On Weighted Estimates of High-Order Riesz-Bessel Transformations Generated by the Generalized Shift Operator

The Boundedness of High Order Riesz-Bessel Transformations Generated by the Generalized Shift Operator in Weighted L p,ω,γ -spaces with General Weights

“…The basic goal of this paper is to establish weighted L p,ω,γ -estimates with general weights for the norms of the high order Riesz-Bessel transformations generated by a generalized shift operator [4] …”

Boundedness of Riesz Potential Generated by Generalized Shift Operator on Ba Spaces

“…We note that this convolution satisfies the property f 1 * f 2 = f 2 * f 1 (see [1], [2], [9]- [11]). The Riesz potential I α is defined by…”

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]). …”

“…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.…”