Vector-valued extensions for fractional integrals of Laguerre expansions

We prove some vector-valued inequalities for fractional integrals defined in the context of two different orthonormal systems of Laguerre functions. Our results are based on estimates of the corresponding kernels with precise control of the parameters involved. As an application, mixed norm estimates for the fractional integrals related to the harmonic oscillator are deduced.Comment: 21 pages. Revised versio

The Riesz transform for the harmonic oscillator in spherical coordinates

In this paper we show weighted estimates in mixed norm spaces for the Riesz transform associated with the harmonic oscillator in spherical coordinates. In order to prove the result we need a weighted inequality for a vector-valued extension of the Riesz transform related to the Laguerre expansions which is of independent interest. The main tools to obtain such extension are a weighted inequality for the Riesz transform independent of the order of the involved Laguerre functions and an appropriate adaptation of Rubio de Francia's extrapolation theorem.Comment: 19 pages. To appear in Constructive Approximatio

A weighted transplantation theorem for Jacobi coefficients

We present a transplantation theorem for Jacobi coefficients in weighted spaces. In fact, by using a discrete vector-valued local Calder\'{o}n-Zygmund theory, which has recently been furnished, we prove the boundedness of transplantation operators from $\ell^p(\mathbb{N},w)$ into itself, where $w$ is a weight in the discrete Muckenhoupt class $A_{p}(\mathbb{N})$. Moreover, we obtain weighted weak $(1,1)$ estimates for those operators.Comment: 16 page

Maximal estimates for a generalized spherical mean Radon transform acting on radial functions

We study a generalized spherical means operator, viz. generalized spherical mean Radon transform, acting on radial functions. As the main results, we find conditions for the associated maximal operator and its local variant to be bounded on power weighted Lebesgue spaces. This translates, in particular, into almost everywhere convergence to radial initial data results for solutions to certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations. Moreover, our results shed some new light to the interesting and important question of optimality of the yet known $L^p$ boundedness results for the maximal operator in the general non-radial case. It appears that these could still be notably improved, as indicated by our conjecture of the ultimate sharp result.Comment: 20 pages, 2 figures. Sharpness results added and minor things improved or corrected. Accepted for publication in Annali di Matematica Pura ed Applicat

Riesz transforms on compact Riemannian symmetric spaces of rank one

In this paper we prove mixed norm estimates for Riesz transforms related to Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one. These operators are closely related to the Riesz transforms for Jacobi polynomials expansions. The key point is to obtain sharp estimates for the kernel of the Jacobi--Riesz transforms with uniform control on the parameters, together with an adaptation of Rubio de Francia's extrapolation theorem. The latter results are of independent interest.Comment: 19 pages. To appear in Milan Journal of Mathematic