# An Intersection Representation for a Class of Anisotropic Vector-valued Function Spaces

## TL;DR

This paper establishes an intersection representation for a broad class of anisotropic vector-valued function spaces, improving understanding and tools for boundary value problems in partial differential equations.

## Contribution

It introduces a new intersection representation for weighted anisotropic mixed-norm Besov and Lizorkin-Triebel spaces, enhancing the classical Fubini property.

## Key findings

- Provides an intersection representation for these function spaces.
- Improves the classical Fubini property for Lizorkin-Triebel spaces.
- Applications to weighted maximal regularity in parabolic boundary value problems.

## Abstract

The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting \`a la Hedberg$\&$Netrusov, which includes weighted anisotropic mixed-norm Besov and Lizorkin-Triebel spaces. In the special case of the classical Lizorkin-Triebel spaces, the intersection representation gives an improvement of the well-known Fubini property. The main result has applications in the weighted $L_{q}$-$L_{p}$-maximal regularity problem for parabolic boundary value problems, where weighted anisotropic mixed-norm Lizorkin-Triebel spaces occur as spaces of boundary data.