Weight characterization of the boundedness for the riemann-liouville discrete transform

Alexander Meskhi
Department of Mathematical Analysis, I. Javakhishvili Tbilisi State University,
Tbilisi, Georgia,
Abdus Salam School of Mathematical Sciences, GC University, 68-B New Muslim Town,
Lahore, Pakistan.
Ghulam Murtaza
Department of Applied Sciences, National Textile University, Faisalabad, Pakistan.

\(^{1}\)Corresponding Author: meskhi@rmi.ge

Copyright © 2013 Alexander Meskhi, Ghulam Murtaza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: December, 2013.

Abstract

We establish necessary and sufficient conditions on a weight sequence \({v_j}^{∞}_{j}=1\) governing the boundedness for the Riemann-Liouville discrete transform \(I_α\) from \(l^p (\mathbb{N})\) to \(l^{q}_{vj}(N)\) (trace inequality), where \(0 < α < 1\). The derived conditions are of \(D\). Adams or Maz’ya–Verbitsky (pointwise) type.

Keywords:

Riemann–Liouville discrete transform with product kernels, discrete Hardy operator, discrete potentials, weighted inequality, trace inequality.