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Differentiation theorems for BV functions of several variables, and applications

Xianrui Zhang

TL;DR

The work develops a multivariable differentiation theory for functions of bounded variation by introducing joint increments, joint derivatives, and joint monotonicity. It establishes a BV Lebesgue differentiation theorem with an optimal $L^1$ bound by the total variation, and a Jordan decomposition expressing BV functions as differences of jointly monotone components. A Fubini-type differentiation theorem for infinite series of jointly monotone functions extends classical differentiation results to the multivariable setting. The authors also connect AC, BV, and derivative integrals through complementary corollaries, and provide a Riesz-type result for the interchange of derivatives with sums. Collectively, this yields a cohesive framework for differentiation in multiple variables with precise a.e. and integrability conclusions, suitable for analysis and geometric measure theory applications.

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.

Differentiation theorems for BV functions of several variables, and applications

TL;DR

The work develops a multivariable differentiation theory for functions of bounded variation by introducing joint increments, joint derivatives, and joint monotonicity. It establishes a BV Lebesgue differentiation theorem with an optimal bound by the total variation, and a Jordan decomposition expressing BV functions as differences of jointly monotone components. A Fubini-type differentiation theorem for infinite series of jointly monotone functions extends classical differentiation results to the multivariable setting. The authors also connect AC, BV, and derivative integrals through complementary corollaries, and provide a Riesz-type result for the interchange of derivatives with sums. Collectively, this yields a cohesive framework for differentiation in multiple variables with precise a.e. and integrability conclusions, suitable for analysis and geometric measure theory applications.

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 of bounded variation, the joint derivative exists almost everywhere, its norm is bounded by the total variation of , 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.
Paper Structure (11 sections, 182 equations, 4 figures)

This paper contains 11 sections, 182 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proofs of the Theorems
  3. Proof of Theorem 1.1
  4. Proof of Proposition 1.2
  5. Proof of Corollary 1.3
  6. Proof of Theorem 1.4
  7. Proof of Theorem 1.5
  8. Proof of Corollary 1.6
  9. Proof of Theorem 1.7
  10. Proof of Corollary 1.8
  11. Proof of Corollary 1.9

Figures (4)

  • Figure 1.1: An illustration of Definition 1.4 in two dimensions. The sum of the absolute values of the joint increments of $f$ over the sub-rectangles is uniformly bounded.
  • Figure 1.2: An illustration of Definition 1.5 in three dimensions. Provided that the total volume of the shaded sub-rectangles is sufficiently small, the sum of the absolute values of the joint increments of the function over these sub-rectangles is sufficiently small.
  • Figure 2.1: (a) An illustration of the two-dimensional case. The side lengths of the four rectangles are $\frac{1}{N}$; (31) represents the sum of the integrals of $\widetilde{f}$ over the two blue rectangles minus the sum of the integrals of $\widetilde{f}$ over the two red rectangles. (b) An illustration of the three-dimensional case. The side lengths of the eight rectangles are $\frac{1}{N}$; (31) represents the sum of the integrals of $\widetilde{f}$ over the four blue rectangles minus the sum of the integrals of $\widetilde{f}$ over the four red rectangles.
  • Figure 2.2: An illustration of the two-dimensional case. The blue point in the middle is the newly added partition point $\boldsymbol{c}$. Only part of rectangles are affected by the new point; they are divided into smaller rectangles.