Equivalent Characterization on Besov Space

Cong He,Jingchun Chen,Micah Osilike

doi:10.1155/2021/6688250

Abstract

In this paper, we give an equivalent characterization of the Besov space. This reveals the equivalent relation between the mixed derivative norm and single-variable norm. Fourier multiplier, real interpolation, and Littlewood-Paley decomposition are applied.

Highlights

In Sobolev spaces, it is known that ∥f ∥H2ðR2Þ ~ ∥f ∥L2ðR2Þ + ∑2i=1∥∂2 f /∂x2i ∥L2ðR2Þ, where ∥f ∥H2ðR2Þ ≕ ∥f ∥L2ðR2Þ + ∥∂x1 ∂x2 f ∥ L2ðR2Þ+∥∂2x1 f ∥L2ðR2Þ+∥∂2x2 f ∥L2ðR2Þ: Note that on the right hand side of the definition ∥f ∥H2ðR2Þ, it contains the mixed derivative norm ∥∂x1 ∂x2 f ∥L2ðR2Þ: This mixed derivative norm would make the calculation more complicated or even infeasible to estimate partial differential equations with some anisotropy property, like Vlasov-Poisson equation [1, 2], in fractional Sobolev space [3]
We have the following equivalent norm theorem in Sobolev spaces
The methods could be adapted to the weighted Sobolev spaces and weighted Besov space, or even in the anisotropic function space

Summary

Introduction

In Sobolev spaces, it is known that ∥f ∥H2ðR2Þ ~ ∥f ∥L2ðR2Þ + ∑2i=1∥∂2 f /∂x2i ∥L2ðR2Þ, where ∥f ∥H2ðR2Þ ≕ ∥f ∥L2ðR2Þ + ∥∂x1 ∂x2 f ∥ L2ðR2Þ+∥∂2x1 f ∥L2ðR2Þ+∥∂2x2 f ∥L2ðR2Þ: Note that on the right hand side of the definition ∥f ∥H2ðR2Þ, it contains the mixed derivative norm ∥∂x1 ∂x2 f ∥L2ðR2Þ: This mixed derivative norm would make the calculation more complicated or even infeasible to estimate partial differential equations with some anisotropy property, like Vlasov-Poisson equation [1, 2], in fractional Sobolev space [3]. We aim to prove ∥f ∥Bsp,rðRnÞ ~ ∑nj=1 ∥f ∥Bsp,r,xj ðRnÞ which realizes the separation, i.e., the right hand side does not contain the “mixed derivative” term, it only contains fractional derivative with respect to a single variable for each term. When it comes to estimate ∥f ∥Bsp,rðRnÞ in solving partial differential equations, it is equivalent to estimate ∥f ∥Bsp,r,xj ðRnÞ individually. For the other equivalent characterizations for Besov spaces, refer to [4,5,6,7] and the references therein

Preliminaries

Equivalent Characterization