$L^p$-asymptotic behaviour of solutions of the heat equation on Riemannian symmetric spaces of noncompact type
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914101290074112 |
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| author | Naik, Muna Ray, Swagato K. Sarkar, Jayanta |
| author_facet | Naik, Muna Ray, Swagato K. Sarkar, Jayanta |
| contents | For Riemannian symmetric spaces $X=G/K$ of noncompact type, we show that for all left $K$-invariant $f\in L^1(X)$, the functions $\|h_t\|_{L^p(X)}^{-1}(f\ast h_t-M_p(f)h_t)$ (with $h_t$ being the heat kernel of $X$) converges to zero in $L^p(X)$, $p\in [1,\infty]$, as $t\to\infty$, with the constant $M_p(f)$ depending only on $p$ and $f$. We also prove an analogous result for the fractional heat kernels $h_t^α$, $α\in (0,1)$. The above results have recently been proved for the important special cases $p=1$ and $α= 1,\frac 12$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^p$-asymptotic behaviour of solutions of the heat equation on Riemannian symmetric spaces of noncompact type Naik, Muna Ray, Swagato K. Sarkar, Jayanta Classical Analysis and ODEs Primary 43A85, Secondary 22E30, 58J35, 35B40 For Riemannian symmetric spaces $X=G/K$ of noncompact type, we show that for all left $K$-invariant $f\in L^1(X)$, the functions $\|h_t\|_{L^p(X)}^{-1}(f\ast h_t-M_p(f)h_t)$ (with $h_t$ being the heat kernel of $X$) converges to zero in $L^p(X)$, $p\in [1,\infty]$, as $t\to\infty$, with the constant $M_p(f)$ depending only on $p$ and $f$. We also prove an analogous result for the fractional heat kernels $h_t^α$, $α\in (0,1)$. The above results have recently been proved for the important special cases $p=1$ and $α= 1,\frac 12$. |
| title | $L^p$-asymptotic behaviour of solutions of the heat equation on Riemannian symmetric spaces of noncompact type |
| topic | Classical Analysis and ODEs Primary 43A85, Secondary 22E30, 58J35, 35B40 |
| url | https://arxiv.org/abs/2404.09985 |