Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian
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| Format: | Preprint |
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2025
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| author | Bui, The Anh Mukherjee, Suman |
| author_facet | Bui, The Anh Mukherjee, Suman |
| contents | We establish fractional Leibniz rules for the Dunkl Laplacian $Δ_k$ of the form $$\|(-Δ_k)^s(fg)\|_{L^p(dμ_k)} \lesssim \|(-Δ_k)^s f\|_{L^{p_1}(dμ_k)} \|g\|_{L^{p_2}(dμ_k)} + \|f\|_{L^{p_1}(dμ_k)} \|(-Δ_k)^s g\|_{L^{p_2}(dμ_k)}.$$ Our approach relies on adapting the classical paraproduct decomposition to the Dunkl setting. In the process, we develop several new auxiliary results. Specifically, we show that for a Schwartz function $f$, the function $(-Δ_k)^s f$ satisfies a pointwise decay estimate; we establish a version of almost orthogonality estimates adapted to the Dunkl framework; and we investigate the boundedness of Dunkl paraproduct operators on the Lebesgue spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10042 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian Bui, The Anh Mukherjee, Suman Functional Analysis Analysis of PDEs 42B20, 42B25, 26A33, 44A15 We establish fractional Leibniz rules for the Dunkl Laplacian $Δ_k$ of the form $$\|(-Δ_k)^s(fg)\|_{L^p(dμ_k)} \lesssim \|(-Δ_k)^s f\|_{L^{p_1}(dμ_k)} \|g\|_{L^{p_2}(dμ_k)} + \|f\|_{L^{p_1}(dμ_k)} \|(-Δ_k)^s g\|_{L^{p_2}(dμ_k)}.$$ Our approach relies on adapting the classical paraproduct decomposition to the Dunkl setting. In the process, we develop several new auxiliary results. Specifically, we show that for a Schwartz function $f$, the function $(-Δ_k)^s f$ satisfies a pointwise decay estimate; we establish a version of almost orthogonality estimates adapted to the Dunkl framework; and we investigate the boundedness of Dunkl paraproduct operators on the Lebesgue spaces. |
| title | Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian |
| topic | Functional Analysis Analysis of PDEs 42B20, 42B25, 26A33, 44A15 |
| url | https://arxiv.org/abs/2507.10042 |