On embeddings of spaces of bivariate functions of bounded $p$-variation

TL;DR

This paper establishes precise bounds for the total p-variation of bivariate functions using mixed modulus of continuity and explores embeddings of mixed norm spaces with bounded p-variation in their sections.

Contribution

It provides sharp estimates for total p-variation and investigates embeddings of mixed norm spaces with bounded p-variation of linear sections.

Findings

Sharp estimates of total p-variation in terms of mixed modulus of continuity.

Characterization of embeddings for mixed norm spaces with bounded p-variation.

Analysis of the structure of bivariate functions with bounded p-variation.

Abstract

We obtain sharp estimates of the Hardy-Vitali type total $p$-variation of a function of two variables in terms of its mixed modulus of continuity in $L^p([0,1]^2)$. We also investigate various embeddings for mixed norm spaces of bivariate functions whose linear sections have bounded $p$-variation in the sense of Wiener