Sun, Limin Some estimates related to fractal measures and Laplacians on manifolds. (English) Zbl 0839.58054 Ark. Mat. 33, No. 1, 173-182 (1995). Summary: Let \(\Delta\) be the Laplace-Beltrami operator on an \(n\)-dimensional complete \(C^\infty\) manifold \(M\). In this paper, we establish an estimate of \(e^{t \Delta} (d \mu)\) valid for all \(t > 0\), where \(d \mu\) is a locally uniformly \(\alpha\)-dimensional measure on \(M\), \(0 \leq \alpha \leq n\). The result is used to study the mapping properties of \((I - t \Delta)^{-\beta}\) considered as an operator from \(L^p (M,d \mu)\) to \(L^p (M,dx)\), where \(dx\) is the Riemannian measure on \(M\), \(\beta > (n - \alpha)/2p'\), \(1/p + 1/p' = 1\), \(1 \leq p \leq \infty\). MSC: 58J35 Heat and other parabolic equation methods for PDEs on manifolds Keywords:fractal measures; Laplacians; Laplace-Beltrami operator × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Li, P.; Yau, S. T., On the parabolic kernel of Schrödinger operator, Acta Math, 156, 153-201 (1986) · Zbl 0611.58045 · doi:10.1007/BF02399203 [2] Schoen, R andYau, S T Analysis on manifolds,Lectures presented in the Insti tute for Advanced Study, School of Mathematics, Princeton, N J, 1984 [3] Setti, A. G., Gaussian estimates for the heat kernel of the weighted Laplacian and fractal measures, Canad J Math, 44, 1061-1078 (1992) · Zbl 0772.58058 · doi:10.4153/CJM-1992-065-4 [4] Strichartz, R. S., Analysis of the Laplacian on the complete Riemannian manifold, J Funct Anal, 52, 48-79 (1983) · Zbl 0515.58037 · doi:10.1016/0022-1236(83)90090-3 [5] Strichartz, R. S., Spectral asymptotics of fractal measures on Riemannian mani folds, J. Funct Anal, 102, 176-205 (1991) · Zbl 0741.28006 · doi:10.1016/0022-1236(91)90140-Z [6] Watson, G. N., A Treatise on the Theory of Bessel Function (1922), Cambridge: Cambridge Univ Press, Cambridge · JFM 48.0412.02 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.