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Weighted \(L^p\) estimates for the area integral associated with self-adjoint operators on homogeneous space. (English) Zbl 1248.42021
The authors extend, to the case of spaces of homogeneous type \(X\), the classical weighted \(L^p\) estimates of S. Y. A. Chang, J. M. Wilson and T. H. Wolff [Comment. Math. Helv. 60, 217–246 (1985; Zbl 0575.42025)] and S. Chanillo and 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 \(L^2(X)\):
\[ S_Pf(x)=\bigg(\int_{d(x,y)<t}|t\sqrt{L}e^{-t\sqrt{L}}f(y)|^2\,\frac{d\mu(y)}{V(y,t)}\frac{dt}t\bigg)^{1/2}, \]
\[ S_Hf(x)=\bigg(\int_{d(x,y)<t}|t^2 Le^{-t^2 L}f(y)|^2\,\frac{d\mu(y)}{V(y,t)}\frac{dt}t\bigg)^{1/2}. \]
In particular, if \(T\) is either \(S_P\) or \(S_H\):
(a) \(\int_XT(f)^pw\,d\mu(x)\leq c(X,p)\int_X|f|^p(Mw)\,d\mu(x),\quad 1<p\leq2,\)
(b) \(\int_{\{T(f)>\lambda\}} w\,d\mu(x)\leq \frac{c(X)}{\lambda}\int_X|f|(Mw)\,d\mu(x),\quad \lambda>0,\)
(c) \(\int_XT(f)^pw\,d\mu(x)\leq c(X,p)\int_X|f|^p(Mw)^{p/2}w^{-(p/2-1)}d\mu(x),\quad 2<p<\infty.\)
As a corollary, using the method of R. Fefferman and J. Pipher [Am. J. Math. 119, No. 2, 337–369 (1997; Zbl 0877.42004)] they can prove that \[ \| Tf\|_{L^2(X,w\,d\mu)}\leq C\| w\|_{A_1}^{1/2}\| f\|_{L^2(X,w\,d\mu)} \] and \[ \| Tf\|_{L^p(X)}\leq Cp^{1/2}\| f\|_{L^p(X)},\quad\text{as }p\to\infty. \]

MSC:
42B25 Maximal functions, Littlewood-Paley theory
43A85 Harmonic analysis on homogeneous spaces
26D15 Inequalities for sums, series and integrals
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