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

Abstract

In this study, the boundedness of the high order Riesz-Bessel transformations generated by generalized shift operator in weighted L p,ω,γ -spaces with general weights is proved.

The Two-Weighted Inequalities for Sublinear Operators Generated by B Singular Integrals in Weighted Lebesgue Spaces

“…Remark 3.2 Note that, the Theorem 3.1 in the case k = 1 was proved in [8] and the Corollary 3.1 for the high order Riesz-Bessel transformations was proved in [6].…”

“…with a constant C 6 , independent on f , to hold, it is necessary and sufficient that the following condition be satisfied:…”

“…, 0) and the generalized shift operator T y f (x) is the usual shift operator f (x − y) was first introdused by Soria and Weiss in [21]. The condition (1.2) is satisfied by many interesting operators in harmonic analysis, such as the B singular integral operators (for example, for k = 1 see [2,7,8,15,16,19,20]), the B Hardy-Littlewood maximal operator ( [12], for n = k = 1 see [22], for k = 1 see [11] and for k = n see [9]), the high order Riesz-Bessel transformations (see [6,19]) and so on.…”

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

“…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.…”