Fractional Hardy's inequality for half spaces in the Heisenberg group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910743740284928 |
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| author | Rawat, Rama Roy, Haripada |
| author_facet | Rawat, Rama Roy, Haripada |
| contents | We establish the following fractional Hardy's inequality $$\int_{\mathbb{H}^n_+}\frac{|f(ξ)|^p}{x_1^{sp}|z|^α}dξ\leq C\int_{\mathbb{H}^n_+}\int_{\mathbb{H}^n_+}\frac{|f(ξ)-f(ξ')|^p}{d(ξ^{-1}\circ ξ')^{Q+sp}|z'-z|^α}dξ'dξ,\ \ \forall\,f\in C_c(\mathbb{H}^n_+)$$ for the half space $\mathbb{H}^n_+:=\{ξ=(z,t)=(x_1,x_2,\ldots, x_n, y_1,y_2,\ldots,y_n,t)\in\mathbb{H}^n:x_1>0\}$ in the Heisenberg group $\mathbb{H}^n$ under the conditions $sp>1$ and $α\geq (2n+sp)/2$. We also provide an alternate proof of a fractional Hardy's inequality in $\mathbb{H}^n$ established in an earlier work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_07439 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional Hardy's inequality for half spaces in the Heisenberg group Rawat, Rama Roy, Haripada Analysis of PDEs 46E35, 26D15, 43A80 We establish the following fractional Hardy's inequality $$\int_{\mathbb{H}^n_+}\frac{|f(ξ)|^p}{x_1^{sp}|z|^α}dξ\leq C\int_{\mathbb{H}^n_+}\int_{\mathbb{H}^n_+}\frac{|f(ξ)-f(ξ')|^p}{d(ξ^{-1}\circ ξ')^{Q+sp}|z'-z|^α}dξ'dξ,\ \ \forall\,f\in C_c(\mathbb{H}^n_+)$$ for the half space $\mathbb{H}^n_+:=\{ξ=(z,t)=(x_1,x_2,\ldots, x_n, y_1,y_2,\ldots,y_n,t)\in\mathbb{H}^n:x_1>0\}$ in the Heisenberg group $\mathbb{H}^n$ under the conditions $sp>1$ and $α\geq (2n+sp)/2$. We also provide an alternate proof of a fractional Hardy's inequality in $\mathbb{H}^n$ established in an earlier work. |
| title | Fractional Hardy's inequality for half spaces in the Heisenberg group |
| topic | Analysis of PDEs 46E35, 26D15, 43A80 |
| url | https://arxiv.org/abs/2412.07439 |