Sharp estimates for potential operators associated with Laguerre and Dunkl-Laguerre expansions

We study potential operators associated with Laguerre function expansions of convolution and Hermite types, and with Dunkl-Laguerre expansions. We prove qualitatively sharp estimates of the corresponding potential kernels. Then we characterize those $1 \le p,q \le \infty$, for which the potential operators are $L^p-L^q$ bounded. These results are sharp analogues of the classical Hardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and Dunkl-Laguerre settings.


Introduction
In recent years the study of potential theory for 'Laplacians' associated with classical orthogonal expansions attracted considerable attention. The model case of the Riesz potentials I σ = (− ) −σ , where denotes the Euclidean Laplacian in R d , d ≥ 1, is related to continuous expansions with respect to the system {exp(2πi ·, ξ ) : ξ ∈ R d }.
The L p − L q boundedness of I σ , 0 < σ < d/2, is characterized by the celebrated Hardy-Littlewood-Sobolev theorem.
There is a wide variety of papers and results pertaining to potential operators in numerous orthogonal expansion settings, both continuous and discrete. For instance, Anker [1] studied potential operators in the framework of non-compact symmetric spaces and his analysis was based on sharp pointwise estimates of the corresponding kernels. In [3] Bongioanni and Torrea investigated potential operators related to the harmonic oscillator H = − + x 2 , which plays the role of a Laplacian in the context of multi-dimensional Hermite function expansions. Some complementary comments on that research are contained in [9,Section 2]. More recently, in [10] it was shown that the L p − L q bounds obtained in [3] for the potential operator I σ = H −σ are in fact sharp in the sense of admissible p and q. This was achieved as a consequence of qualitatively sharp estimates for the integral kernel of I σ established also in [10]. A thorough study of potential operators associated with classical one-dimensional Jacobi and Fourier-Bessel expansions was furnished by Nowak and Roncal [7]. The corresponding analysis is also based on sharp estimates of the potential kernels proved in that work. Similar questions in the settings of Hankel and Hankel-Dunkl transforms have just been studied by the authors in [11].
Potential operators related to multi-dimensional Laguerre operators in R d + , and to the Dunkl harmonic oscillator in R d with the underlying reflection group isomorphic to Z d 2 , were investigated by the authors in [9]. Recall that the latter 'Laplacian' is a differentialdifference operator, and its eigenfunctions express via certain Laguerre functions. Hence the associated expansions are sometimes referred to as Dunkl-Laguerre expansions. The aim of [9] was to prove L p − L q bounds for the considered potential operators for a possibly wide range of p and q. Another objective was to obtain in a similar spirit two-weight L p − L q bounds, with power weights involved. All these results in [9] were derived as indirect and somewhat tricky consequences of analogous theory for I σ , and under the restriction α ∈ [−1/2, ∞) d on the multi-parameter of type.
The present paper is motivated by the natural question to what extent the results of [9] are optimal in the sense of admissible p and q. Further motivation comes from a related problem, but certainly of independent interest, of describing the behavior of the relevant potential kernels via pointwise estimates. Finally, yet another motivation follows from a desire to get rid of the above mentioned restriction on α. All these inspirations found a positive outcome. For technical reasons, we consider only d = 1 and thus work in dimension one, otherwise the analysis we present would become much more sophisticated. Then we investigate the settings from [9], that is, according to the terminology used in [14], the situations of Laguerre function expansions of convolution and Hermite types, and Dunkl-Laguerre expansions (see Section 2 for the definitions), with no artificial restrictions on α imposed. We prove qualitatively sharp estimates for the relevant potential kernels (Theorems 2.1 and 2.4). Then we characterize those 1 ≤ p, q ≤ ∞, for which the potential operators are L p − L q bounded (Theorems 2.2, 2.3 and 2.6). In particular, it follows that the unweighted L p − L q bounds from [9] are in fact sharp, at least in the one-dimensional case.
It is remarkable that our present results enable further research which is no doubt of interest, but beyond the scope of this work. Let us mention here the following issues: • characterization of weak type and restricted weak type inequalities for the potential operators (see [ We note that the results of this paper concerning the analogues of the classical Riesz potentials contain implicitly parallel results for counterparts of the classical Bessel potentials (Id − ) −σ . This is because spectra of the 'Laplacians' considered are already separated from 0. By tracing the proofs of Theorems 2.1 and 2.4 it is straightforward to see that integral kernels of Bessel potentials in the Laguerre and Dunkl-Laguerre settings also satisfy the qualitatively sharp bounds stated there. Consequently, Theorems 2.2, 2.3 and 2.6 remain true if the Laguerre and Dunkl-Laguerre Riesz potentials are replaced by analogues of Bessel potentials in these contexts.
Finally, we remark that although the present framework is one-dimensional, it has, at least in the setting of Laguerre expansions of convolution type, a multi-dimensional background. More precisely, if α = n − 1, n ≥ 1, then the context of Laguerre function expansions of convolution type is related to a 'radial' analysis in C n equipped with twisted convolution, see [13,14] for details. Continuing this line of thought, we note that the system of Laguerre functions of Hermite type also has a multi-dimensional connection, since it consists of eigenfunctions of the Hankel transform.
The paper is organized as follows. In Section 2 we briefly introduce the settings to be investigated and state the main results (Theorems 2.1-2.4 and 2.6). The corresponding proofs are contained in the two succeeding sections. In Section 3 we show qualitatively sharp estimates for the relevant potential kernels. Section 4 is devoted to characterizing L p − L q boundedness of the Laguerre and Dunkl-Laguerre potential operators.
Throughout the paper we use a standard notation, which is consistent with that used in [9,10]. In particular, we write X Y to indicate that X ≤ CY with a positive constant C independent of significant quantities. We shall write X Y when simultaneously X Y and Y X. Furthermore, X Y exp(−cZ) means that there exist positive constants C, c 1 , c 2 , independent of significant quantities, such that In a number of places we will use natural and self-explanatory generalizations of the " " relation, for instance, in connection with certain integrals involving exponential factors. In such cases the exact meaning will be clear from the context. By convention, " " is understood as " " whenever no exponential factors are involved.
We treat positive kernels and integrals as expressions valued in the extended half-line [0, ∞]. Similar remark concerns expressions occurring in various estimates, with the natural limiting interpretations like, for instance, (0 + ) β = ∞ when β < 0.

Preliminaries and Statement of Results
We will consider two interrelated settings corresponding to one-dimensional Laguerre function expansions of convolution type and of Hermite type. Also, we will study the onedimensional context of Dunkl-Laguerre expansions associated with the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to Z 2 . The latter situation may be regarded as an extension of that of Laguerre function expansions of convolution type, see Section 2.3 below. All the three frameworks in question have deep roots in the existing literature. In particular, in the last decade they were widely investigated from the harmonic analysis perspective. For all the facts (tacitly) invoked in what follows we refer to [9] and references given there.

Laguerre Function Setting of Convolution Type
Let α > −1. The Laguerre functions of convolution type are given by where c α n > 0 are the normalizing constants, and L α n , n ≥ 0, are the classical Laguerre polynomials. The system { α n : n ≥ 0} is an orthonormal basis in L 2 (dμ α ), where μ α is the measure on the half-line R + = (0, ∞) defined by The α n are eigenfunctions of the Laguerre 'Laplacian' we have L α α n = (4n+2α+2) α n . We denote by the same symbol L α the natural self-adjoint extension whose spectral resolution is given by the α n . The integral kernel G α t (x, y) of the Laguerre heat semigroup {exp(−tL α )} can be expressed explicitly in terms of the modified Bessel function I α . More precisely, x,y >0.
(1) Given σ > 0, we consider the potential operator where the potential kernel is defined as We will prove the following general and qualitatively sharp estimates of K α,σ (x, y).
(ii) If x + y > 1, then Thus, among other things, we see that the kernel behaves in an essentially different way, depending on whether (x, y) is close to the origin of R 2 or far from it. We remark that under the restrictions α ≥ −1/2 and σ < α + 1, an upper bound for K α,σ (x, y) was obtained recently in [4,Proposition 5.1].
The description of K α,σ (x, y) from Theorem 2.1 enables a direct analysis of the potential operator I α,σ . In particular, it allows us to characterize those 1 ≤ p, q ≤ ∞, for which I α,σ is L p − L q bounded, see also Fig. 1 below.
(a) If α ≥ −1/2, then I α,σ is bounded from L p (dμ α ) to L q (dμ α ) if and only if Note that the sufficiency part of Theorem 2.2 (a) was essentially known to the authors earlier, even in the multi-dimensional case, see [9,Theorem 4.1]. Here, however, we give a direct proof which offers a better insight into the structure of I α,σ . The necessity part, as well as item (b) in Theorem 2.2, is new. It seems a bit surprising that the conditions in (a) and (b) are different, since many known results related to the system { α n } are homogeneous in α > −1, without any 'phase shift' at α = −1/2; see for instance [12,

Laguerre Function Setting of Hermite Type
This Laguerre context is derived from the previous one by modifying the Laguerre functions α n so as to make the resulting system orthonormal with respect to Lebesgue measure dx in R + . Thus, given a parameter α > −1, we consider the functions ϕ α n (x) = x α+1/2 α n (x), x > 0.
Then the system {ϕ α n : n ≥ 0} is an orthonormal basis in L 2 (dx). The associated 'Laplacian' is and we have L H α ϕ α n = (4n + 2α + 2)ϕ α n . The Laguerre heat semigroup {exp(−tL H α )}, generated by means of the natural self-adjoint extension of L α H in this context, has an integral representation. The associated heat kernel is (xy) α+1/2 G α t (x, y), x, y > 0, see Eq. 1. For σ > 0, consider the potential operator Because of this simple link between the two potential kernels, Theorem 2.1 gives qualitatively sharp estimates also for K α,σ H (x, y). Then, taking into account the behavior of the kernel for α < −1/2 and x and y small, it is not hard to see that I α,σ H can be well defined on L p only if 1 H denotes the natural domain of the integral operator I α,σ H ). Note that this case is qualitatively different from the case α ≥ −1/2, see the comments preceding [9, Theorem 3.1].
The following result gives a complete and sharp description of L p − L q boundedness of I α,σ H . It reveals that for α ≥ −1/2, I α,σ H behaves exactly like I −1/2,σ , see Fig. 1 (a) with α = −1/2, and thus like the Hermite potential operator I σ (see the case of I −1/2,σ D in Theorem 2.6 below). On the other hand, for α < −1/2 the L p −L q behavior of I α,σ H is more subtle, and partially this is caused by the restriction on p mentioned above. In particular, the region characterizing those ( 1 p , 1 q ) for which I α,σ H is L p − L q bounded may take various peculiar shapes, see Fig. 2 below.

Dunkl-Laguerre Setting
Let α > −1. The generalized Hermite functions are given by where α n are the Laguerre functions of convolution type naturally extended to R as even functions. The system {h α n : n ≥ 0} is an orthonormal basis in L 2 (dw α ), where w α is the even extension of μ α , Notice that expansions of even functions with respect to {h α n } reduce to expansions with respect to the Laguerre system { α n }. The h α n are eigenfunctions of the one-dimensional Dunkl harmonic oscillator (notice that this is a differential-difference 'Laplacian') and one has L D α h α n = (2n + 2α + 2)h α n . We use the same symbol to denote the natural, in this situation, self-adjoint extension of L D α . For α = −1/2 this setting coincides with that of classical Hermite function expansions. Note that the parameter α represents the so-called (in the Dunkl theory) multiplicity function. This function is trivial when α = −1/2, positive when α > −1/2, and negative for α < −1/2. The latter case is exotic in the sense that positivity of the multiplicity function is crucial in several important aspects of the Dunkl theory. As we shall see, the positivity turns out to be significant also in our developments.
As in the previous settings, for σ > 0 we consider the potential operator where the potential kernel is expressed via G α,D t (x, y) as In view of Eqs. 5 and 6, we have These two relations deliver enough information to obtain a characterization of L p − L q boundedness of I α,σ D . Nevertheless, the question of an exact description of K α,σ D (x, y) is an important problem in its own right. The result below provides qualitatively sharp estimates of K α,σ D (x, y) for α > −1/2 (the case of a positive multiplicity function).
(A) Assume that xy ≥ 0, i.e. x and y have the same sign.
(B) Assume that xy < 0, i.e. x and y have opposite signs.
For the sake of completeness, we recall that (see [10,Theorem 2.4]) uniformly in x, y ∈ R. When xy ≥ 0, this agrees with the estimates of Theorem 2.4 (A) taken with α = −1/2. On the other hand, in case xy < 0 the exponential factor in Eq. 9 possesses essentially better decay if compared with the exponential factor in Theorem 4 (B2). In particular, on the line y = −x the right-hand side in Eq. 9 has an exponential decay, which is not the case of the right-hand side in (B2). This reflects a discontinuity in the behavior of α (u) as α → (−1/2) + (exponential growth/decay when u → −∞), see Section 3.2 below. When α < −1/2 (the case of a negative multiplicity function), no general sharp estimates in the spirit of Theorem 2.4 are possible, because K α,σ D (x, y) attains also negative values. In fact, we have the following result (the proof can be found in Section 3.2). Proposition 2.5 Let −1 < α < −1/2 and σ > 0 be fixed. There exists an unbounded set D ⊂ {(x, y) : xy < 0} ⊂ R 2 of positive Lebesgue measure such that Finally, we establish a sharp description of L p − L q boundedness of I α,σ D . It occurs that I α,σ D behaves exactly in the same way as I α,σ , see (b) If α < −1/2, then I α,σ D is bounded from L p (dw α ) to L q (dw α ) if and only if Note that the result in the Hermite case α = −1/2 was known earlier, see [10] and references given there. Essentially, also the sufficiency part of Theorem 2.6 (a) was known before, even in the multi-dimensional setting, see [9, Theorem 6.1]. The rest of the theorem is new.

Remark 2.7
The estimates of Theorem 2.1 specified to α = ±1/2 and the estimates Eq. 9 for the harmonic oscillator potential kernel are consistent in the following way. Theorem 2.1 implies Eq. 9 for xy > 0, as can be verified by means of the relation x, y > 0.

Estimates of the Potential Kernels
In this section we prove Theorems 2.1 and 2.4, and also Proposition 2.5. We begin with two auxiliary technical results that provide sharp description of the behavior of the integrals J A (T , S) and E A (T , S) defined below. These are essentially [10, Lemmas 2.1-2.3]. Here we give slightly more general statements, nevertheless their proofs are almost the same as those in [10]. Let uniformly in T , S ≥ 0.

Estimates of the Laguerre Potential Kernels
Proving Theorem 2.1 requires some further preparation. The plan is to estimate K α,σ (x, y) first in terms of J A (T , S) and E A (T , S), and then to apply Lemmas 3.1 and 3.2. Let be the heat kernel associated with one-dimensional Hermite function expansions; note that y). The behavior of G α t (x, y) can be described in a sharp way in terms of G t (x, y).
Proof Elementary exercise based on the asymptotics Eq. 4.
where c 1 < c 2 are positive constants, independent of x and y, that may be different in the lower and upper estimate.
Proof In view of Lemma 3.3, To proceed, we get rid of the maximum above by splitting the integral according to the point p(xy), where the function p(r) is defined by the identity sinh 2p(r) = r, r > 0. Notice that p(xy) xy, xy ≤ 1, log 2xy, xy > 1.

Taking into account the explicit form of G t (x, y), we can write
We first prove (a). To this end assume that xy ≤ 1. Since p(xy) xy, we have Changing the variable of integration, we get Since p(xy) xy, we conclude that Next, we estimate I ∞ . We split this integral according to C satisfying 2p(r) ≤ Cr, r ≤ 1. This bound actually holds with C = 1, in view of the inequality sinh t > t, t > 0. Then we obtain The treatment of I ∞,2 is obvious since the integral over (1, ∞) is a finite constant. To deal with I ∞,1 we change the variable of integration and find that c(x 2 +y 2 ) s α−σ e −s ds.
Since p(xy) xy and x 2 + y 2 (x + y) 2 , we infer that where c 1 < c 2 are positive constants that may differ in the lower and upper estimate. Item (a) follows.
To prove (b), assume that xy > 1. Consider first I ∞ . Since p(xy) log 2xy, we can write But t σ −1 e −2(α+1)t e −ct for t > p(1), so, taking into account that actually p(xy) log 2xy, we see that Finally, we analyze I 0 . We split this integral getting Clearly, Further, changing the variable of integration we see that Altogether, the above estimates justify item (b).
We are now in a position to prove Theorem 2.1.
Proof of Theorem 2. 1 We distinguish three cases. Case 1: xy > 1. Notice that in this case x + y > 1, so we must show that By Lemma 3.4 we know that Here E σ −3/2 can be estimated by means of Lemma 3.2, we get In this sum the first term can be neglected since, as easily verified, it contributes to the relation no more than the second one. Further, the factor (xy) −α−1/2 can be replaced by (x + y) −2α−1 . This is clear when x and y are comparable. In the opposite case, it suffices to take into account the bounds (x + y) 2 |x − y|(x + y) xy, recall that xy > 1 and use the exponential decay. The conclusion follows. Case 2: xy ≤ 1 and x + y > 3. Again, our aim is to prove Eq. 11. Observe that in this case x and y are non-comparable. For symmetry reasons, we may assume that x > 2y. Thus the estimate to be shown is But x > 1, so it is enough to check that By Lemma 3.4, Here U 1 agrees with the right-hand side of Eq. 12, so it suffices to bound suitably U 2 and U 3 from above. Observe that U 2 can be estimated from above by replacing the second argument of J α−σ by ∞. Then, taking into account that Thus this term also fits to Eq. 12 contributing in the sense of no more than the first one. Finally, U 3 can be estimated from above by replacing the second argument of E σ −3/2 by 0. Then, with the aid of Lemma 3.2 and the relations |x − y| x + y x > 1 we get the last estimate being a consequence of the inequalities xy ≤ 1 and x > 1. Now Eq. 12 follows. Case 3: xy ≤ 1 and x + y ≤ 3. In this case x ≤ 3 and y ≤ 3. Since the estimates of (i) and (ii) of Theorem 2.1 essentially coincide for x and y separated from 0 and ∞, what we need to prove is (13) We keep using the description of K α,σ in terms of U 1 , U 2 and U 3 , see above. Observe that U 1 (x, y) 1.
To estimate U 2 we apply Lemma 3.1 (b)-(d). After some elementary manipulations, taking into account that x and y stay bounded, we get Considering U 3 , by the boundedness of x and y and the structure of the integral E σ −3/2 we may assume that its second argument is 0. Then, in view of Lemma 3.2, we have We claim that, excluding the exponential factor, all the products xy here can be replaced by (x + y) 2 . Indeed, this is clear when x and y are comparable. In the opposite case, say when x > 2y, log + is controlled by a constant, so its argument can be replaced by (x + y) 2 /(x − y) 2 1. Further, we have (x − y) 2 /xy x/y, and given any γ ∈ R and C > 0 fixed The claim follows and we conclude that Assume that x y. Then the exponential factor on the right-hand side above is roughly a constant. Moreover, U 3 (x, y) (x + y) 2σ −2α−2 . Therefore, in view of Eqs. 14 and 15, for comparable x and y Notice that the right-hand side here is separated from 0, and this remains true even without the second term. Thus U 1 (x, y) U 2 (x, y) + U 3 (x, y) and the second term can be replaced by χ {σ =α+1} log + 1 x+y . We see that Eq. 13 holds when x y. Finally, let x and y be non-comparable. For symmetry reasons, we may assume that x > 2y. Then the desired estimate Eq. 13 takes the form On the other hand, from Eqs. 14 and 15 we have Observe that the fourth term on the right-hand side here is controlled by the other terms, so it may be neglected. Moreover, the sum of the first three terms is separated from 0 and thus controls U 1 (x, y). This means that Here we can neglect χ {σ <α+1} since x 2σ −2α−2 1 for σ ≥ α + 1. After that one can also replace 1 + log + (1/x) by log + (1/x) since x 2σ −2α−2 ≡ 1 when σ = α + 1. Thus we arrive at Eq. 16. The proof of Theorem 2.1 is complete.

Estimates of the Dunkl Potential Kernel
We first focus our attention on the Dunkl heat kernel G α,D t (x, y). Recall that this kernel is defined by means of the auxiliary function α (u) = |u| −α I α (|u|) + sgn(u)I α+1 (|u|) .
As we saw in Section 2.3, α (u) u −α I α (u), u ≥ 0, with the value at u = 0 understood in a limiting sense. However, for u < 0 the situation is more subtle, because of the cancellation occurring in the difference of the Bessel functions. Thus we now analyze the function For α = −1/2 this has an explicit form (cf. [5, (5.8.5)]) and we have notice the exponential decay. Further, when α < −1/2, it is not difficult to see that α (u) is negative for sufficiently large u. Indeed, by the standard large argument asymptotics for the Bessel function (cf. [5, (5.11.10) here ∼ means the asymptotic equality, i.e. that the ratio of the two quantities tends to 1 as u → ∞. Finally, in case α > −1/2 we use [6, Theorem 2] (specified to L ν,1,0 and U ν,2,0 ; see [6, p. 10]) getting This implies Here, in contrast with the Hermite case α = −1/2, we have an exponential growth as u → ∞.
From the above considerations we draw the following conclusions. The behavior of G α,D t (x, y) is qualitatively different in the singular case α = −1/2 (trivial multiplicity function). The case α < −1/2 (negative multiplicity functions) is exotic in the sense that the heat kernel takes also negative values. Indeed, taking into account Eq. 17, we have G α,D t (x, y) < 0 when xy < 0 and |xy|/ sinh 2t is large enough. On the other hand, the case α > −1/2 is more standard. With the aid of Lemma 3.3 and Eq. 1 we can describe the behavior of G α,D t (x, y) in terms of the Hermite heat kernel G t (x, y) = G −1/2,D t (x, y).  (b) If xy < 0, then Proof Assume first that xy > 0. Using the same arguments as those justifying Eq. 6 we get G α,D t (x, y) G α t (|x|, |y|) for x and y in question. Then the conclusion follows from Lemma 3.3. The case when xy = 0 is even simpler. Recall that α (0) is a positive constant (depending on α) and observe that the desired relation is an immediate consequence of Eq. 3.
Let now xy < 0. By Eq. 18 Combining this with Eqs. 3 and 1 we see that An application of Lemma 3.3 finishes the proof.
Note that item (a) in Proposition 3.5 will not be needed for the proof of Theorem 2.4, but we stated it for the sake of completeness. On the other hand, Proposition 3.5 (b) is essential, together with good estimates of the resulting auxiliary kernel (a) If xy ≤ 1, then where c 1 < c 2 are positive constants, independent of x and y, that may be different in the lower and upper estimate.
Proof Using the notation of the proof of Lemma 3.4 and recalling the explicit formulas for G t (x, y), we can write Here I ∞ is the same as in the proof of Lemma 3.4, and from that proof we know that Thus it remains to analyze I 0 .
In case (a) we have p(xy) xy ≤ 1, so The right-hand side here coincides with the right-hand side in Eq. 10 after replacing α by α + 1 and σ by σ + 1. Thus we already know that Considering (b), when xy > 1 we have As in the proof of Lemma 3.4, Moreover, since e 2p(xy) xy, The conclusion follows.
Proof of Theorem 2.4 Let us first assume xy ≥ 0. Observe that K α,σ D (x, y) = K α,σ D (−x, −y), so it is enough to consider the case x, y ≥ 0. If x, y > 0, then we easily get the desired estimates by means of Eq. 8 and Theorem 2.1. If x = 0 or y = 0, then Eq. 8 still holds, with a limiting understanding of the values of K α,σ (x, y) and, implicitly, G α t (x, y). Tracing the proof of Theorem 2.1, one can ensure that the asserted bounds for K α,σ (x, y) remain true for all x, y ≥ 0, hence the conclusion again follows.
Assume next that xy < 0. Taking into account Proposition 3.5 (b), we infer that K α,σ D (x, y) K α,σ (|x|, |y|). On the other hand, the estimates of Lemma 3.6 coincide with the estimates of Lemma 3.4 with α replaced by α + 1 and σ replaced by σ + 1. Thus the behavior of K α,σ (x, y) is the same as the behavior of K α+1,σ +1 (x, y) in the sense of the bounds from Theorem 2.1. Now the conclusion follows by observing that |x|+|y| = |x −y| and ||x| − |y|| = |x + y| when xy < 0.
Proof of Proposition 2.5 Let x, y ∈ R 2 be such that xy < 0. By the asymptotics Eq. 17 it follows that provided that |xy|/ sinh 2t is sufficiently large. We then focus on x and y such that Eq. 19 holds uniformly in t ≤ p(1), and we may assume that |xy| > 1. As in the proof of Lemma 3.6 (b), we infer that for x, y in question, with some positive constants c i , i = 1, . . . , 4. The right-hand side here is certainly negative when y = −x and |x| is large enough, say |x| ≥ C > 0, as can be seen from Lemma 3.2. For continuity reasons, the same must be true for (x, y) lying in a neighborhood of the set {(x, y) : x = −y, |x| ≥ C}.

L p − L q Estimates
This section is devoted to the proofs of Theorems 2.2, 2.3 and 2.6. Given 1 ≤ p ≤ ∞, we denote by p its conjugate exponent, 1/p + 1/p = 1.

L p − L q Estimates in the Laguerre Setting of Convolution Type
Theorem 2.2 follows immediately from the two lemmas below that describe sharply L p −L q behavior of two auxiliary operators (with non-negative kernels) into which I α,σ splits naturally. These operators are interesting in their own right, so for the sake of completeness the lemmas provide slightly more information than actually needed to conclude Theorem 2.2. We split I α,σ according to the kernel splitting and denote the resulting integral operators by I α,σ 0 and I α,σ ∞ , respectively.
The first of these lemmas follows essentially from the recent results of Nowak and Roncal [7] for potential operators in the setting of Jacobi expansions.
Proof of Lemma 4.1 In view of Theorem 2.1, K α,σ 0 (x, y) satisfies the sharp estimates of Theorem 2.1 (i) in the square 0 < x, y ≤ 2, and vanishes outside this square. Comparing to [7,Theorem 2.2], we see that the behavior of K α,σ 0 (x, y) for x, y ≤ 2 is exactly the same as the behavior of the Jacobi potential kernel K α,β σ (θ, ϕ) in the Jacobi trigonometric polynomial setting on the interval (0, π). More precisely, for any fixed β > −1, hold for 1 ≤ p ≤ ∞ when σ > 1/2 and for 1 ≤ p < 1 1−2σ when σ ≤ 1/2. Moreover, for σ ≤ 1/2 and Actually, only Eq. 20 will be used in the sequel. However, we include also Eq. 21 to show that Eq. 20 is optimal in the sense of the range of admissible parameters.
Proof of Lemma 4.3 By Theorem 2.1, K α,σ ∞ (x, y) satisfies the estimates of Theorem 2.1 (ii) outside the square 0 < x, y ≤ 2, and vanishes inside this square. Therefore, it is convenient to consider separately the cases σ < 1/2, σ = 1/2 and σ > 1/2. In what follows we treat the case σ < 1/2 leaving a similar analysis for the remaining cases to the reader. We only mention that in the case σ = 1/2 it is convenient to split further the kernel according to the summands in the factor 1 + log + 1 |x−y|(x+y) . Then the part related to the 1 can be included into the discussion of the case σ < 1/2 to give Eq. 20, while the part coming from the log + does not make worse the upper bound in Eq. 20 and is decisive for Eq. 21. Finally, we observe that considering 0 < x < 1 and x > 4 is enough for the proof of Eq. 20 since for 1 ≤ p < ∞ each of the two functions where f σ,α (x, y) denotes the expression on the right-hand side of " " in Theorem 2.1 (ii), is continuous on (0, ∞); as for p = ∞, the same is true for x → sup y>a f σ,α (x, y), a = 0, 2, provided that σ > 1/2. This may be checked in detail with the aid of the dominated convergence theorem when p < ∞, or directly otherwise.
Proof of Lemma 4.2 The structure of the proof is as follows. The upper estimate of Lemma 4.3 readily enables us to establish L p − L 1 and L 1 − L q boundedness of I α,σ ∞ for the admissible p and q. This, together with a duality argument based on the symmetry of the kernel, K α,σ ∞ (x, y) = K α,σ ∞ (y, x), and the Riesz-Thorin interpolation theorem, gives L p − L q bounds for p and q satisfying where the first inequality should be replaced by a weak one in case η > 1/2. The case when σ < η = 1/2 and 1 p − σ η = 1 q , 2σ < 1 p < 1, is more subtle and will be treated by different methods. Finally, the lack of L p − L q boundedness for the relevant p and q will be shown by giving explicit counterexamples. To simplify the notation, in what follows · p denotes the norm in the Lebesgue space L p (R + , dμ α ).