On maximal functions generated by H\'ormander-type spectral multipliers
| Autor: | Chen, Peng, Lin, Xixi, Wu, Liangchuan, Yan, Lixin |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $(X,d,\mu)$ be a metric space with doubling measure and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. We assume that there exists an $L$-harmonic function $h$ such that the semigroup $\exp(-tL)$, after applying the Doob transform related to $h$, satisfies the upper and lower Gaussian estimates. In this paper we apply the Doob transform and some techniques as in Grafakos-Honz\'ik-Seeger \cite{GHS2006} to obtain an optimal $\sqrt{\log(1+N)}$ bound in $L^p$ for the maximal function $\sup_{1\leq i\leq N}|m_i(L)f|$ for multipliers $m_i,1\leq i\leq N,$ with uniform estimates. Based on this, we establish sufficient conditions on the bounded Borel function $m$ such that the maximal function $M_{m,L}f(x) = \sup_{t>0} |m(tL)f(x)|$ is bounded on $L^p(X)$. The applications include Schr\"odinger operators with inverse square potential, Scattering operators, Bessel operators and Laplace-Beltrami operators. Comment: 37 pages |
| Databáze: | arXiv |
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