New variable weighted conditions for fractional maximal operators over spaces of homogeneous type (2408.04544v2)

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.