Potential operators associated with Hankel and Hankel-Dunkl transforms

Abstract

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q ≤ ∞, for which the potential operators satisfy L p − L q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L p − L q estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L p − L q bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Nápoli, Drelichman and Durán, and Duoandikoetxea. A. NOWAK AND K. STEMPAKMoreover, in case of the Riesz potentials, we also determine those 1 ≤ p, q ≤ ∞, for which two powerweight L p −L q bounds hold, see Theorems 2.5, 2.10 and 2.15. All these results are Hankel or Hankel-Dunkl counterparts of a series of recent sharp results concerning potential operators in several classic settings related to discrete orthogonal expansions: Hermite function expansions [28], Jacobi and Fourier-Bessel expansions [26], and Laguerre function and Dunkl-Laguerre expansions [29]. The frameworks studied in this paper correspond to continuous orthogonal expansions, and the general approach elaborated in the above mentioned articles applies here as well. However, the present L p − L q results have somewhat different flavor.An interesting by-product of our results is an alternative proof of a radial analogue of the celebrated two power-weighted L p − L q estimates for the classical Riesz potentials due to Stein and Weiss [34]. Such an analogue was obtained by Rubin [32] in the eighties of the last century. Rubin's result, being apparently overlooked, was recently reinvestigated and refined by De Nápoli, Drelichman and Durán [15], and Duoandikoetxea [17]. In fact, Corollary 2.6 below slightly extends the above mentioned results, see the related comments following the statement.Crucial aspects of our results are their sharpness and completeness. The latter means, in particular, that in each of the contexts we treat the full admissible range of the associated parameter of type, which in the Hankel-Dunkl setting manifests in including an 'exotic' case of negative multiplicity functions. Some parts of our results were obtained earlier, by various authors, which is always commented in the relevant places according to our best knowledge. In this connection, we mention again the article of Muckenhoupt and Stein [25], and the works of Gadjiev and Aliev [18] where Riesz and Bessel potentials in the context of the modified Hankel transform were investigated, Thangavelu and Xu [36] where Riesz and Bessel potentials for the Dunkl transform were introduced and studied, Hassani, Mustapha and Sifi [21] where for the Riesz potentials the subject was continued, Betancor, Martínez and Rodríguez-Mesa [10] where Riesz potentials for the Hankel transfo...

On potential spaces related to Jacobi expansions

“…We point out that intimately connected to potential spaces are potential operators, and in the above-mentioned contexts they were studied intensively and thoroughly in the recent past. We refer the interested readers to [13,[17][18][19] and also to references given in these works. In particular, [13] delivers a solid ground for our developments.…”

Two-Weight Mixed Norm Estimates for a Generalized Spherical Mean Radon Transform Acting on Radial Functions

“…So one has to look at the Hardy type operators H η 0 , H η ∞ , H log 0 , H log ∞ . Arguing as in [13] we find that the condition…”

“…Therefore we now analyze each of these operators. We will argue similarly as in [13,Section 4.1]. Our main tool will be the following characterization of two power-weight L p − L q inequalities for the Hardy operator and its dual; see e.g.…”

“…Here we may assume that (C1) and (C2) are satisfied, and ρ > −1. We can then invoke the analysis of T and S performed in [13,Section 4.1], with the quantity ρ+1 2r playing the role of σ from [13]. To be precise, here the bound ρ+1 2r < α + 1 may not be satisfied, but that does not affect the arguments in question.…”

“…Our main tool will be the following characterization of two power-weight L p − L q inequalities for the Hardy operator and its dual; see e.g. [2,18] and [13,Lemma 4.1].…”