Growth Estimates for Solutions to the Wave Equation on Damek--Ricci Spaces
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908671549636608 |
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| author | Wang, Yunxiang Yan, Lixin Zhang, Hong-Wei |
| author_facet | Wang, Yunxiang Yan, Lixin Zhang, Hong-Wei |
| contents | Let $\mathcal{L}$ be the left-invariant distinguished Laplacian, and let $\mathrm{d}ρ$ denote the right Haar measure on a Damek--Ricci space $S$. Let $u(t,x)$ denote the solution to the wave equation $\partial_t^2 u-\mathcal{L} u=0$ with initial data $(u,\partial_t u)|_{t=0}=(f,g)$. In this paper, we establish the sharp-in-regularity $L^p$ bounds \begin{align*} \|u(t,\cdot)\|_{L^p(S ,\mathrm{d}ρ)} \lesssim_p (1+|t|)^{2|\frac{1}{p}-\frac{1}{2}|}\|(\mathrm{Id}+\mathcal{L})^{\frac{α_0}{2}}\!f\|_{L^p(S ,\mathrm{d}ρ)}+(1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\frac{α_1}{2}}\!g\|_{L^p(S,\mathrm{d}ρ)} \end{align*} for all $t\in\mathbb{R}^*$ and $1<p<\infty$, where the exponents $α_0 = n\left|1/p-1/2\right|$ and $α_1 = n\left|1/p-1/2\right| -1$ attain their critical values. This result settles, in full generality, the conjecture raised by Müller, Thiele, and Vallarino. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_18995 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Growth Estimates for Solutions to the Wave Equation on Damek--Ricci Spaces Wang, Yunxiang Yan, Lixin Zhang, Hong-Wei Classical Analysis and ODEs Analysis of PDEs 43A85, 22E30, 42B15, 35L20 Let $\mathcal{L}$ be the left-invariant distinguished Laplacian, and let $\mathrm{d}ρ$ denote the right Haar measure on a Damek--Ricci space $S$. Let $u(t,x)$ denote the solution to the wave equation $\partial_t^2 u-\mathcal{L} u=0$ with initial data $(u,\partial_t u)|_{t=0}=(f,g)$. In this paper, we establish the sharp-in-regularity $L^p$ bounds \begin{align*} \|u(t,\cdot)\|_{L^p(S ,\mathrm{d}ρ)} \lesssim_p (1+|t|)^{2|\frac{1}{p}-\frac{1}{2}|}\|(\mathrm{Id}+\mathcal{L})^{\frac{α_0}{2}}\!f\|_{L^p(S ,\mathrm{d}ρ)}+(1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\frac{α_1}{2}}\!g\|_{L^p(S,\mathrm{d}ρ)} \end{align*} for all $t\in\mathbb{R}^*$ and $1<p<\infty$, where the exponents $α_0 = n\left|1/p-1/2\right|$ and $α_1 = n\left|1/p-1/2\right| -1$ attain their critical values. This result settles, in full generality, the conjecture raised by Müller, Thiele, and Vallarino. |
| title | Growth Estimates for Solutions to the Wave Equation on Damek--Ricci Spaces |
| topic | Classical Analysis and ODEs Analysis of PDEs 43A85, 22E30, 42B15, 35L20 |
| url | https://arxiv.org/abs/2511.18995 |