Boundedness of the discrete Hilbert transform on discrete weighted Morrey spaces
Rashid Aliev, Amil Jabiyev
TL;DR
The paper investigates the boundedness of the discrete Hilbert transform on discrete weighted Morrey spaces. It introduces discrete Morrey spaces $m_{\lambda,p}$ and their weighted counterparts $m_{\lambda,p,w}$, along with weight classes $\tilde{A}_p$ and $\tilde{\Delta}_2$, and proves key lemmas establishing structural properties of these weights. The main results show that for $1\le p<\infty$ and $0\le \lambda \le 1/p$, if $\{w_k\}\in\tilde{A}_p$, then the discrete Hilbert transform $H$ is well defined on $m_{\lambda,p,w}$ and bounded, with $\|H(b)\|_{m_{\lambda,p,w}} \le C_{\lambda,p,w}\|b\|_{m_{\lambda,p,w}}$ for $b\in m_{\lambda,p,w}$. The proofs combine discretized analogues of Morrey-space techniques with weighted estimates, leveraging the boundedness of associated operators like the discrete singular integral $S$ and the Hardy–Littlewood maximal operator $M$. Overall, the work extends boundedness results from continuous to discrete weighted Morrey spaces and provides a framework for analyzing discrete data in Morrey-type norms, with relevance to digital signal processing and discrete harmonic analysis.
Abstract
The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. The Hilbert transform was the motivation for the development of modern harmonic analysis. Its discrete version is also widely used in many areas of science and technology and plays an important role in digital signal processing. The essential motivation behind thinking about discrete transforms is that experimental data are most often not taken in a continuous manner but sampled at discrete time values. Since much of the data collected in both the physical sciences and engineering are discrete, the discrete Hilbert transform is a rather useful tool in these areas for the general analysis of this type of data. In this paper, we discuss the discrete Hilbert transform on discrete Weighted Morrey spaces and obtain its boundedness in these spaces.