Generalized fractional maximal functions in Lorentz spaces

Abstract

In this paper we give the complete characterization of the boundedness of the generalized fractional maximal operator $$M_{\varphi ,{\Lambda }^{\alpha }(b)}f(x):=\underset{Q\ni x}{\sup }\frac{\Vert f{\chi }_Q{\Vert }_{{\Lambda }^{\alpha }(b)}}{\varphi (|Q|)}(x\in {\mathbb{R}}^n),$$ between the classical Lorentz spaces ${\Lambda }^p(v)$ and ${\Lambda }^q(w)$ for appropriate functions $\varphi$, where $0<p,q<\infty$, $0<\alpha \le r<\infty$, $v,w,b$ are weight functions on $(0,\infty )$ such that $0<B(x):={\int }_0^{x}b<\infty$, $x>0$, $B\in {\Delta }_2$ and $B(t)/t^{\alpha /r}$ is quasi-increasing.

arXiv comment18 pages

Cited by in corpus (1)

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