Power weighted L p -inequalities for Laguerre–Riesz transforms

c 2008 by Institut Mittag-Leffler. All rights reserved

Power weighted L ^p -inequalities for Laguerre–Riesz transforms

Eleonor Harboure, Carlos Segovia^†, Jos´e L. Torrea and Beatriz Viviani

Abstract.In this paper we give a complete description of the power weighted inequalities, of strong, weak and restricted weak type for the pair of Riesz transforms associated with the Laguerre function system{L^α_k}, for any givenα>−1. We achieve these results by a careful estimate of the kernels: near the diagonal we show that they are local Calder´on–Zygmund operators while in the complement they are majorized by Hardy type operators and the maximal heat-diffusion operator.

We also show that in all the cases our results are sharp.

1. Introduction

The purpose of this article is to study boundedness properties in L^p((0,∞), x^δdx) for the Riesz transforms associated with the orthogonal systems of Laguerre functionsL^α_k,α>−1, started in [4], see also [17]. There, following the ideas developed in [12], the appropriate Riesz transforms are introduced and, using a transference method from Hermite systems along with Kanjin’s transplantation theorem for Laguerre expansions, continuity results in L^p((0,∞), x^δdx) were ob- tained for δranging in a certain interval depending on pandα. Recently (see [6]

and [3]), sharper results of this kind have been given for the maximal heat operator related to such expansions. There, the authors not only obtain strong type inequal- ities but they analyse the behaviour at the extreme points, proving weak type or restricted weak type inequalities that, as they show, are sharp.

†During the revision process of this article, Carlos Segovia unexpectedly died. The other authors are deeply touched by the loss of such an outstanding colleague and friend. We are also indebted for all we have learnt and enjoyed working in his company. Undoubtedly, his clearness of thinking and his ability for computations are vividly present throughout this manuscript.

Partially supported by Ministerio de Educaci´on y Ciencia (Spain), MTM2005–08350-C03- 01, Proyecto IALE (UAM-Banco Santander Central-Hispano), and grants from CONICET and Universidad Nacional del Litoral (Argentina).

The fourth author was also supported by grant SAB2003-0024 from MEC (Spain).

Our aim in this paper is to perform an analogous analysis for the case of the Riesz operators. Our approach this time is through a careful study of their kernels. We make use of a technique, that nowadays might be called classical, of splitting the kernel into its “global” and “local” parts (see (3.1) and (3.2) for their precise definitions). The local parts of the Riesz transforms are shown to be local Calder´on–Zygmund operators in the sense defined in [10] (see Theorem 3), which gives no restriction on the exponent δ. On the other hand, the global parts are estimated by a sum of positive operators, namely a variation of the maximal heat operator and some Hardy type operators. We combine both estimates to give our main result stated as Theorem 1.

We remark at this point that our boundedness results for the Riesz transforms are a little worse than those for the maximal heat operator. Not only do we not get boundedness onL^∞, as expected, but we also get a weaker result at one of the end points, when the range ofpis a finite interval. Nevertheless it turns to be sharp.

As we just mentioned, the main key to achieve our results is a careful analysis of the kernels. Since they involve modified Bessel functions and their derivatives, part of the proof becomes highly technical. However, such precise knowledge of the behaviour of the kernels allows us to get really sharp inequalities. In fact, in Section 6, we show that our boundedness results on L^p

(0,∞), x^δdx

cannot be improved any further (see Theorem 2).

Let us remark that this technique of splitting the kernel according to the local and global regions, goes back to Muckenhoupt and Stein (see [7] and [9]) in con- nection with problems concerning Poisson summation and conjugacy for different systems of orthogonal polynomials, like those of Jacobi, Hermite or Laguerre.

Later, even though it is not pointed out explicitly, K. Stempak in [13] did also make use of such technique in order to bound the heat and Poisson maximal oper- ators for various systems of Laguerre expansions. Finally, very recently, A. Nowak and K. Stempak in [11], have obtained boundedness results for Riesz transforms associated to other Laguerre orthonormal functions, using again this type of par- tition of the kernel. We include, as a final section, a brief discussion on how their results are related to ours.

Given a real numberα>−1, we consider the Laguerre second order differential operatorL_αdefined by

L_α=−y d² dy²− d

dy+y 4+α²

4y, y >0.

It is well known thatL_αis non-negative and selfadjoint with respect to the Lebesgue measure on (0,∞), furthermore its eigenfunctions are the Laguerre functions L^α_k

defined by

L^α_k(y) =

Γ(k+1)

Γ(k+α+1) 1/2

e^−y/2y^α/2L^α_k(y),

whereL^α_k are the Laguerre polynomials of typeαsee [15, p. 100] and [16, p. 7]. The orthogonality of Laguerre polynomials with respect to the measuree^−yy^αleads to the orthonormality of the family{L^α_k}^∞_k=0in L²((0,∞), dy).

Given the second order non negative and selfadjoint differential operatorL_α and its heat semigroupT_t=e^−tLα, following [12], we shall consider the Riesz poten- tials

L^−σf(x) = 1 Γ(σ)

_∞

0

t^σ−1T_tf(x)dt, σ >0, which can be derived from the identitys^−σ=Γ(σ)⁻¹_∞

0 t^σ−1e^−tsdt.

In order to define the corresponding Riesz transforms, the following first order derivatives can be introduced, see [4],

δ_y^α=√ y d

dy+1 2

√ y− α

√y

and ∂_y^β=−√ y d

dy+1 2

√ y− β

√y

= (δ^β−_y ¹)^∗. The action forα>−1 and β >0 on the Laguerre functions is given by

δ_y^α(L^α_k) =−√

kL^α_k−⁺¹₁ and ∂_y^β(L^β_k) =−√

k+1L^β−_k+1¹. Moreover

L_α− α+1

2

= (δ_y^α)^∗δ^α_y=∂_y^α+1δ^α_y.

Hence the Riesz transforms for the Laguerre function expansions are defined by R^α₊=δ^α_y(L_α)^−1/2, α >−1, and R^β₋=∂_y^β(L_β)^−1/2, β >0,

that is

R^α₊(L^α_k) =−

√k

k+(α+1)/2L^α_k−⁺¹1 and R^β₋(L^β_k) =−

√k+1

k+(β+1)/2L^β−_k+1¹. One of the main interests in studying Riesz transforms lies in their intimate con- nection with potential and Sobolev spaces. In our case it is easy to check that R^α+1₋ R^α₊=T_m,the multiplier operator associated with the sequence

m_k= k

(k+(α+1)/2)(k+α/2).

The boundedness of T_m andT_m−1 in L²((0,∞), dx) is obvious, but moreover [16, Theorem 6.3.4] gives the continuity in L^p((0,∞), dx). From this observation and the boundedness of the above Riesz transforms inL^p((0,∞), dx) given in Theorem 1 below, we may write

fp=T_m−1R^α₋⁺¹R^α₊fp≤CR^α₊fp=Cδ^α_y(L_α)^−1/2fp≤Cfp, forf satisfying_∞

0 f(y)L^α0(y)dy=0. Therefore for good enough functions, we get δ_y^αfp∼ (L_α)^1/2fp.

This equivalence is the key to define the Sobolev space associated to this Laplacian in terms of theδ^α_y derivative.

The above observation leads us to give the following definition.

Definition1. In what follows we denote by R^α the Riesz transform vector associated to L_α, that is

R^α= (R^α₊, R₋^α+1).

Since we are interested inL^p-boundedness of both Riesz transforms simultaneously we shall state our results in terms of the operator

R^α(f) = (|R^α₊f|²+|R^α+1₋ f|²)^1/2. Our main result is the following.

Theorem 1. Let α>−1 and δ be a real number. Then the operator R^α satisfies.

(a)R^αis of strong type(p, p)with respect tox^δdxas long as−αp/2<δ+1<

(α+2)p/2 and1<p<∞.

(b) R^α is of weak type (1,1)with respect to x^δdx as long as −α/2≤δ+1≤ (α+2)/2 ifα=0and0<δ+1≤1whenα=0.

(c) If α=0 and −αp/2=δ+1 for some 1<p<∞, then R^α is of restricted weak type (p, p)with respect to the measure x^δdx.

(d)If(α+2)p/2=δ+1for some1<p<∞, thenR^αis of restricted weak type (p, p)with respect tox^δdx.

We notice that our results include negative values ofδ+1 when α>0. In fact if we fixαandδthe range ofpmay be described as follows.

Forα>0:

Ifδ≤−1−α/2 it is of strong type for−2(δ+1)/α<p<∞, and ifp=−2(δ+1)/α it is of restricted weak type if p>1 and weak type whenp=1.

If−1−α/2<δ≤α/2 it is of strong type for 1<p<∞and weak type (1,1).

Ifδ >α/2 it is of strong type if 2(δ+1)/(2+α)<p<∞and restricted weak type whenp=2(δ+1)/(2+α).

Forα<0:

Ifδ≤−α/2−1 there are no boundedness results except forδ=−α/2−1 when we have weak type (1,1).

If−α/2−1<δ≤α/2 it is of strong type if 1<p<−2(δ+1)/α, weak type (1,1) and restricted weak type forp=−2(δ+1)/α.

If δ >α/2 it is of strong type for 2(δ+1)/(2+α)<p<−2(δ+1)/α and of re- stricted weak type at both end pointsp=2(δ+1)/(2+α) andp=−2(δ+1)/α.

Forα=0:

Ifδ≤−1 there are no results.

If−1<δ≤0 it is of strong type 1<p<∞and of weak type forp=1.

If δ >0 it is of strong type for δ+1<p<∞ and of restricted weak type for p=δ+1.

At this point we would like to remark that if we compare our results for the Riesz transforms with those obtained for the maximal operator of the heat semi- group ([6], [3]) we notice that besides the expected difference at p=∞ we also obtain a weaker result at the end pointp=−2(δ+1)/α. However we will show that it cannot be improved.

In fact the next theorem states that our results are sharp.

Theorem 2. Let α>−1andδ∈R. Then

(a)R^α is not strong type(1,1) with respect tox^δdx for any δ.

(b) If α=0 and −αp/2=δ+1 for some p>1, then R^α is not of weak type (p, p)with respect tox^δdx.

(c)If(α+2)p/2=δ+1for some1<p<∞, thenR^αis not of weak type (p, p) with respect tox^δdx.

2. Preliminaries

The heat diffusion semigroupe^−tLα is given by W^α(f, t, x) =

_∞

0

U_α(s, x, z)f(z)dz, wheres=(1−e^−t/2)/(1+e^−t/2),andU_α(s, x, z) is

1 2

1−s²

2s e^{−1/4(s+1/s)(x1/2−z1/2)2}e^{−1/2(s+1/s)(xz)1/2}I_α 1−s²

2s (xz)^1/2

. (2.1)

I_α is the modified Bessel function of order α(see [13] and [6]). We introduce the following notation which we will use frequently

M(s, x, z) =1 2

1−s²

2s e^{−1/4(s+1/s)(x1/2−z1/2)2}e^{−1/2(s+1/s)(xz)1/2},

and, if we also set w=((1−s²)/2s)(xz)^1/2, the kernel for the semigroup can be written as

U_α(s, x, z) =M(s, x, z)I_α(w).

To get the kernel of the Riesz–Laguerre transformR⁺_α we write R₊^α(x, z) =δ^α_x(L^α)^−1/2(x, z) =δ^α_x 1

Γ₁

2

_∞

0

e^−tLα(x, z)t^1/2dt t

=Cδ^α_x 1

0

e^{−2 log((1+s)/(1−s))Lα}(x, z)

log 1+s

1−s ₋1/2

ds 1−s². Now, using the equalityI_α(r)=αI_α(r)/r+I_α+1(r), the kernel of the Riesz transform can be written as

R₊^α(x, z) =C √

x d dx+1

2 √

x− α

√x

(L^α)^−1/2(x, z)

=C 1

0

1 2

1−s²

2s z^1/2U_α+1(s, x, z)−1 2

(1−s)²

2s x^1/2U_α(s, x, z)

×

log 1+s

1−s ₋1/2

ds 1−s²

=C 1

0

K₊^α(s, x, z)

log1+s 1−s

₋1/2

ds 1−s².

From Theorems 3 and 4, in the next section, and following the ideas of the proof of Proposition 3.2 in [14], it can be shown that for f, g∈C_c^∞((0,∞)) with disjoint compact supports, we have

R^α_±f, g = _∞

0

_∞

0

R^α_±(x, z)f(x)g(z)dx dz.

(2.2)

We leave it to the reader to check the details yielding that the kernel for the other Riesz transform is given by

R^β₋(x, z) =C 1

0

K₋^β(s, x, z)

log 1+s

1−s ₋1/2

ds 1−s²,

where

K₋^β=−1 2

1−s² 2s

z^1/2U_β+1−1+s 1−sx^1/2U_β

− β x^1/2U_β. (2.3)

3. Proof of the main result

We split the kernels of the Riesz transforms into their “local” and “global”

parts. By the local part we mean the restriction of the kernel to the set {(x, z) :x/4< z <4x}

and by the global part its restriction to the complementary set. In what follows we state some results for each of these parts and we show our main result, assuming they are true. The proofs of these theorems will be given in Sections 4 and 5, respectively.

Theorem 3. Let α>−1. We denote byK(x, z)either the kernelR^α₊(x, z)or R^α₋⁺¹(x, z). Then forxandz satisfyingx/4<z <4xwe have

(a)|K(x, z)|≤C/|x−z|; (b)

∂

∂xK(x, z) +

∂

∂zK(x, z) ≤ C

|x−z|².

Theorem 4. Let α>−1. We denote byK(x, z)either the kernelR^α₊(x, z)or R^α₋⁺¹(x, z).If we denote byT_glob the operator given by

Tglobf(x) =

(0,∞)

K(x, z)χ_{{z≤x/4}∪{z≥4x}}f(z)dz, (3.1)

then, we have

(i)Tglob is of strong type(p, p)with respect tox^δdxas long as −αp/2<δ+1<

(α+2)p/2 and1<p<∞;

(ii) T_glob is of weak type (1,1) with respect to x^δdx as long as −α/2≤δ+1≤ (α+2)/2 if α=0 and0<δ+1≤1when α=0;

(iii) if α=0 and −αp/2=δ+1 for some 1<p<∞, then T_glob is of restricted weak type(p, p)with respect to the measure x^δdx;

(iv)if(α+2)p/2=δ+1for some1<p<∞, thenT_glob is of restricted weak type (p, p)with respect tox^δdx.

Proof of Theorem 1. It is enough to prove the statements for each of the components of R^α. We consider the operator

R^α_+,locf=R^α₊f−R^α_+,globf, (3.2)

where

R₊^α_,globf= _∞

0

R^α₊(x, z)χ_{{z≤x/4}∪{z≥4x}}f(z)dz.

By Plancherel’s theorem, R^α₊ is bounded on L²((0,∞), dx). From Theorem 4 the operatorR^α_+,globis also bounded onL²((0,∞), dy). ThereforeR^α_+,locis bounded on L²((0,∞), dx), for anyα>−1. Moreover by (2.2) we have that forf, g∈C_c^∞((0,∞)) with disjoint compact supports

R^α_+,locf, g = _∞

0

4x x/4

R^α₊(x, z)f(x)g(z)dx dz.

Therefore by using Theorem 3 we have that R^α_+,loc is a “local Calder´on–Zygmund operator” as defined in [10]. Hence we can apply Theorem 4.3 (a) in [10] to conclude that R^α_+,loc is bounded onL^p((0,∞), x^δdx) for anyδ∈R and 1<p<∞. Applying again Theorem 4 we get the boundedness ofR^α₊as stated in (a), (c) and (d).

Moreover asR₊^α_,locis a “local Calder´on–Zygmund operator” we can apply The- orem 4.3 (b) in [10] getting that R^α_+,loc is bounded from L¹((0,∞), x^δdx) into L^1,∞((0,∞), x^δdx) for any δ∈R. Therefore by using Theorem 4 (ii) we get (b).

All the arguments are also valid forR^α+1₋ .

4. Local region. Proof of Theorem 3

As we have seen in (2.1) the kernelU_α(s, x, z) can be expressed by U_α(s, x, z) =M(s, x, z)I_α

1−s² 2s (xz)^1/2

, (4.1)

where M(s, x, z)=¹₂((1−s²)/2s)e^{−1/4(s+1/s)|x1/2−z1/2|2}e^{−1/4(s+1/s)(xz)1/2}.

As in the local regionx∼z, the last exponential is equivalent to e^−c(s+1/s)x. Now we state some basic inequalities about the behaviour of I_α that we shall use often, (see for example [5]).

(a) If 0≤w≤1, then I_α(w)w^α. (b) Ifw≥1 then I_α(w)e^ww^−1/2.

(c) Ifα≥−¹₂ thenI_α(w)≤C_αe^ww^−1/2 for anyw≥0.

(d) If−1<β≤α, thenI_α(w)≤CI_β(w).

From (b) it is easy to see that forw=((1−s²)/2s)(xz)^1/2≥1 andx∼z, U_α(s, x, z)≤C

1−s² 2s

1/2

e^{−1/4(s+1/s)|x1/2−z1/2|2}e^−csxx^−1/2. (4.2)

Before proceeding to prove Theorem 3 we state two lemmas which give some esti- mates useful to obtain the conditions on the size of the kernels and their derivatives.

Lemma 1. Let γ >¹₂ andε≥0. Assume that L(s, x, z) =|z^1/2−x^1/2|^{2((γ−1)+ε)}

x^1/2+εs^γ e^{−(1/4s)|x1/2−z1/2|2}. Then there exists a constant C such that forxandz in the local region

1/2

0 L(s, x, z)

log1+s 1−s

₋1/2

ds≤ C

|x−z|.

Proof. Using that log(1+s)/(1−s)∼sfor 0≤s≤1/2 and thatx∼zwe have 1/2

0

L(s, x, z)

log 1+s

1−s ₋1/2

ds

≤|z^1/2−x^1/2|^{2[(γ−1)+ε]}

x^1/2+ε

1/2 0

e^{−(1/4s)|x1/2−z1/2|2} s^γ−1/2

ds s

≤C |z^1/2−x^1/2|^{2[(γ−1)+ε]}

|x^1/2−z^1/2|^2γ−1x^1/2+ε _∞

0

e^−uu^γ−1/2−1du

≤C |z^1/2−x^1/2|^2ε

|x^1/2−z^1/2|x^1/2+ε

=C|z^1/2−x^1/2|^2ε|z^1/2+x^1/2|

|x−z|x^1/2+ε ≤ C

|x−z|, as we wanted. This completes the proof of the lemma.

Lemma 2. Let σ≥0andγ such that γ−σ >¹₂. Assume that L(s, x, z) =|z^1/2−x^1/2|^2(γ−3/2)

s^γ−σx^1+σ e^{−(1/4s)|x1/2−z1/2|2}. Then, in the local region x∼z we have

1/2 0

L(s, x, z)

log 1+s

1−s ₋1/2

ds≤ C

|x−z|².

Proof. By the estimate log((1+s)/(1−s))∼s and the fact that in the local regionx∼z, we get

1/2 0

L(s, x, z)

log 1+s

1−s ₋1/2

ds

≤C|z^1/2−x^1/2|^2(γ−3/2) x^1+σ

1/2 0

e^{−(1/4s)|x1/2−z1/2|2} s^{γ−σ−1/2}

ds s

≤C |z^1/2−x^1/2|^2(γ−3/2) x^1+σ|z^1/2−x^1/2|^{2(γ−σ−1/2)}

_∞

0

e^−uu^{γ−σ−1/2}du u

≤C |z^1/2−x^1/2|^2σ x^1+σ|x^1/2−z^1/2|²

≤C C

|x−z|²,

which completes the proof of the lemma.

Proof of Theorem 3. As we have seen in (2.3), given α>−1, the kernels of both Riesz transforms before integration in sare:

K₊^α=1 2

1−s² 2s

z^1/2U_α+1−1−s 1+sx^1/2U_α

. and

K₋^α+1=−1 2

1−s² 2s

z^1/2U_α+2−1+s

1−sx^1/2U_α+1

−α+1 x^1/2U_α+1.

Therefore, if we call b_s=¹₂(1−s²)/2s and a_s either (1−s)/(1+s) or (1+s)/(1−s), we get that we should analyse

H1=b_s(z^1/2U_β+1−a_sx^1/2U_β) and H2=−α+1 x^1/2U_α+1,

since choosing in H1, a_s=(1−s)/(1+s) and β=α, we get K₊^α, while K₋^α+1 is ob- tained taking a_s=(1+s)/(1−s), β=α+1 in H1 and then performing −H1+H2. Therefore it would be enough to get the desired estimates for those kernels. Setting ψ(s)=(log((1−s)/(1+s)))^−1/2/(1−s²) we must show that in the local regionx∼z,

(a)1

0 |H_i(s, x, z)|ψ(s)ds≤C/|x−z|,i=1,2, (b)1

0 |(∂/∂x+∂/∂z)H_i(s, x, z)|ψ(s)ds≤C/|x−z|²,i=1,2.

We split the first integral in 0≤s≤¹₂ and ¹₂≤s≤1. In the latter case we observe that all the terms in H1 and H2 are bounded in absolute value by Cx^1/2U_α with αthat may be assumed to be negative. In fact for H₁ we just use thatU_γ≤CU_γ

whenγ≤γ and thatx∼z, while forH₂ we use thatI_α+1(w)≤CwI_α(w), and so 1

x^1/2U_α+1(s, x, z) = 1

x^1/2M(s, x, z)I_α+1

1−s² 2s (xz)^1/2

≤ C

x^1/2U_α(s, x, z)1−s²

2s (xz)^1/2≤Cx^1/2U_α(s, x, z).

Now, forw=((1−s²)/2s)(xz)^1/2≤1 and ¹₂≤s≤1 we have x^1/2U_α≤C(1−s²)x^1/2

1−s² 2s (xz)^1/2

_α

e^{−c|x1/2−z1/2|2}e^−cx

≤C(1−s²)x^1/2+αe^{−c|x1/2−z1/2|2}e^−cx≤C(1−s²)^1+α

x^1/2 e^{−c|x1/2−z1/2|2}, (4.3)

where we used thatx∼zande^−cx≤Cx^−(1+α).

Similarly forw≥1, using (4.2), we get for someC >0, x^1/2U_α≤C(1−s²)^1/2e^{−c|x1/2−z1/2|2}e^−cx≤(1−s²)^1/2

x^1/2 e^{−c|x1/2−z1/2|2}. (4.4)

Therefore, lettingγ=min α+1,¹₂

, we have 1

1/2

x^1/2U_α(s, x, z)ψ(s)ds≤ C

x^1/2e^{−c|x1/2−z1/2|2} 1

1/2

(1−s²)^γ

log1+s 1−s

₋1/2

ds 1−s²

≤ C

x^1/2|x^1/2−z^1/2|≤C|x^1/2+z^1/2| x^1/2|x−z| ≤ C

|x−z|.

Now we consider 0≤s≤¹₂. This time we observe that all terms in H1 andH2 are bounded by (C/s)x^1/2U_α. For this kernel, in the regionw=((1−s²)/2s)(xz)^1/2≤1, using thatI_α(w)∼w^α, we have

C

sx^1/2U_α(s, x, z)≤ C sx^1/2

x s

1+α

e^−cx/se^{−(1/4s)|x1/2−z1/2|2}. Therefore, applying Lemma 1 withε=0 andγ=1 we are done.

The most delicate part concernsH₁ in the regionw≥1. Here we will use the cancellation by means of the asymptotic formula ([5], p. 123)

I_α(w) = e^w (2πw)^1/2

1+O_α

1 w

. First we writeH₁ as

H₁(s, x, z) =b_s(z^1/2−x^1/2)U_β+1+b_sx^1/2(U_β+1−a_sU_β) =H₁¹+H₁².

The first term follows easily since using the behavior ofI_βat infinity,H₁¹is bounded by

C|z^1/2−x^1/2| s²

s^1/2

(xz)^1/4e^{−(1/4s)|x1/2−z1/2|2}≤C|z^1/2−x^1/2|

x^1/2s^3/2 e^{−(1/4s)|x1/2−z1/2|2}, and we may apply Lemma 1 with γ=³₂ and ε=0. For the second we use the asymptotic formula to get

|I_β+1(w)−a_sI_β(w)| ≤ e^w (2πw)^1/2

(1−a_s)+C w

.

But, for both possible values ofa_s it holds that 1−a_s≤Csfor 0≤s≤¹₂. Therefore since in our situationwx/s, we get

|H₁²| ≤Cx^1/2

s |U_β+1−a_sU_β| ≤C

se^{−(1/4s)|x1/2−z1/2|2}e^−csxx^1/2 s

s x

s+s

x

≤Ce^{−(1/4s)|x1/2−z1/2|2} e^−csx

s^1/2 + 1 s^1/2x

≤Ce^{−(1/4s)|x1/2−z1/2|2} sx^1/2 , (4.5)

where we used that e^−csx≤C(sx)^−1/2 and that in the region w≥1,since w∼x/s, we have 1/x^1/2≤C/s^1/2. An application of Lemma 1 with ε=0 andγ=1 leads to the desired bound forH₁².

Finally we take care of H₂ in 0≤s≤¹₂ and w≥1. But, using that for w≥1, I_α+1(w)≤Ce^w, we get just as above

H2(s, x, z)≤ C

sx^1/2e^{−(1/4s)|x1/2−z1/2|2}, and we are done.

Now we turn to the boundedness of the derivatives ofH₁andH₂.

First we analyze the derivatives with respect toz. Straightforward calculations show that we have to deal with the following type of terms:

Case 1.

β+3 4

1−s² 2s

1 z^1/2U_β+1. Case 2.

1−s² 4s

−1−s²

4s x^1/2U_β−1 4

s+1

s

z^1/2U_β+1

. Case 3.

1−s² 4s

1 4a_s

s+1

s

x^1/2U_β−1−s²

4s a_sxz^−1/2U_β+1

.

Case4.

−1−s² 4s

β 2a_sx^1/2

z U_β. Case5.

−(α+1) 4

s²+1 s

1

x^1/2U_α+1+(α+1)1−s² 2s

1

z^1/2U_α+2+(α+1)² 1

zx^1/2U_α+1. With such notation, (∂/∂z)K₊^α is the sum of the first four terms with a_s=(1−s)/(1+s) and β=α, while (∂/∂z)K₋^α+1 is the sum of Case 5 with all the others with a_s=(1+s)/(1−s) and β=α+1. As before, for ¹₂≤s≤1, all the terms above are bounded by one of the type

C

x^1/2U_α(s, x, z)e^εx,

forε>0 small enough and we may assume thatα<0. This is clear for terms having x^−1/2,z^−1/2 orx^1/2z⁻¹. For those withx^1/2,z^1/2orxz^−1/2 we use thatx∼zand that x^1/2≤Ce^εx/x^1/2. Finally, the last term of Case 5 follows from the inequality I_α+1(w)≤CwI_α(w) valid for anyw≥0.

Now, for the above kernel we have that for w=((1−s²)/2s)(xz)^1/2≤1, x∼z and ¹₂≤s≤1,

1

x^1/2U_αe^εx≤C1−s²

x^1/2 e^εxe^−cxe^{−c|x1/2−z1/2|2}(1−s²)^αx^α

≤C(1−s²)^1+αx^1/2e^−cx x^1−α

1

|x^1/2−z^1/2|^2(1+α)

≤C(1−s²)^1+α x^1−α

|x^1/2−z^1/2|^−2α

|x−z|² |x^1/2+z^1/2|²

≤C(1−s²)^1+α

|x−z|² , as long as 0<ε<cand−1<α≤0.

On the other hand ifw≥1 we get 1

x^1/2U_αe^εx≤C(1−s²)^1/2

x e^{−c|x1/2−z1/2|2}e^εxe^−cx

≤C(1−s²)^1/2 x

1

|x^1/2−z^1/2|²≤C(1−s²)^1/2

|x−z|² .

In both cases we arrive at the desired estimate since the integrals againstψ(s) are finite. Now we proceed with the case 0≤s≤¹₂. Here again we notice that all the