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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.16626 |
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| _version_ | 1866914400676347904 |
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| author | Biliatto, Victor Coacalle, Joel Picon, Tiago |
| author_facet | Biliatto, Victor Coacalle, Joel Picon, Tiago |
| contents | Let $A(D)$ be an elliptic homogeneous linear differential operator with complex constant coefficients, $ μ$ be a vector-valued Borel measure and $w$ be a positive locally integrable function on $\mathbb{R}^N$. In this work, we present sufficient conditions on $μ$ and $w$ for the existence of solutions in the weighted Lebesgue spaces $L^p_w$ for the equation $A^{*}(D)f=μ$, for $ 1\leq p<\infty $. Those conditions are related to a certain control of the Riesz potential of the measure $μ$. We also present sufficient conditions for the solvability when $p=\infty$ adding a canceling condition on the operator. Our method is based on a new weighted $L^1$ Stein-Weiss type inequality on measures for a special class of vector fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16626 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solvability of elliptic homogeneous linear equations with measure data in weighted Lebesgue spaces Biliatto, Victor Coacalle, Joel Picon, Tiago Analysis of PDEs 47F05 35A23 35B45 35J48 28A12 Let $A(D)$ be an elliptic homogeneous linear differential operator with complex constant coefficients, $ μ$ be a vector-valued Borel measure and $w$ be a positive locally integrable function on $\mathbb{R}^N$. In this work, we present sufficient conditions on $μ$ and $w$ for the existence of solutions in the weighted Lebesgue spaces $L^p_w$ for the equation $A^{*}(D)f=μ$, for $ 1\leq p<\infty $. Those conditions are related to a certain control of the Riesz potential of the measure $μ$. We also present sufficient conditions for the solvability when $p=\infty$ adding a canceling condition on the operator. Our method is based on a new weighted $L^1$ Stein-Weiss type inequality on measures for a special class of vector fields. |
| title | Solvability of elliptic homogeneous linear equations with measure data in weighted Lebesgue spaces |
| topic | Analysis of PDEs 47F05 35A23 35B45 35J48 28A12 |
| url | https://arxiv.org/abs/2504.16626 |