GENERALIZED HARDY’S INEQUALITIES WITH NONLINEAR INTEGRATION LIMITS

Abstract

In this paper, we establish two integral inequalities arising in the study of weighted norm estimates. First, we consider a sequence of measurable sets forming a partition of ℝm and derive an upper bound for a sum involving weighted integrals of a function g, controlled by a sequence of positive numbers. Second, we prove an integral inequality involving a non-decreasing function h with sup(h(x)/x) < ∞ and a power-weighted integral of f. Higher dimensional analogue of this inequality are also established. © 2025 Elsevier B.V., All rights reserved.