,d (f )(z). m(z)2 + η(1 + |z|2 )s W

(ii) We have ∗ kfη,g kHsα,β =

Z Rd+1 +

1/2 |m(z)|4 α,d 2 s |F (f )(z)| dµ (x) β,d [m(z)2 + η(1 + |z|2 )s ]2 W

12

A. GASMI, A. EL GARNA

≤

Z Rd+1 +

≤

EJDE-2017/124

1/2 |m(z)|2 α,d 2 s |F (f )(z)| dµ (x) β,d 4η(1 + |z|2 )s W

kmkL∞ (dµβ,d ) kf kHsα,β . √ 2 η

(iii) We have ∗ kfη,g

− f kHsα,β = =

Z Rd+1 +

Z Rd+1 +

1/2 |m(z)|2 α,d 2 2 s − 1| |F (f )(z)| dµ (x) β,d W m(z)2 + η(1 + |z|2 )s 1/2 η 2 (1 + |z|2 )2s α,d |FW (f )(z)|2 dµsβ,d (x) . 2 2 s 2 [m(z) + η(1 + |z| ) ]

|

Since

η 2 (1 + |z|2 )2s ≤ 1, [m(z)2 + η(1 + |z|2 )s ]2 we obtain the result with the dominated convergence theorem. (iv) It is easy to see that α,d ∗ FW (fη,g − f )(z) =

−η(1 + |z|2 )s F α,d (f )(z). m(z)2 + η(1 + |z|2 )s W

So ∗ fη,g (y) − f (y) =

Z Rd+1 +

−η(1 + |z|2 )s F α,d (f )(z)Λα,d (−y, z)dµα,d (z), m(z)2 + η(1 + |z|2 )s W

and sup y∈Rd+1 +

∗ |fη,g (y)

Z − f (y)| ≤ Rd+1 +

η(1 + |z|2 )s |F α,d (f )(z)|dµα,d (z). m(z)2 + η(1 + |z|2 )s W

Hence the needed result is a consequence of the dominated convergence theorem and the inequality η(1 + |z|2 )s ≤ 1. m(z)2 + η(1 + |z|2 )s  ¨ rmander multiplier theorem for the operator Tα,α,m 5. Ho The aim of this section is to prove an analogue of the famous H¨ormander multiplier theorem for the operator Tα,α,m , which will be denoted as Tα,m . The theorem is stated as follows. Theorem 5.1. Let ` the least integer greater than α + 1 and m be a bounded C∗2` -function on Rd+1 \ {0}, satisfying the H¨ ormander condition 1/2 Z q d+1 ∂ m (5.1) |∆sd ( q )(ξ)|2 dµα,d (ξ) ≤ CRα+ 2 −(2s+q) , ∂ξ R/2≤|ξ|≤2R d+1 for all R > 0, where C is a constant independent of R and s ∈ {0, 1, . . . , `} and q ∈ {0, 1, . . . , 2`}. Then the operator Tα,m can be extended to a bounded operator from Lp (dµα,d ) into itself for 1 < p < ∞. We need to establish some results associated with the Weinstein analysis to prove Theorem5.1.

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WEINSTEIN TRANSFORM

13

d+1 Theorem 5.2. Let K be a measurable function on {(x, y) ∈ Rd+1 : |x| = 6 |y|} + ×R+ 2 and T be a bounded operator from L (dµα,d ) into itself such that Z T (f )(x) = K(x, y)f (y)dµα,d (y), (5.2) Rd+1 +

for all compactly supported f in L2 (dµα,d ) and for a.e. x ∈ Rd+1 / | supp(f )| = + , |x| ∈ {|y|, y ∈ supp(f )}. If K satisfies Z |K(x, y) − K(x, z)| dµα,d (x) ≤ C , ∀y, z ∈ Rd+1 (5.3) + , ||x|−|y||>2|y−z|

then T extends to a bounded operator from Lp (dµα,d ) into itself for 1 < p ≤ 2. Proof. We proceed by the same manner as in the proof of the classical theorem of singular integral given in [27, Chp. I], by considering here the doubling measure dµα,d and proving that T is a weak-type (1, 1).  Before proving Theorem 5.1 we need the following lemmas. d+1 Lemma 5.3. Let ϕ ∈ S∗ (Rd+1 we have + ), then for all x, y ∈ R+

kτxα (ϕ) − τyα (ϕ)kL1 (dµα,d ) ≤ C|xd+1 − yd+1 |k

∂ϕ kL1 (dµα,d ) ∂zd+1

(5.4)

Proof. In view of (2.15) we have Z aα π τxα ϕ(z) = ϕ(x0 + z 0 , ψθ (xd+1 , zz+1 ))dν(θ), 2 0 q 2 + 2xd+1 zd+1 cos θ and dν(θ) = (sin θ)2α dθ. where ψθ (xd+1 , zd+1 ) = x2d+1 + zd+1 By the mean value theorem, it follows that τxα (ϕ)(z) − τyα (ϕ)(z) Z 1Z π zt + zd+1 cos θ ∂ϕ aα = (xd+1 − yd+1 ) (x0 + z 0 , Ψθ (zt , z))dν(θ)dt, 2 Ψθ (zt , zd+1 ) ∂zd+1 0 0 where zt = xd+1 + t(yd+1 − xd+1 ). Now, using the inequality |zt + zd+1 cos θ| ≤ 1, Ψθ (zt , zd+1 ) and making the change of variable u → Ψθ (zt , zd+1 ), we obtain |τxα (ϕ)(z) − τyα (ϕ)(z)| Z 1Z π aα ∂ϕ ≤ |xd+1 − yd+1 | | (x0 + z 0 , Ψθ (zt , z))|dν(θ)dt 2 ∂zd+1 0 0 Z 1Z aα ∂ϕ ≤ |xd+1 − yd+1 | | (x0 + z 0 , u)|qα (zt , zd+1 , u)u2α+1 du dt. 2 ∂z d+1 0 R+ So, (5.4) follows from Fubini’s theorem and (2.16).



Lemma 5.4 (Berstein’s lemma). Let f ∈ L1 (dµα,d ) and λ > 0 and suppose that α,d d+1 supp(FW (f )) ⊂ {x ∈ Rd+1 + , |x| < λ}. Then for all x, y ∈ R+ , we have kτxα (f ) − τyα (f )kL1 (dµα,d ) ≤ Cλ|xd+1 − yd+1 |kf kL1 (dµα,d ) .

14

A. GASMI, A. EL GARNA

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α,d d+1 Proof. Choose φ ∈ S∗ (Rd+1 + ) such that |FW (φ)(x)| = 1 for all x ∈ {x ∈ R+ , |x| < α,d α,d 1} and put φλ (x) = λ2α+2+d φ(λx). Then |FW (φλ )(x)| = |FW (φ)( λx )| = 1 in {x ∈ Rd+1 + , |x| < 1} and we can write

τxα (f )(z) − τyα (f )(z) = φλ ∗W (τxα (f ) − τyα (f ))(z) = f ∗W (τxα (φλ ) − τyα (φλ ))(z). Using (5.4) we get kτxα (f ) − τyα (f )k1,α ≤ Ckf kL1 (dµα,d ) kτxα (φλ ) − τyα (φλ )kL1 (dµα,d ) α α ≤ Ckf kL1 (dµα,d ) kτxλ (φ) − τyλ (φ)kL1 (dµα,d )

≤ Cλ|xd+1 − yd+1 |kf kL1 (dµα,d ) , which gives the desired result.



Lemma 5.5. If m satisfies (5.1). Then there exists a locally integrable function k on Rd+1 \ {0} such that for all |x| ∈ / | supp(f )|, + Z Tα,m (f )(x) = K(x, y)f (y)dµα,d (y), Rd+1 +

where K is given on {(x, y) ∈ Rd+1 × Rd+1 : |x| = 6 |y|}, by + + K(x, y) = τxα (k)(−y 0 , yd+1 ).

(5.5)

Proof. Let ϕ ∈ D∗ (Rd+1 ), supported in { 21 ≤ |ξ| ≤ 2} and satisfying : +∞ X

ϕ(2−j ξ) = 1,

ξ 6= 0.

j=−∞ α,d −1 ) (mj ). Let us prove first the estimate Put mj (ξ) = m(ξ)ϕ(2−j ξ) and kj = (FW s j(α+ k(∆α,d W ) mj kL2 (dµα,d ) ≤ Cs 2

d+1 2 −s)

,

s = 0, 1, . . . , `.

(5.6)

In fact by induction we can show that there exist constants bα,r , r ∈ {0, 1, . . . , s}, depending only on α, satisfying s (∆α,d W ) mj (ξ) =

s X

Csj

j=0

2j X

r−s s−j bαr ξd+1 ∆d

r=1

∂ r mj (ξ), r ∂ξd+1

ξ 6= 0.

Leibniz formula gives r X ∂ r mj ∂q m ∂ r−q ϕ (ξ) = Crq 2j(q−r) q (ξ) r−q (2−j ξ). r ∂ξd+1 ∂ξd+1 ∂ξd+1 q=0

Hence, s (∆α,d W ) mj (ξ) =

s X j=0

Csj

2j X

r−s bα,r ξd+1

r=1

Using (5.4), we obtain Z r r−s p ∂ mj |ξd+1 ∆d r (ξ)|2 dµα,d (ξ) ∂ξd+1 Rd+1 +

r X q=0

Crq 2j(q−r) ∆ds−j (

∂ q m ∂ r−q ϕ q r−q ). ∂ξd+1 ∂ξd+1

(5.7)

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WEINSTEIN TRANSFORM

2j(r−s)

≤ C2

r X

2j(q+2p−r)

2

(Crq )2

q=0

≤ Cr,s 22j(α+

d+1 2 −s)

Z 2j−1 ≤|ξ|≤2j+1

|∆pd (

15

∂q m )(ξ)|2 dµα,d (ξ) q ∂ξd+1

.

So, the inequality (5.6) that we seek is a consequence of (5.7). Now, applying Plancherel’s theorem and using (5.6), we obtain s j(α+ k(−1)s |x|2s kj (x)|L2 (dµα,d ) = k(∆α,d W ) mj kL2 (dµα,d ) ≤ Cs 2

d+1 2 −s)

,

for s = 0, 1, . . . , `. Applying this formula with s = 0 and s = `, we get that the series −1 +∞ X X kkj (x)kL2 (dµα,d ) , k(ix)` kj (x)kL2 (dµα,d ) j=−∞

j=0

P+∞

are convergent and the series j=−∞ |kj (x)| is convergent for a.e. x 6= 0. By the Cauchy-Schwarz inequality it follows that Z −1 X α |τx (f )(y)| |kj (−y 0 , yd+1 )|dµα,d (y) Rd+1 +

≤

j=−∞ −1 X

kτxα (f )kL2 (dµα,d )

kkj kL2 (dµα,d ) < ∞,

j=−∞

Z Rd+1 +

≤

|τxα (f )(y)|

∞ X

|kj (−y 0 , yd+1 )|dµα,d (y)

j=0 ∞ α X τ (f )(y) ky ` kj (−y 0 , yd+1 )kL2 (dµα,d ) kL2 (dµα,d ) k x ` y j=0

< ∞,

for |x| ∈ / | supp(f )| (which implies that 0 ∈ / supp(τx (f ))). Thus we concluded that Z +∞ X |τxα (f )(y)| |kj (y)|dµα,d (y) < ∞, |x| ∈ / | supp(f )|. Rd+1 +

j=−∞

P+∞ This allows us to take k = j=−∞ kj and one can write for |x| ∈ / | supp(f )|, Z Z T (f )(x) = k(z)τxα (f )(−z 0 , zd+1 )dµα,d (z) = K(x, y)f (y)dµα,d (y). Rd+1 +

Rd+1 +

Which completes the proof.



∗ Proof of Theorem 5.1. First, we note that the adjoint operator Tm is the multiplier operator associated with m and Z ∗ Tm (f )(x) = K(y, x)f (y)dµα,d (y); |x| ∈ / | supp(f )|, Rd+1 +

where K is given by (5.5). Using the duality argument, it is sufficient to show that the function K satisfies the condition (5.3) of the Theorem 5.2, which follows from ∞ Z X |τyα (kj )(x) − τzα (kj )(x)|dµα,d (x) ≤ C, (5.8) j=−∞

||x|−|y||>2|y−z|

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A. GASMI, A. EL GARNA

EJDE-2017/124

for all y, z ∈ Rd+1 + . To prove (5.8) we need the following two estimates Z Z d+1 |kj (x)|dµα (x) ≤ C, |kj (x)|dµα,d (x) ≤ C(2j t)α+ 2 −s , Rd+1 +

t > 0.

|x|≥t

(5.9) Cauchy-Schwarz inequality, Plancherel’s theorem and (2.9), give Z |kj (x)|dµα (x) ≤ k(1 + 2j |x|2 )−` kL2 (dµα,d ) k(1 + 2j |x|2 )` |kj (x)|kL2 (dµα,d ) Rd+1 +

≤ C2−j(α+

d+1 2 )

` X

q 2jq C`q k(∆α,d W ) mj kL2 (dµα,d ) ≤ C.

q=0

Hence the first inequality of (5.9) is proved. The second inequality of (5.9) follows in similar way. Let us remark that if ||x| − |y|| > 2|y − z| and |u| > ||x| − |z|| then |u| > |y − z|. Therefore, in view of (2.15), (2.16), (5.9) and Fubini-Tonelli’s theorem Z |τyα (kj )(x) − τzα (kj )(x)|dµα,d (x) ||x|−|y||>2|y−z| Z Z α ≤ |τy (kj )(x)|dµα,d (x) + |τz (kj )(x)|dµα,d (x) ||x|−|y||>2|y−z| ||x|−|y||>2|y−z| Z ≤2 |kj (z)||dµα,d (z) |u|>|y−z|

≤ C(2j |y − z|)α+

d+1 2 −`

.

On the other hand, Lemma 5.4 and (5.9) give Z |τyα (kj )(x) − τzα (kj )(x)|dµα,d (x) ||x|−|y||>2|y−z|

≤ kτyα (kj ) − τzα (kj )kL1 (dµα,d ) ≤ C2j |yd+1 − zd+1 |. Thus to get (5.8), we write ∞ Z X j=−∞

≤C



|τyα (kj )(x) − τzα (kj )(x)|dµα,d (x)

||x|−|y||>2|y−z|

X

(2j |y − z|)α+

d+1 2 −`

{2j |y−z|≥1}

+

X

 2j |yd+1 − zd+1 | ≤ C.

{2j |y−z|