Estimates for pseudo-differential operators on the torus revisited. III
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909791554633728 |
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| author | Cardona, Duván Martínez, Manuel Alejandro |
| author_facet | Cardona, Duván Martínez, Manuel Alejandro |
| contents | This paper finishes the goal of the authors started in two previous manuscripts dedicated to revisiting the continuity properties of toroidal pseudo-differential operators with symbols in the Hörmander classes. Here we prove pointwise estimates in terms of the Fefferman-Stein sharp maximal function and of the Hardy-Littlewood maximal function. Combining these estimates with the properties of Muckenhoupt's weight class $A_p$ we obtain boundedness theorems for pseudo-differential operators between weighted Lebesgue spaces on the torus $L^p(w)$. These results are given in the context of the global symbolic analysis defined on $\mathbb{T}^n\times \mathbb{Z}^n$ as developed by Ruzhansky and Turunen by using discrete Fourier analysis, and extend those of Park and Tomita available in the Euclidean case. Moreover, we include continuity results on Sobolev spaces $W^s_p$ and on Besov spaces $B^s_{p,q}$ on the torus. Our techniques are taken from Park and Tomita \cite{park-tomita} and we consider its toroidal extension here for the completeness of the boundedness of toroidal pseudo-differential operators with respect to the current literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_13338 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimates for pseudo-differential operators on the torus revisited. III Cardona, Duván Martínez, Manuel Alejandro Analysis of PDEs 22E30, 58J40 This paper finishes the goal of the authors started in two previous manuscripts dedicated to revisiting the continuity properties of toroidal pseudo-differential operators with symbols in the Hörmander classes. Here we prove pointwise estimates in terms of the Fefferman-Stein sharp maximal function and of the Hardy-Littlewood maximal function. Combining these estimates with the properties of Muckenhoupt's weight class $A_p$ we obtain boundedness theorems for pseudo-differential operators between weighted Lebesgue spaces on the torus $L^p(w)$. These results are given in the context of the global symbolic analysis defined on $\mathbb{T}^n\times \mathbb{Z}^n$ as developed by Ruzhansky and Turunen by using discrete Fourier analysis, and extend those of Park and Tomita available in the Euclidean case. Moreover, we include continuity results on Sobolev spaces $W^s_p$ and on Besov spaces $B^s_{p,q}$ on the torus. Our techniques are taken from Park and Tomita \cite{park-tomita} and we consider its toroidal extension here for the completeness of the boundedness of toroidal pseudo-differential operators with respect to the current literature. |
| title | Estimates for pseudo-differential operators on the torus revisited. III |
| topic | Analysis of PDEs 22E30, 58J40 |
| url | https://arxiv.org/abs/2508.13338 |