New variable weighted conditions for fractional maximal operators over spaces of homogeneous type

TL;DR

This paper develops new variable weighted conditions for fractional maximal operators on spaces of homogeneous type, establishing sharp estimates and characterizations in the context of dyadic analysis and variable exponent spaces.

Contribution

It introduces a novel class of variable multiple weights and a new variable testing condition, enabling comprehensive weighted estimates for multilinear fractional maximal operators.

Findings

Established strong and weak type weighted estimates for multilinear fractional maximal operators.

Proved quantitative two-weighted estimates using a new variable testing condition.

Developed a variable weighted characterization framework over spaces of homogeneous type.

Abstract

Based on the rapid development of dyadic analysis and the theory of variable weighted function spaces over the spaces of homogeneous type $(X,d,\mu)$ in recent years, we systematically consider the quantitative variable weighted characterizations for fractional maximal operators. On the one hand, a new class of variable multiple weight $A_{\vec{p}(\cdot),q(\cdot)}(X)$ is established, which enables us to prove the strong and weak type variable multiple weighted estimates for multilinear fractional maximal operators ${{{\mathscr M}_{\eta }}}$. More precisely, \[ {\left[ {\vec \omega } \right]_{{A_{\vec p( \cdot ),q( \cdot )}}(X)}} \lesssim {\left\| \mathscr{M}_\eta \right\|_{\prod\limits_{i = 1}^m {{L^{p_i( \cdot )}}({X,\omega _i})} \to {L^{q( \cdot )}}(X,\omega )({WL^{q( \cdot )}}(X,\omega ))}} \le {C_{\vec \omega ,\eta ,m,\mu ,X,\vec p( \cdot )}}. \] On the other hand, on account of the classical Sawyer's condition $S_{p,q}(\mathbb{R}^n)$, a new variable testing condition $C_{{p}(\cdot),q(\cdot)}(X)$ also appears in here, which allows us to obtain quantitative two-weighted estimates for fractional maximal operators ${{{M}_{\eta }}}$. To be exact, \begin{align*} \|M_{\eta}\|_{L^{p(\cdot)}(X,\omega)\rightarrow L^{q(\cdot)}(X,v)} \lesssim \sum\limits_{\theta = \frac{1}{{{p_{\rm{ - }}}}},\frac{1}{{{p_{\rm{ + }}}}}} {{{\left( {{{[\omega ,v]}_{C_{p( \cdot ),q( \cdot )}^2(X)}} + {{[\omega ]}_{C_{p( \cdot ),q( \cdot )}^1(X)}}{{[\omega ,v]}_{C_{p( \cdot ),q( \cdot )}^2(X)}}} \right)}^\theta }}. \end{align*} The implicit constants mentioned above are independent on the weights.