Abstract
In the present work, we revisit several structural properties of almost automorphic and compact almost automorphic functions from the Euclidean space \({\mathbb {R}}^m\) (\(m \ge 1\)) with values in a Banach space \({\mathbb {X}}\). When \({\mathbb {X}}\) is a Banach algebra, it is proven that the spaces formed by these functions are also Banach algebras. As applications, first we prove regularity of almost automorphic solution of Poisson’s equation; that is, we prove that bounded continuous functions with weak (distributional) almost automorphic Laplacian are compact almost automorphic; then, we prove the compact almost automorphy in space variable of classical solutions to heat equation with almost automorphic initial datum.
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Acknowledgements
The authors would like to express their gratitude to the anonymous referees for their careful reading and helpful comments.
Funding
Alan Chávez is supported by Grant 038-2021-Fondecyt Perú. Manuel Pinto is partially supported by Grant 038-2021-Fondecyt Perú and grand 1170466 Fondecyt-Chile.
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Communicated by Anton Abdulbasah Kamil.
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Chávez, A., Khalil, K., Pereyra, A. et al. Compact Almost Automorphic Solutions to Poisson’s and Heat Equations. Bull. Malays. Math. Sci. Soc. 47, 43 (2024). https://doi.org/10.1007/s40840-023-01637-5
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DOI: https://doi.org/10.1007/s40840-023-01637-5