Weighted \(L^p\) estimates for the area integral associated with self-adjoint operators on homogeneous space (Q432418)

The authors extend, to the case of spaces of homogeneous type \(X\), the classical weighted \(L^p\) estimates of \textit{S. Y. A. Chang}, \textit{J. M. Wilson} and \textit{T. H. Wolff} [Comment. Math. Helv. 60, 217--246 (1985; Zbl 0575.42025)] and \textit{S. Chanillo} and \textit{R. L. Wheeden} [Indiana Univ. Math. J. 36, 277--294 (1987; Zbl 0598.34019)] for some area integral operators associated to a non-negative self-adjoint operator \(L\) on \(L^2(X)\):NEWLINENEWLINENEWLINE\[NEWLINE S_Pf(x)=\bigg(\int_{d(x,y)<t}|t\sqrt{L}e^{-t\sqrt{L}}f(y)|^2\,\frac{d\mu(y)}{V(y,t)}\frac{dt}t\bigg)^{1/2}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE S_Hf(x)=\bigg(\int_{d(x,y)<t}|t^2 Le^{-t^2 L}f(y)|^2\,\frac{d\mu(y)}{V(y,t)}\frac{dt}t\bigg)^{1/2}. NEWLINE\]NEWLINENEWLINENEWLINEIn particular, if \(T\) is either \(S_P\) or \(S_H\):NEWLINENEWLINENEWLINENEWLINE (a) \(\int_XT(f)^pw\,d\mu(x)\leq c(X,p)\int_X|f|^p(Mw)\,d\mu(x),\quad 1<p\leq2,\)NEWLINENEWLINE(b) \(\int_{\{T(f)>\lambda\}} w\,d\mu(x)\leq \frac{c(X)}{\lambda}\int_X|f|(Mw)\,d\mu(x),\quad \lambda>0,\)NEWLINENEWLINE(c) \(\int_XT(f)^pw\,d\mu(x)\leq c(X,p)\int_X|f|^p(Mw)^{p/2}w^{-(p/2-1)}d\mu(x),\quad 2<p<\infty.\)NEWLINENEWLINENEWLINENEWLINE As a corollary, using the method of \textit{R. Fefferman} and \textit{J. Pipher} [Am. J. Math. 119, No. 2, 337--369 (1997; Zbl 0877.42004)] they can prove that NEWLINE\[NEWLINE \| Tf\|_{L^2(X,w\,d\mu)}\leq C\| w\|_{A_1}^{1/2}\| f\|_{L^2(X,w\,d\mu)} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \| Tf\|_{L^p(X)}\leq Cp^{1/2}\| f\|_{L^p(X)},\quad\text{as }p\to\infty. NEWLINE\]