The convolution in anisotropic Besov spaces

Abstract

We study the boundedness of the convolution operator in Nikol'skii-Besov anisotropic spaces $B_{\bf p\boldsymbol\tau}^{\boldsymbol\alpha\bf q}$. These spaces are constructed on the basis of anisotropic Lorentz spaces $L_{\bf p\boldsymbol\tau}$, where $\bf p$ и $\boldsymbol\tau$ are vector parameters. The properties of anisotropic Nikol'skii-Besov spaces are investigated.
The main goal of the paper is to solve the following problem: let $f$ and $g$ be functions from some classes of the Nikol'skii-Besov space scale. It is necessary to determine which space belongs to their convolution $f*g$. We prooved the inequality of different Nikol'skii metrics for trigonometric polynomials with spectrum in binary blocks in anisotropic Lorentz spaces $L_{\bf p\boldsymbol\tau}$. Conditions are obtained in terms of the corresponding vector parameters $\boldsymbol\alpha$, $\bf p$, $\bf q$, $\boldsymbol\tau$, $\bf r$, $\boldsymbol\mu$, $\boldsymbol\beta$, $\boldsymbol\eta$, $\bf h$, $\boldsymbol\nu$, $\boldsymbol\gamma$, $\boldsymbol\xi$, which are necessary and sufficient conditions for embeddings
\begin{equation*}
B_{\bf r\boldsymbol\mu}^{\boldsymbol{\beta\eta}}*B_{\bf h\boldsymbol\nu}^{\boldsymbol{\gamma\xi}}\hookrightarrow B_{\bf p\boldsymbol\tau}^{\boldsymbol\alpha\bf q}.
\end{equation*}
This statement is an analogue of O'Neil inequality for Lorentz spaces. 
In particular, the classical O'Neil inequality follows from the proved results. The obtained criterion is generalized by the results of Burenkov and Batyrov, who considered this problem in Besov spaces with scalar parameters.