Abstract
In this work, we present new results on solvability of the equation \(A^{*}(D)f=\mu \) for \(f \in L^{p}\) and positive measure data \(\mu \) associated to an elliptic homogeneous linear differential operator A(D) of order m. Our method is based on (m, p)-energy control of \(\mu \) giving a natural characterization for solutions when \(1\le p < \infty \). We also obtain sufficient conditions in the limiting case \(p=\infty \) using new \(L^{1}\) estimates on measures for elliptic and canceling operators.
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Acknowledgements
The authors would like to thank Prof. Pablo de Nápoli for some discussions on two weighted inequalities and the referee for their careful reading and useful suggestions and comments.
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Victor Biliatto was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES—Grant 88882.441243/2019-01) and the Tiago Picon by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq—Grant 311430/2018-0) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP—Grant 18/15484-7).
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Biliatto, V., Picon, T. A Note on Lebesgue Solvability of Elliptic Homogeneous Linear Equations with Measure Data. J Geom Anal 34, 22 (2024). https://doi.org/10.1007/s12220-023-01457-w
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DOI: https://doi.org/10.1007/s12220-023-01457-w
Keywords
- Divergence-measure vector fields
- Lebesgue solvability
- \(L^{1}\) estimates
- Elliptic equations
- Canceling operators