The modulus of $p$-variation and its applications

Abstract

In this note, we introduce the notion of modulus of $p$-variation for a function of a real variable, and show that it serves in at least two important problems, namely, the uniform convergence of Fourier series and computation of certain $K$-functionals. To be more specific, let $\nu$ be a nondecreasing concave sequence of positive real numbers and $1\leq p<\infty$. Using our new tool, we first define a Banach space, denoted $V_p[\nu]$, that is intermediate between the Wiener class $BV_p$ and $L^\infty$, and prove that it satisfies a Helly-type selection principle. We also prove that the Peetre $K$-functional for the couple $(L^\infty,BV_p)$ can be expressed in terms of the modulus of $p$-variation. Next, we obtain equivalent sharp conditions for the uniform convergence of the Fourier series of all functions in each of the classes $V_p[\nu]$ and $H^\omega\cap V_p[\nu]$, where $\omega$ is a modulus of continuity and $H^\omega$ denotes its associated Lipschitz class. Finally, we establish optimal embeddings into $V_p[\nu]$ of various spaces of functions of generalized bounded variation. As a by-product of these latter results, we infer embedding results for certain symmetric sequence spaces.