MAXIMAL OPERATORS FOR THE HOLOMORPHIC LAGUERRE SEMIGROUP

MAXIMAL OPERATORS FOR THE HOLOMORPHIC LAGUERRE SEMIGROUP

EMANUELA SASSO

Abstract

For eachpin [1,∞)letEp denote the closure of the region of holomorphy of the Laguerre semigroup{Mt^α:t >0}onL^pwith respect to the Laguerre measureµα. We prove weak type and strong type estimates for the maximal operatorf →sup{|M_z^αf|:z∈E_p}. In particular, we give a new proof for the weak type 1 estimate for the maximal operator associated toMt^α. Our starting point is the well-known relationship between the Laguerre semigroup of half-integer parameter and the Ornstein-Uhlenbeck semigroup.

1. Introduction

The purpose of this paper is to analyse a class of maximal operators associ- ated to the holomorphic Laguerre semigroup on the half-lineR₊. We shall be working with the Laguerre probability measureµα of type α, with α ≥ 0, whose density isµα(x)= _(α+^xαe−x₁₎ with respect to the Lebesgue measure. The Laguerre semigroup is the symmetric diffusion semigroup{M_t^α : t ≥ 0}on (R₊, µα), whose infinitesimal generator is the differential operator

Lαu(x)= −x ∂²

∂x²u(x)−(1+α−x) ∂

∂xu(x), i.e.u(x, t)=e^−tLαf (x)satisfies the Cauchy problem





∂

∂tu= −Lαu, u(x,0)=f (x).

By virtue of Stein’s maximal theorem [12] the operator M^αf (x)=sup

t≥0|M_t^αf (x)|

is bounded on L^p(µα), for 1 < p ≤ ∞. B. Muckenhoupt proved thatM^α is of weak type(1,1)in the one-dimensional case [9]. This result has been

Received February 13, 2004; in revised form September 27, 2004.

extended by Dinger to the Laguerre semigroup, of orderα=(α1, . . . , αd)on R^d₊, withd >1, defined as the tensor product ofdone-dimensional Laguerre semigroups of orderαi, withi=1, . . . , d [3].

The Laguerre semigroup can be extended to complex values of the parameter t. This is achieved by analytic continuation of theL²(µα)-functiont →M_t^αf to the half-plane{z ≥0}. Since M_t^α has an integral kernelmα,t(x, y)with respect to the Laguerre measure, it is equivalent to substitutet by a complex variable in the expression of the kernel (for more details, see Section 2). The resulting operatorM_z^α is well defined as a bounded operator onL²(µα)for all z, withz≥0 andα≥0. For anyp, with 1≤p≤ ∞, define

E_p= {x+iy:|siny/2| ≤tanφpsinhx/2},

whereφp=arccos|1−2/p|. The above represents the region of holomorphy of M_z^αacting onL^p(µα), i.e.M_z^αis bounded onL^p(µα)if and only ifzbelongs to E_p(see [10]). The mapz→M_z^αfromE_pto the space of the bounded operators onL^p(µα)is continuous in the strong operator topology and its restriction to the interior ofE_pis analytic. Therefore the maximal operator

(1) Mα,pf (x)= sup

z∈E_p|M_z^αf (x)|

is well defined. The aim of this paper is to investigate the boundedness prop- erties inL^q(µα)of this maximal operator, wheneverα ≥0. It is well known that weak type estimates for Mα,p are a key tool to investigate the almost everywhere convergence ofM_z^αf toM_z^α₀f asztends toz0, forf inL^q(µα).

We remark thatMα,1is the maximal operator for the heat kernelmα,t, which is known to be of weak type 1 and of strong typepfor eachp > 1 by the aforementioned results of Stein and Muckenhoupt. For 1 < p <2 we shall prove that the operatorMα,p is of strong typeq for eachq in(p, p)and of weak typepifp < _α+^2α+_3/²₂. On the other hand forp > _α+^2α+_3/²₂it is not of weak typep. MoreoverMα,2is not of weak type 2.

We follow the same strategy adopted by [6] to study the maximal oper- ators associated to the holomorphic Ornstein-Uhlenbeck semigroup. By the periodicity properties of the semigroupM_z^α, we may restrict the parameterz to the regionF_p = {z ∈ E_p : 0 ≤ z ≤ π}. To obtain the positive results we decompose the operator into a “local” part, whose kernel is supported in a sort of neighbourhood of the diagonal, and in a “global” part. For the negative result, we provide counterexamples by analysing the behaviour ofmα,zon the boundary ofE_p. To be more specific, a critical point iszp= |log(p−1)|+iπ. Therefore it is natural to investigate the smaller maximal operator defined by taking (1) only over the set obtained deleting fromE_pa small neighbourhood of

the pointzp. Observe that at this point the operatorM_z^α_p, withα∈N/2−1, may be reduced to the Fourier transform fromL^p(Rⁿ, dx)to Ł^p(Rⁿ, dx)acting on radial functions.

The paper is organized as follows. In Section 2 we recall some basic prop- erties of the Laguerre semigroup and we decompose the maximal operators in

“local” and “global” parts. In Section 3 we estimate the local part while Sec- tion 4 is devoted to showing the positive results regarding the global parts of the maximal operators. Finally the negative results will be proved in Section 5.

2. Preliminaries and statement of results

The Laguerre semigroup onR₊is the family of integral operators{M_t^α :t ≥0} defined by the following kernel expressed in terms of the standard Bessel functionJα.

(2) mα,t(x, y)=(α+1)(1−e^−t)⁻¹exp

−e^−t(x+y) 1−e^−t

(−xye^−t)^−α/2Jα

2(−xye^−t)^1/2 1−e^−t

,

with respect to the Laguerre measureµα (see, for instance, [3]). Since this kernel admits an analytic continuation on the half-plane z ≥ 0, it is easy to see that the Laguerre semigroup has analytic continuation to a family of operators{M_z^α :z≥0}fromD(R₊)toD(R₊)such that

(3) M_z+^α 2iπf (x)=M_z^αf (x), M_z^αf (x)=M_z^αf (x).

By [10] the operatorM_z^α extends to a bounded operator onL^p(µα), for 1 ≤ p ≤ ∞ andα ≥ 0, if and only if z belongs to the set E_p, defined in the previous section. The setE_pis a closed 2πi-periodic subset of the right half- plane. Moreover, if 1/p+1/p = 1, thenE_p = E_p andE_p ⊂ E_q, for each 1< p < q <2. In particular, the end-point cases areE1 = {x+ikπ : x ≥ 0, k∈Z}andE₂= {z:z≥0}.

Our purpose is to investigate the boundedness of the maximal operatorMα,p, defined in (1), onL^q(µα), for 1 ≤ q ≤ ∞. It turns out that we may restrict the parameterzto the “fundamental domain”F_p = {z ∈E_p : 0≤ z ≤π}. Indeed, by (3) and the properties of the region E_p, it is easy to see that the maximal operator

M_α,p^∗ f (x)= sup

z∈F_p|M_z^αf (x)|, andMα,pare simultaneously of weak or strong type.

Letµ˜αbe the Borel measure onR₊with densityµ˜α(x)= 2^x2α+1e−x

2

(α+1) , with respect to the Lebesgue measure, and consider the map defined on test functions by

(4) f (x)=f (x²).

The mapis an isometry betweenL^q(µα)andL^q(µ˜α)and betweenL^q,∞(µα) andL^q,∞(µ˜α). DefineM_z^α =M_z^α−1. It is quite straightforward to see that

mα,z(x, y)=mα,z(x², y²)is the integral kernel ofM_z^α. Clearly we may reduce the problem to the study of the boundedness ofM^∗_α,ponL^q(µ˜α), defined by

M^∗_α,pf (x)= sup

z∈F_p|M_z^αf (x)|.

More generally, we shall consider the family of maximal operatorsM^∗_α,p,σ defined as follows. Letzpdenote the point on the boundary ofF_pwith imagin- ary partπ. For eachσ, with 0≤σ <|zp|, letF_p,σ = {z∈F_p:|z−zp| ≥σ}. Define

M^∗_α,p,σf (x)= sup

z∈F_p,σ|M_z^αf (x)|.

We are now ready to state our results. SinceE_p =E_p, we only need study the boundedness ofM^∗_α,p,σ for 1≤p≤2.

Theorem2.1. Forα ≥0, the following hold:

(1) The operatorM^∗_α,1is of weak type 1 and of strong typeqfor everyq in (1,∞];

(2) Let1 < p < 2. The operatorM^∗_α,pis of strong type q wheneverp <

q < p;

(3) Let1< p <2. The operatorM^∗_α,pis of weak typep, whenp < _α+^2α+_3/²₂, but it is not of weak typep, whenp > _α+^2α+_3/²₂;

(4) If1< p <2and0< σ <|zp|, the operatorM^∗_α,p,σ is of weak typep andp, but not of strong typep;

(5) Ifα /∈ ^2N₂⁻¹, the operatorM^∗_α,2,σ, with0≤σ < π, is not of weak type 2.

Remark2.2. We shall see that the results of statements (1) and (2) can be extended to thed-dimensional case (see Remark 4.6). Moreover, every negative result holds also in higher dimension. Indeed, by restricting the operators to functions which depend only on one variable inR^d₊, one sees that it suffices to consider the one-dimensional cased = 1. In particular, by (3),M^∗_α,pcannot be of weak typepwheneverp >min_i=1,...,d_2αi+2

αi+3/2

, withα=(α1, . . . , αd).

Remark2.3. Recall that the holomorphic Ornstein-Uhlenbeck semigroup acting on square integrable functions on Rⁿ, with respect to the Gaussian measure dγ (x)=π^−n/2e^−|x|2dx, is the family of operatorsHz, withz≥0, defined by

Hzf (x)=

Rⁿhz(x, y)f (y)dγ (y), if z=iπZ, Hikπf (x)=f ((−1)^kx), for k∈Z.

Here

hz(x, y)=(1−e^−2z)^−n/2exp

−|e^−zx−y|² 1−e^−2z

e^|y|2

is the Mehler kernel with respect to the Gaussian measureγ. It is well known that the region of holomorphy of the Ornstein-Uhlenbeck semigroup onL^p(γ ) is 2E_p(see [6]). The maximal operators

H_p^∗f (x)= sup

z∈2E_p

|Hzf (x)|,

H_p,σ^∗ f (x)= sup

z∈2F_p,σ|Hzf (x)|

have been investigated in [6] and [11]. In particular, in [6] the authors have proved the analogue of our Theorem 2.1 for the operatorsH_p^∗andH_p,σ^∗ . Note however that, while the operatorM_α,p^∗ is of weak typepforp < _α+^2α+_3/²₂, the operatorH_p^∗is never of weak typep, for allp >1. Since whenα=n/2−1 andn >1, the Laguerre operator can be interpreted asHzacting on the radial functions onRⁿ (see [7]), we obtain that forp < _n+²ⁿ₁, the maximal operator H_p^∗, restricted to the space of radial functions, is of weak typep.

To investigateM^∗_α,p, we essentially adopt the same techniques of [6]. We split the operator in a “local” part and the remaining or “global” part. To describe this decomposition, first we must write the kernel ofM_z^αas an average over [−1,1] of a family of kernels mα,z(·,·, s) depending on an additional variabless in [−1,1]. Indeed, by using the integral form of Bessel functions, forα ≥ 0 (see, for instance, [4, p. 15]), and by considering the action of the isometry, we may write the kernel ofM_z^α as

mα,z(x, y)= ¹

−1mα,z(x, y, s)ds, where

(5)

mα,z(x, y, s)=(1−e^−z)^−α−1e^{12ez/12+1(x2+y2+2xys)−21ez/12−1(x2+y2−2xys)}$α(s)

and $α(s) = _(α+^(α+11)

2)√π(1 − s²)^α−1/2. It will be useful to observe that

[−1,1,]$α(s)ds = 1. To split the operator we introduce two sets: the “local”

region L=

(x, y, s)∈R₊×R₊×[−1,1] :

(x²+y²−2xys)≤min

1, 1

x²+y²+2xys

,

and its complementG, the “global” region. This choice is suggested by the de- scription of the corresponding local region in polar coordinates in the Ornstein- Uhlenbeck case [6]. Observe that the diagonal{(x, x,1)}, is contained inL, i.e.Lis a neighbourhood of the diagonal inR₊×R₊× {1}. The local and global parts of the operatorM^∗_α,p,σ are defined by

M_α,p,σ^loc f (x)= sup

z∈F_p,σ

_R

+

1

−1

mα,z(x, y, s)χL(x, y, s)dsf (y)dµ˜α(y) , M_α,p,σ^gl f (x)= sup

z∈F_p,σ

_R

+

1

−1

mα,z(x, y, s)χG(x, y, s)dsf (y)dµ˜α(y) . Clearly M^∗_α,p,σf (x)≤M_α,p,σ^loc f (x)+M_α,p,σ^gl f (x),

we shall study the operatorsM_α,p,σ^loc andMα,p,σ^gl separately. First, however, it is convenient to simplify the expression ofmα,z(x, y)by means of the change of variable

z=τ(ζ )=2 log1+ζ 1−ζ.

The same change of variable, without the factor 2 in front of logarithm, was introduced in [6], to which we refer the reader for the properties of the mapτ. Here we only recall thatτ is a biholomorphic transformation of a neighbour- hood of 0, which maps the rayR₊e^iφp onto∂E_p∩ {z ∈ C : 0 ≤ z < 2π}. Therefore the heat kernel becomes

mα,τ(ζ )(x, y)=(α+1)(1+ζ )^2+2α (4ζ)^1+α exp

−(x²+y²)(1−ζ )² 4ζ

ixy

4 1

ζ −ζ _−α

Jα

ixy

2 1

ζ −ζ

.

Forα≥0, we may also write

mα,τ(ζ )(x, y)= ¹

−1

mα,τ(ζ)(x, y, s)ds,

where (6)

mα,τ(ζ)(x, y, s)= (1+ζ )^2+2α

(4ζ)^1+α e^x2+y22e⁻

ζ

4(x²+y²+2xys)+_4ζ¹(x²+y²−2xys)

$α(s).

Remark2.4. We prove now some identities which will be useful in the sequel. Assume thatf, g are in L¹(R₊, mα), wheremα is the measure with density

(7) mα(y)=e^y2µ˜α(y).

Observe that the measuremα is simply proportional to a power of y times Lebesgue measure.

We define thegeneralized translationas τ_y^αf (x)=

[−1,1]f

x²+y²−2xys

$α(s)ds,

and thegeneralized convolutionoff andgas (8) f#_αg(x)=

R₊ [−1,1]

f

x²+y²−2xys

g(y)$α(s)dsdmα(y).

Forα = ⁿ₂ −1, these correspond to the average over the sphere of a radial funcion and to the convolution of radial functions inRⁿ, respectively. It is well known that generalized translations and generalized convolution share many of the properties of ordinary translations and convolution inRⁿ[8]. In particular

(i) the functiony→τ_y^αf is continuous inL^p(mα); (ii) f#_αg=g#_αf.

Namely, by the change of variables=cosθ, we have (9)

f#_αg(x)=

R+ [0,π]f

x²+y²−2xycosθ

g(y)(sinθ)^2αdsdmα(y).

For eachx¯ in R², letx denote the absolute value ofx¯. Ifx¯ andy¯ are inR², letθ be the angle between the nonzero vectorsxandy. Interpreting (9) as an integral onR²in spherical coordinates, we obtain that

f#_αg(x)=

R²f (| ¯x− ¯y|)g(| ¯y|)

1− x·y

|x||y|

2_α

|y|^2α2 dy (α+1/2)√

π.

Now by the further change of variables,y−x=w, we have f#_αg(x)=

R²f (| ¯w|)g(| ¯x− ¯w|)

1−

x·w

|x||w|

2_α

|w|^2α2 dw (α+1/2)√

π

=g#_αf (x).

Namely, by the sine theorem

1− x·y

|x||y|

2

|y|²=

1−

x·w

|x||w|

2

|w|².

This concludes the proof of item (ii).

Moreover, by choosingg=1,f#_αgis well defined for a.e.x∈R₊and

R₊ [−1,1]

f

x²+y²−2xys

$α(s)dsdmα(y)=

R₊f (y)dmα(y).

3. Results for the “local” part

In this section we shall prove that M_α,p,σ^loc is of weak type 1 and of strong typeq, for eachq in (1,∞] andσ ≥ 0. SinceM_α,p,σ^loc ≤ M_α,p,^loc0 = M_α,p^loc, it is enough to consider the latter operator. In the following, we will use the measuremα defined in (7). Moreover, for eachα≥0 andt >0, letkα,t(x, y) be the function

(x, y)→ ¹

−1

kα,t(x, y, s)$α(s)ds,

wherekα,t(x, y, s)=(4t)^−α−1exp

−^x2+y2₄⁻_t^2xys .

Lemma3.1. Suppose thatα≥0. Let{T_t^α :t >0}be the family of integral operators, defined by

T_t^αf (x)=

R+

kα,t(x, y)f (y)dmα(y),

on C_c^∞(R₊). Then {T_t^α : t > 0} is a diffusion semigroup on (R₊, mα). Moreover the maximal operator

T^∗f (x)=sup

t>0|T_t^αf (x)|

is of weak type(1,1).

Proof. LetH¹(R₊, mα)denote the space of all functionsf, such that both f and its distributional derivativefare inL²(mα). LetQα be the quadratic form, defined forf inH¹(R₊, mα)by

Qα(f )= ^∞

0

d dxf

²(x)dmα(x).

The formQα with dense domainH¹(R₊, mα) is closed, soQα is the form of a self adjoint operator −1α ≥ 0, which onC_c²(R₊) coincides with the differential operator

− d²

dx² − (2α+1) x

d dx.

We claim that−1α is the infinitesimal generator of{T_t^α : t >0}. Indeed, by Remark 2.4 we have that

(10)

R₊kα,t(x, y)dmα(y)=(4t)^−α−1

R₊ [−1,1]

exp

−w² 4t

$α(s)dsdmα(w)

=

R+

e^−w2dmα(w)=1,

and for eachδ >0 it is quite simple to prove that

(11) lim

t→0⁺

∞

δ kα,t(x, y)dmα(y)=0. By (10) and (11), it is easy to prove that for everyf

t→lim0⁺ R₊kα,t(x, y)f (y)dmα(y)=f (x)

inL²(mα). So the claim is proved once we verify thatut(x)=T_t^αf (x)solves the Cauchy problem

∂tut = −1αut

u⁰=f.

We have that ∂tkα,t(x, y) = −1αxkα,t(x, y). Indeed, by a straightforward calculation, we get that

[−1,1]

[∂tkα,t(x, y, s)+1αxkα,t(x, y, s)]$α(s)ds=0.

Now by Lebesgue dominated convergence theorem we can differentiate under the integral sign. Hence∂tut = −1αut. This proves the claim.

We observe that, by (10)T_t^α can be extended to a contraction onL¹(mα) and, by duality, to a contraction onL^∞(mα). By interpolation we get thatT_t^α is a diffusion semigroup. Notice thatt → T_t^αf is a continuous function on R₊for each f ∈ L²(mα). Thus, since the supremum overt > 0 coincides with the supremum over all rational t > 0, T^∗f is a measurable function.

By the Littlewood-Paley-Stein theory, the maximal operatorT^∗ is bounded onL^p(mα), for 1 < p < ∞. It remains now to prove the weak type(1,1) boundedness. LetEt be the ergodic means of{T_t^α :t >0}, i.e.

Etf = 1 t

t

0 T_σ^αfdσ,

onL^p(mα)∩L²(mα)and witht >0. The associated ergodic maximal operator is defined by

E^∗f =sup

t>0|Etf|,

onL^p(mα)∩L²(mα). The Hopf-Dunford-Schwartz ergodic maximal theorem asserts that

(12) mα{x∈R₊:E^∗f (x) > λ} ≤ 2

λf1, ∀λ >0, ∀f ∈L¹(mα).

Forf ≥0, sinceT_t^αis positivity preserving, by Fubini’s theorem we have that E2tf (x)≥ 1

2t

2t

t T_σ^αfdσ

= 1 2t

2t

t R₊kα,σ(x, y)f (y)dmα(y)dσ

=

R+

1 2t

2t

t kα,σ(x, y)f (y)dσdmα(y)

≥

R+

1 2t

2t

t (4σ )^−α−1dσ

[−1,1]

e^{−x2+y24t−2xys}$α(s)dsf (y)dmα(y)

=CαT_t^αf,

for some positive constantCα. So, for anyf inL¹(mα),

(13) T^∗f ≤sup

t>0T_t^α|f| ≤Cαsup

t>0Et|f|.

Now the weak type(1,1)estimate forT^∗follows from (12) and (13) and this concludes the proof.

Remark3.2. By the definition of generalized convolution,T_t^αf =f#_αk, withk(x)= (4t)^−α−1e^−x4t2. So whenα = ⁿ₂ −1, withn∈ Nandn > 1, the operatorT_t^α corresponds to the heat semigroup acting on radial functions of Rⁿ.

Lemma3.3. For eachp ∈ [1,2), there exists a constant C such that for everyt in(0,1]and(x, y, s)inL

sup

|φ|≤φp

|mα,τ(te^iφ)(x, y, s)|

≤Ct^−α−1e^y2exp

−cosφp

4t (x²+y²−2xys)

$α(s).

Proof. By (6) we have

|mα,τ(te^iφ)(x, y, s)| ≤Ct^−α−1e^x2+y22 exp

−cosφ

4t (x²+y²−2xys)

$α(s).

Since, if(x, y, s)∈L, then x²−y²≤

(x²−y²)²+4(1−s²)x²y²

≤

(x²+y²+2xys)(x²+y²−2xys)

≤1,

then we may majorisee^{x2+y2 2} byCe^y2 and the result follows.

Theorem3.4. For eachp ∈ [1,2), the operatorM_α,p^loc is of weak type1 and of strong typeq, whenever1< q≤ ∞.

Proof. For any fixedf ≥0, Lemma 3.3 yields (14)

M_α,p^locf (x)≤Csup

t>0t^−α−1

R₊ [−1,1]

exp

−cosφp

4t (x²+y²−2xys)

χL(x, y, s)$α(s)dsf (y)dmα(y).

We claim thatM_α,p^loc is bounded onL^∞(µ˜α). Namely, by Remark 2.4, we get

|M_α,p^locf (x)|

≤Cf∞sup

t>0t^−α−1

R+ [−1,1]

exp

−cosφp

4t w²

$α(s)dsdmα(w).

Sinceα ≥0, the integral t^−α−1

R+ [−1,1]

exp

−cosφp

4t w²

$α(s)dsdmα(w)

is finite and bounded by a constant independent ofxandt. This concludes the proof of our claim.

It only remains to prove thatM_α,p^loc is also of weak type 1. By Lemma 3.3, our operator is bounded by

Wf (x)=sup

t>0

t^−α−1

R+

e^{−cosφpx2+y24t−2xys}χL(x, y, s)$α(s)dsf (y)dmα(y), whose kernel is supported in the local region. Since, by Lemma 3.1,W is of weak type(1,1)with respect to the measuremα,M_α,p^loc is of weak type(1,1), with respect to the same measure. We consider the vector-valued operatorS given by

Sf (x)=

R+ [−1,1]mα,t(x, y, s)χL(x, y, s)dsf (y)dµ˜α(y)

t∈Q

.

It is clear that M_α,p^locf (x) = Sf (x)5∞. In order to prove that M_α,p^loc maps L¹(µ˜α) into weak-L¹(µ˜α), it is enough to prove that S maps L¹(µ˜α) into weak-L¹₅^∞(µ˜α). Now the conclusion follows by applying toSa vector-valued version of the arguments in Section 5.2 of [10] (see also [5, Section 5]).

4. Results for the “global” part

In this section we shall estimate the global maximal operators Mα,p,σ^gl , for 1≤p <2 andσ ≥0. Our estimates are a consequence of the inequality (15) sup

|φ|≤φp

|m_α,τ(te^iφ₎(x, y, s)|

≤Ct^−α−1e^{x2+y2 2−cos4φp(t(x2+y2+2xys)+1t(x2+y2−2xys))}$α(s), for allt in(0,1] and all(x, y, s)inR₊×R₊×[−1,1], which follows easily from (6). First we give two different expressions for the right hand side of this inequality. Since cosφp= _p² −1, we have that

e^{x2+y22−cos4φp}(^{t(x2+y2+2xys)2+1}t(x²+y²−2xys))

=exp x²

p +y²

p − cosφp

4t Qt(x, y, s)

=exp x²

p +y²

p − cosφp

4t Q−t(x, y, s)

,

whereQτ(x, y, s)is a quadratic form inx, ydefined by

Qτ(x, y, s)=(1+τ)²x²−2xys(1−τ²)+(1−τ)²y².

For eachxinR₊, consider the sectionG(x)= {(y, s):(x, y, s)∈G}and for every fixedδ >0, define

J±(x, t)=

G(x)exp

−δ

tQ±t(x, y, s)

$α(s)dsdmα(y).

Lemma4.1. For eachδ >0and(x, y, s)∈G, (i) there exists a constantCsuch that

sup

0<t≤1t^−α−1exp

−δ

tQt(x, y, s)

≤C

(1+x)^2α+2∧(x²(1−s²))^−α−1

; (ii) for eachpin(1,∞)and eachηin(0,1), there exists a constantCsuch

that sup

0<t≤1−ηt^−p(α+1)exp

−δ

tQ±t(x, y, s)

J_±^p/p(x, t)

≤C

(1+x)^2α+2∧(x²(1−s²))^−α−1 .

Proof. We claim that for eachηin(0,1)there exists a positive constantC such that for allx, y, sandt ≥ −1+η

(16) Qt(x, y, s)≥Cx²(1−s²).

Moreover for all(x, y, s)∈Gandt < (1+x)⁻²/8

(17) Qt(x, y, s)≥C 1

(1+x)².

Assuming the claim for the moment, we prove the lemma. To obtain (i) first, we observe that (16) implies

t^−α−1exp

−δ

tQt(x, y, s)

≤C(Qt(x, y, s))^−α−1≤C(x²(1−s²))^−α−1. Next, we observe that on the one hand, ift ≥ (1+x)⁻²/8, it is enough to majorise the exponential by 1, to get

t^−α−1exp

−δ

tQt(x, y, s)

≤C(1+x)^2α+2.

On the other hand, ift < (1+x)⁻²/8, by (17) t^−α−1exp

−δ

tQt(x, y, s)

≤C(1+x)^2α+2

and this concludes the proof of (i). Next we prove (ii). By Remark 2.4, we have

(18) J±(x, y)≤C

R₊e^{−δt(1−t)2w2}dmα(w)≤Ct^α+1, since 1∓t > η. Hence by (16)

t^−p(α+1)exp

−δ

tQ±t(x, y, s)

J_±^p/p(x, t)≤C(x²(1−s²))^−α−1. To prove the other inequality, we majorise the exponential by 1 and we consider separately the casest ≥(1+x)⁻²/8 andt < (1+x)⁻²/8. In the first case, by (18) we obtain that

t^−p(α+1)J_±^p/p(x, t)≤C(1+x)^2α+2. In the second case, by (17) and Remark 2.4 we get that

J_±^p/p(x, t)≤C

w>C(1+x)⁻¹e^−w2/tdmα(w) _p/p

≤Ct^p/p(α+1)√

t(1+x)2α+2

, which, again, implies (18).

We must finally prove the claims. To prove (16), consider two vectorsv¯and

¯

winR², such that|¯v| =(1+t)x,| ¯w| =(1−t)yand the angle betweenv¯and

¯

wis arccoss. Then|Qt(x, y, s)|^1/2 = |¯v− ¯w|is minorised by the length of the projection ofv¯on the direction orthogonal tow¯. This gives the inequality

|Qt(x, y, s)|^1/2≥(1+t)x(1−s²)^1/2, which implies (16). For (17), it is quite straightforward to verify that, if(x, y, s)∈G, then

(19) (x²+y²−2xys) > 1 4

1 (1+x)².

Namely, wheny≥1+x, this follows fromx²+y²−2xys ≥1. Ify ≤1+x, we have that

(x²+y²+2xys)^1/2≤x+y ≤2(1+x),

and so min(1, (x²+y²+2xys)^1/2) > ¹_2(1+x)¹ . Since(x, y, s)∈G, this implies (19). Now to obtain (17), observe that, ift < (1+x)⁻²/8, then 1−t > ⁷₈and

Qt(x, y, s)^1/2≥(1−t)(x²+y²−2xys)^1/2−2tx

≥ 7 16

1 1+x − 1

4 1

1+x ≥ 3 16

1 1+x,

where the first inequality follows from the geometric interpretation of Qt(x, y, s). This concludes the proof of the claims and of the lemma.

These estimates imply that the operatorM_α,^gl1is of weak type 1. This result is known (see [9]), but here we give a new proof, based on Lemma 4.2 below, which will also be useful to studyMα,p,σ^gl . LetT be the operator onL¹(µ˜α) defined by

Tf (x)=Fα(x)

R₊f (y)dµ˜α(y), where

Fα(x)=e^x2

[−1,1](1+x)^2α+2∧(x²(1−s²))^−α−1$α(s)ds.

Lemma4.2. The operatorT is of weak type1.

Proof. It is enough to prove that the functionFα

x →Fα(x)=e^x2

[−1,1]

(1+x)^2α+2∧(x²(1−s²))^−α−1$α(s)ds

is inL^1,∞(µ˜α). We can choose a constantC0, such thatλ > C0implies that the positive zeror0of the function

ξ →e^ξ2(1+ξ)^2α+2−λ

is greater than 1. Fixλ > 0 and letEλ = {x : Fα(x) ≥ λ}. We must prove thatµ˜α(Eλ) ≤ ^C_λ. Sinceµ˜α is a finite measure it is enough to assume that λ > C0. Moreover, sinceFα(x) ≤ e^x2(1+x)^2α+2, Eλ does not intersect the ballB = {x < r0}. Finally, we need to consider the intersection of Eλ

with the ring R = {r0 < x < 2r0} only. In fact, the elementary relation _∞

M e^−ρ2ρ^2α+1dρ∼e^−M2M^2α forM >1 implies

˜

µα{x >2r⁰} = ^∞

2r0

2x^2α+1

(α+1)e^−x2dx

≤Ce^−4r02r0^2α ≤Ce^−r02(1+r0)^2α ≤Cλ⁻¹.