Boundedness of certain bilinear operators on vanishing generalized Morrey spaces

Abstract

In this article, we consider the bilinear operator $T$ satisfying that there exists a positive constant $C_{(T)}$, depending on $T$, such that, for any measurable functions $f$ and $g$ with compact support, $t\in ℝ$ with , and $x\in {ℝ}^n$ with $0\notin \text{supp}(f(x-t\cdot ))\cap \text{supp}(g(x-\cdot ))$,

$$\left|T(f,g)(x)\right|\le C_{(T)}{\displaystyle {\int }_{{ℝ}^n}\frac{\left|f(x-ty)g(x-y)\right|}{|y{|}^n}dy}.$$

We investigate the boundedness of $T$ on the vanishing generalized Morrey spaces $V_0{L}^{p,\phi }({ℝ}^n)$ and $V_{\infty }{L}^{p,\phi }({ℝ}^n)$, and the boundedness of the subbilinear maximal operator $\mathcal{ℳ}$ on the vanishing generalized Morrey space $V^{(\ast )}{L}^{p,\phi }({ℝ}^n)$, and their applications to some classical (sub)bilinear operators in harmonic analysis. As a byproduct, we also show that $T$ is bounded on generalized Morrey spaces $L^{p,\phi }({ℝ}^n)$. Some typical examples for the main results of this paper are also included.