On maximal functions generated by Hörmander-type spectral multipliers
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| Format: | Preprint |
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2024
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| _version_ | 1866916420609114112 |
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| author | Chen, Peng Lin, Xixi Wu, Liangchuan Yan, Lixin |
| author_facet | Chen, Peng Lin, Xixi Wu, Liangchuan Yan, Lixin |
| contents | Let $(X,d,μ)$ 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-Honzík-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 Schrödinger operators with inverse square potential, Scattering operators, Bessel operators and Laplace-Beltrami operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_01164 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On maximal functions generated by Hörmander-type spectral multipliers Chen, Peng Lin, Xixi Wu, Liangchuan Yan, Lixin Classical Analysis and ODEs 42B15, 42B25, 47F10 Let $(X,d,μ)$ 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-Honzík-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 Schrödinger operators with inverse square potential, Scattering operators, Bessel operators and Laplace-Beltrami operators. |
| title | On maximal functions generated by Hörmander-type spectral multipliers |
| topic | Classical Analysis and ODEs 42B15, 42B25, 47F10 |
| url | https://arxiv.org/abs/2410.01164 |