Description of Pointwise Multipliers in Pairs of Besov Spaces $B_1^k({\mathbb{R}}^n)$

Abstract

Necessary and sufficient conditions for a function to be a multiplier mapping the Besov space $B_1^m({\mathbb{R}}^n)$ into the Besov space $B_1^l({\mathbb{R}}^n)$ with integer $l$ and $m$, $0<l\le m$, are found. It is shown that multipliers between $B_1^m({\mathbb{R}}^n)$ and $B_1^l({\mathbb{R}}^n)$ form the space of traces of multipliers between the Sobolev classes $W_1^{m+1}({\mathbb{R}}_{+}^{n+1})$ and $W_1^{l+1}({\mathbb{R}}_{+}^{n+1})$.