Littlewood-Paley and spectral multipliers on weighted \(L^p\) spaces
The authors consider non-negative self-adjoint operators \(L\) acting on \(L^2(X)\) where \((X,d,\mu)\) is a space of homogeneous type, meaning a metric space with a non-negative Borel doubling measure. For any bounded Borel function \(F:(0,\infty)\to \mathbb C\) the operator \(F(L)=\int_0^\infty F(\lambda)dE(\lambda)\) is well defined and bounded on \(L^2(X)\), where \(E(\lambda)\) denotes the spectral resolution of \(L\). The authors restrict themselves to operators \(L\) such that \(-L\) generates an analytic semigroup \(\{e^{-iL}\}_{t>0}\) which has a kernel \(p_t(x,y)\) satisfying the Gaussian upper bound, i.e. there exist positive constants \(C\) and \(c\) such that \(|p_t(x,y)|\leq \frac{C}{\mu(B(x,\sqrt t)}exp(-\frac{d(x,y)^2}{ct})\) for \(t>0\) and \(x,y\in X\) and try to study properties on \(F\) to get the boundedness on other function spaces. In particular, the paper is devoted to the study of weighted \(L^p\)-norm inequalities for spectral multipliers of \(L\) continuing the line initiated in [\textit{X. T. Duong} et al., J. Funct. Anal. 196, No. 2, 443--485 (2002; Zbl 1029.43006)] and [\textit{X. T. Duong} et al., J. Funct. Anal. 260, No. 4, 1106--1131 (2011; Zbl 1250.42042)]. The authors show that the weighted Hörmander-type spectral multiplier theorem follows from weighted \(L^p\)-norm inequalities for the Lusin and Littlewood-Paley functions, Gaussian heat kernel bounds and appropriate \(L^2\)-estimates of the kernel of the spectral multiplier.
scientific article; zbMATH DE number 6370410
scientific article