Embeddings between weighted Tandori and Cesàro function spaces

Abstract

We characterize the weights for which the two-operator inequality ∥∥∥(∫x0f(t)pv(t)pdt)1p∥∥∥q,u,(0,∞)≤c∥∥∥esssupt∈(x,∞)f(t)∥∥∥r,w,(0,∞)‖(∫0xf(t)pv(t)pdt)1p‖q,u,(0,∞)≤c‖esssupt∈(x,∞)f(t)‖r,w,(0,∞) holds for all non-negative measurable functions on (0,∞)(0,∞), where 0<p<q≤∞0<p<q≤∞ and 0<r<∞0<r<∞, namely, we find the least constants in the embeddings between weighted Tandori and Ces\`{a}ro function spaces. We use the combination of duality arguments for weighted Lebesgue spaces and weighted Tandori spaces with weighted estimates for the iterated integral operators.