Weighted inequalities in generalized Morrey spaces of maximal and singular integral operators on spaces of homogeneous type (Q2718047)

Let \((X,d,\mu)\) be a space of homogeneous type and \(X^+=\{(x,t): x\in X\), \(t\geq 0\}\). Let \(\omega\) be a weight function on \(X\) and \(\beta\) be a positive Borel measure on \(X^+\). Let \(f\) be a locally integrable function on \(X^+\) and denote NEWLINE\[NEWLINE \|f\|_{L^{p,\varphi}(\beta)}=\sup_{x\in X,r>0}\left ( {1\over \varphi(r)}\int_{\widetilde B(x,r)}|f(y,t)|^p d\beta(y,t)\right)^{1/p}, NEWLINE\]NEWLINE for \(1\leq p<\infty\). Here \(\widetilde B=\{(y,t)\in X^+: y\in B\), \(0\leq t\leq r\}\) with \(B=B(x,r)\) a ball on \(X\). We define the generalized Morrey space on \(X^+\) as follows: NEWLINE\[NEWLINE L^{p,\varphi}(X^+,\beta)=\{ f\in L^{p}_{\text{loc}}(X^+): \|f\|_{L}^{p,\varphi}(\beta)<\infty\}. NEWLINE\]NEWLINE Let \(f\) be a locally integrable function on \(X\), we define the generalized maximal operator as follows: NEWLINE\[NEWLINE \widetilde Mf(x,t)=\sup_{(x,t)\in \widetilde B}{1\over \mu(B)}\int_{B} |f(y)|d\mu(y),\quad (x,t)\in X^{+}.NEWLINE\]NEWLINE Here \(d\beta(x,t)=\omega(x) d\mu\). Set \(\widetilde M_qf=(\widetilde Mf^q)^{1/q}\) for \(1\leq q<\infty\). NEWLINENEWLINENEWLINEIn this paper, the author proves that NEWLINE\[NEWLINE\beta(\{(y,t)\in \widetilde B(x,r): \widetilde M_qf(y,t)>\lambda\}) \leq C\lambda^{-p}\varphi(r)\|f\|_{L^{p,\varphi}(\omega)}^pNEWLINE\]NEWLINE for all \(f\in L^{p,\varphi}(X,\omega)\), \(1\leq p<\infty\), \(\lambda>0\) and ball \(B(x,r)\subset X\); NEWLINE\[NEWLINE\|\widetilde M_qf\|_{L^{p,\varphi}(\beta)}\leq C\|f\|_{L^{p,\varphi}(\omega)},NEWLINE\]NEWLINE for all \(f\in L^{p,\varphi}(X,\omega)\), \(1\leq q< p<\infty\). The author also shows that the generalized singular integral operator \(T\) satisfies weak type \((1,1)\) and strong type \((p,p)\), \(1<p<\infty\), properties in Morrey spaces.