Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and bounded variation spaces

TL;DR

This paper develops advanced chain rules for tensor fields and multivariate functions, providing estimates in various function spaces and introducing a generalized Bell polynomial-based chain rule for composition chains.

Contribution

It introduces a novel higher-order chain rule for multivariate functions using nested set partitions and generalized Bell polynomials, extending the Faà di Bruno formula.

Findings

Derived higher-order chain rules for tensor fields and multivariate functions.

Provided estimates for Sobolev-Slobodeckij, Musielak-Orlicz norms, and total variation.

Established regularity conditions for coordinate changes in tensor calculus.

Abstract

We describe higher-order chain rules for multivariate functions and tensor fields. We estimate Sobolev-Slobodeckij norms, Musielak-Orlicz norms, and the total variation seminorms of the higher derivatives of tensor fields after a change of variables and determine sufficient regularity conditions for the coordinate change. We also introduce a novel higher-order chain rule for composition chains of multivariate functions that is described via nested set partitions and generalized Bell polynomials; it is a natural extension of the Fa\`a di Bruno formula. Our discussion uses the coordinate-free language of tensor calculus and includes Fr\'echet-differentiable mappings between Banach spaces.