Differentiation theorems for BV functions of several variables, and applications

TL;DR

This paper extends classical differentiation theorems to functions of several variables with bounded variation, introducing joint derivatives and monotonicity, and establishes key properties and characterizations of these derivatives.

Contribution

It introduces the concepts of joint derivatives and joint monotonicity for BV functions of multiple variables, extending differentiation theorems and their applications.

Findings

Joint derivatives exist almost everywhere for BV functions.

The $L^1$ norm of the joint derivative is bounded by the total variation.

Sum of jointly monotone functions' derivatives equals the derivative of the sum almost everywhere.

Abstract

We extend the classical Lebesgue and Fubini differentiation theorems to functions of several variables, using the notions of joint derivative and joint monotonicity. Our first main result shows that for a function $f$ of bounded variation, the joint derivative exists almost everywhere, its $L^1$ norm is bounded by the total variation of $f$, and equality in this bound characterizes absolute continuity. Our second main result shows that, for a convergent series of jointly monotone increasing functions, the joint derivative of the sum equals the sum of the joint derivatives almost everywhere.