High order Riesz transforms and mean value formula for generalized translate operator

Abstract

In this paper, the mean value formula depends on the Bessel-generalized shift operator corresponding to the solutions of the boundary value problem related to the multidimensional Bessel operator are studied. In addition, Riesz transforms R B related to the multidimensional Bessel operators are studied. Since a Bessel-generalized shift operator is a translation operator corresponding to the multidimensional Bessel operator, we construct a family of R B by using a Bessel-generalized shift operator. Finally, we analyze weighted inequalities involving R B . MSC: Primary 31B05; 35G10; secondary 34B30; 47F05

Fractional Weighted Spherical Mean and Maximal Inequality for the Weighted Spherical Mean and Its Application to Singular Pde

“…Generalized convolution generated by a multi-dimensional generalized translation y x is given by Multi-dimensional Poisson operator x , acts to the integrable function f by the formula Multi-dimensional Hankel transform (2) applied to generalized convolution (6) gives…”

B L m p , ν estimates for the Riesz transforms associated with Laplace-Bessel operator

“…First note that, let Ω(x) = P k (x)|x| −m , K(x) = Ω(x)|x| −n−ν and P k range over the homogeneous harmonic polynomials the latter arise in special case α = 1. Then for α > 1, we call the higher order Riesz-Bessel transform where we refer to α as the degree of the higher order Riesz-Bessel transform [1,6,7]. Since P k is homogeneous B-polynomial of degree k in R n + , we shall say that P k is elliptic if P k (x) vanishes only at the origin.…”

“…. , n) and P k (x) is a homogeneous polynomial of degree k in R n + which satisfies ∆ ν P k = 0 (see [6,7]). …”

“…Analogues of Riesz transforms are defined and studied for the Dunkl transform and Laplace-Bessel operator. Especially on the L p space, several authors have studied different aspects of Riesz transforms [1,5,6,7,12,14]. Riesz transform is a singular integral operator with convolution type.…”

“…The idea of considering Riesz transforms on L p spaces investigated with several papers by authors, see [1,5,6,7,12,14]. In [1,5,6,7], the authors looked at the Riesz transforms associated with the Laplace-Bessel operator on R n . Instead of using the ordinary shift operator to generalized shift operator they used a different RieszBessel operator consisting of functions of the form T y where T y is the generalized shift operator and ∆ B are Laplace-Bessel operator.…”