Dyadic microlocal partitions for position-dependent fiber metrics and Weyl quantization
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| Format: | Preprint |
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2025
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| _version_ | 1866910203765587968 |
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| author | Vergara, Vicente |
| author_facet | Vergara, Vicente |
| contents | We construct a dyadic microlocal partition adapted to a position-dependent fiber metric on phase space. Under uniform ellipticity, the associated fiber norm is equivalent to the Euclidean one; the main effect of the construction is therefore not a new global symbolic order, but the $x$-dependent deformation of the microlocal patches and the derivative losses produced by differentiating the moving normalization. We prove finite-seminorm estimates for the localized symbols, with explicit losses depending on the number of controlled derivatives, and derive corresponding local Weyl quantization bounds through Calderón--Vaillancourt estimates. We also record finite-order Moyal truncation estimates and a semiclassical band normalization. Global recombination is formulated as a conditional Cotlar--Stein criterion with explicit almost-orthogonality hypotheses. Finally, we present two model uses: a patchwise parametrix construction and a compatibility discussion for the Radon transform as a model Fourier integral operator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15183 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dyadic microlocal partitions for position-dependent fiber metrics and Weyl quantization Vergara, Vicente Analysis of PDEs Mathematical Physics 35S05, 35S30 (Primary), 35A18, 44A12, 81Q20, 42B20, 58J40 (Secondary) We construct a dyadic microlocal partition adapted to a position-dependent fiber metric on phase space. Under uniform ellipticity, the associated fiber norm is equivalent to the Euclidean one; the main effect of the construction is therefore not a new global symbolic order, but the $x$-dependent deformation of the microlocal patches and the derivative losses produced by differentiating the moving normalization. We prove finite-seminorm estimates for the localized symbols, with explicit losses depending on the number of controlled derivatives, and derive corresponding local Weyl quantization bounds through Calderón--Vaillancourt estimates. We also record finite-order Moyal truncation estimates and a semiclassical band normalization. Global recombination is formulated as a conditional Cotlar--Stein criterion with explicit almost-orthogonality hypotheses. Finally, we present two model uses: a patchwise parametrix construction and a compatibility discussion for the Radon transform as a model Fourier integral operator. |
| title | Dyadic microlocal partitions for position-dependent fiber metrics and Weyl quantization |
| topic | Analysis of PDEs Mathematical Physics 35S05, 35S30 (Primary), 35A18, 44A12, 81Q20, 42B20, 58J40 (Secondary) |
| url | https://arxiv.org/abs/2510.15183 |