A Note on the Convergence and Boundedness Properties of Discrete Wolff Potential Operators on Weighted Morrey Sequence Spaces

Abstract

We introduce a discrete version of the Wolff potential operator \(\mathcal{W}_{\alpha,\mathrm{p}}\), defined for a sequence \(\mathrm{x}=\{\mathrm{x}(\mathrm{k})\}_{\mathrm{k}\in\mathbb{Z}}\) via an iterated dyadic ball-averaging raised to the power \(1/(\mathrm{p}-1)\), for parameters \(0<\alpha<1\) and \(1<\mathrm{p}<\infty\). The nonlinear averaging structure of the Wolff potential yields substantially different summability properties compared to the classical discrete Riesz potential. We establish sufficient conditions for absolute and uniform convergence of \(\mathcal{W}_{\alpha,\mathrm{p}}\) on \(\ell^{\mathrm{p}}\) and on discrete Morrey spaces, and we prove boundedness from the weighted discrete Morrey space \(\ell_{\mathrm{q}}^{\mathrm{p}}(\omega^{\mathrm{p}},\omega^{\mathrm{q}})\) to \(\ell_{\mathrm{s}}^{\mathrm{q}}(\omega^{\mathrm{q}})\) under suitable parameter conditions, using dyadic decomposition, local-global estimates, and the doubling property of Muckenhoupt weights. An explicit example shows the discrete Riesz potential can diverge while the discrete Wolff potential remains finite, demonstrating the genuine advantage of the nonlinear averaging decay. These results extend fractional integral operator theory to the discrete setting.