On New Decomposition Theorems for Mixed-Norm Besov Spaces with Ingredient Modulus of Smoothness

Abstract

In this paper, we introduce and study the concept of the ingredient modulus of smoothness in component form in Lp→(Rd) and a kind of mixed-norm Sobolev space. We obtain some new properties, inequalities, and auxiliary results in mixed-norm spaces Lp→(Rd). In addition, a new concept of mixed-norm Besov space is presented and a new decomposition theorem for mixed-norm Besov spaces is established.

Keywords

• mixed-norm Lebesgue space
• Sobolev space
• Besov space
• Littlewood–Paley decomposition
• ingredient modulus of smoothness in component form
• mixed-norm Besov space