\(BL_{p,\nu}^{m}\) estimates for the Riesz transforms associated with Laplace-Bessel operator

Volume 11, Issue 6, pp 832--840 http://dx.doi.org/10.22436/jnsa.011.06.09
Publication Date: May 03, 2018 Submission Date: March 28, 2016 Revision Date: March 27, 2018 Accteptance Date: March 28, 2018

Authors

Ismail Ekincioglu - Department of Mathematics Kutahya, Dumlupnar University, Turkey. Cansu Keskin - Department of Mathematics Kutahya, Dumlupnar University, Turkey. Serap Guner - Department of Mathematics Kutahya, Dumlupnar University, Turkey.


Abstract

In this paper, we introduce higher order Riesz-Bessel transforms which we can express partial derivatives of order \(\alpha\) of \(I_{m,\nu}f\) for \(f\in L_{p,\nu}\). In addition, we establish relationship between Riesz potential with higher order Riesz-Bessel transform related to generalized shift operator. By using this relationship, we make some improvements of integral estimates for \(I_{m,\nu}f\) and higher order Riesz-Bessel transform \(R_{\nu}^{m}\) in the Beppo Levi space \(BL_{p,\nu}^{m}\). We prove an estimate for the singular integral operator with convolution type generated by generalized shift operator in the Beppo Levi spaces.


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