Hypoellipticity and Higher Order Gaussian Bounds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917195300208640 |
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| author | Street, Brian |
| author_facet | Street, Brian |
| contents | Let $(\mathfrak{M},ρ,μ)$ be a metric measure space satisfying a doubling condition, $p_0\in (1,\infty)$, and $T(t):L^{p_0}(\mathfrak{M},μ)\rightarrow L^{p_0}(\mathfrak{M},μ)$, $t\geq 0$, a strongly continuous semi-group. We provide sufficient conditions under which $T(t)$ is given by integration against an integral kernel satisfying higher-order Gaussian bounds of the form \[
\left| K_t(x,y) \right| \leq C \exp\left( -c \left( \frac{ρ(x,y)^{2κ}}{t} \right)^{\frac{1}{2κ-1}} \right) μ\left( B_ρ\left(x,ρ(x,y)+t^{1/2κ}\right) \right)^{-1}, \] where $B_ρ$ denotes the metric ball. We also provide conditions for similar bounds on ``derivatives'' of $K_t(x,y)$ and our results are localizable. If $A$ is the generator of $T(t)$ the main hypothesis is that $\partial_t -A$ and $\partial_t-A^{*}$ satisfy a hypoelliptic estimate at every scale, uniformly in the scale. We present applications to subelliptic PDEs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_14456 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypoellipticity and Higher Order Gaussian Bounds Street, Brian Analysis of PDEs Classical Analysis and ODEs Functional Analysis 2020: 35K08 (Primary), 47D06, 35H10, and 35H20 (Secondary) Let $(\mathfrak{M},ρ,μ)$ be a metric measure space satisfying a doubling condition, $p_0\in (1,\infty)$, and $T(t):L^{p_0}(\mathfrak{M},μ)\rightarrow L^{p_0}(\mathfrak{M},μ)$, $t\geq 0$, a strongly continuous semi-group. We provide sufficient conditions under which $T(t)$ is given by integration against an integral kernel satisfying higher-order Gaussian bounds of the form \[ \left| K_t(x,y) \right| \leq C \exp\left( -c \left( \frac{ρ(x,y)^{2κ}}{t} \right)^{\frac{1}{2κ-1}} \right) μ\left( B_ρ\left(x,ρ(x,y)+t^{1/2κ}\right) \right)^{-1}, \] where $B_ρ$ denotes the metric ball. We also provide conditions for similar bounds on ``derivatives'' of $K_t(x,y)$ and our results are localizable. If $A$ is the generator of $T(t)$ the main hypothesis is that $\partial_t -A$ and $\partial_t-A^{*}$ satisfy a hypoelliptic estimate at every scale, uniformly in the scale. We present applications to subelliptic PDEs. |
| title | Hypoellipticity and Higher Order Gaussian Bounds |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 2020: 35K08 (Primary), 47D06, 35H10, and 35H20 (Secondary) |
| url | https://arxiv.org/abs/2410.14456 |