Local $l^\infty$ bounds for eigenfunctions of complex elliptic operators via diophantine problems
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911295613173760 |
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| author | Friedland, Omer Ueberschaer, Henrik |
| author_facet | Friedland, Omer Ueberschaer, Henrik |
| contents | We prove local bounds on the amplitude of eigen- functions of complex constant-coefficient elliptic operators with a smooth potential on an arbitrary open subset of \R^d by estimating it in terms of the number of solutions of a diophantine inequality arising from the symbol of the operator. In the special case of positive elliptic operators, we recover H örmander's classical exponent up to an arbitrarily small loss. We show that a much better exponent may be obtained when the principal symbol of the oper- ator has complex coefficients. We generalize our estimate to any higher-order derivatives of eigenfunctions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10665 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local $l^\infty$ bounds for eigenfunctions of complex elliptic operators via diophantine problems Friedland, Omer Ueberschaer, Henrik Analysis of PDEs Mathematical Physics We prove local bounds on the amplitude of eigen- functions of complex constant-coefficient elliptic operators with a smooth potential on an arbitrary open subset of \R^d by estimating it in terms of the number of solutions of a diophantine inequality arising from the symbol of the operator. In the special case of positive elliptic operators, we recover H örmander's classical exponent up to an arbitrarily small loss. We show that a much better exponent may be obtained when the principal symbol of the oper- ator has complex coefficients. We generalize our estimate to any higher-order derivatives of eigenfunctions. |
| title | Local $l^\infty$ bounds for eigenfunctions of complex elliptic operators via diophantine problems |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2407.10665 |