A Generalization of Littlewood-Paley Type Inequality for Evolution Systems Associated with Pseudo Differential Operators
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866929724054306816 |
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| author | Ji, Un Cig Kim, Jae Hun |
| author_facet | Ji, Un Cig Kim, Jae Hun |
| contents | In this paper, we first prove that the Littlewood-Paley $g$-function, related to the convolution corresponding to the composition of pseudo-differential operator and evolution system associated with pseudo-differential operators, is a bounded operator from $L^{q}((a,b)\times \mathbb{R}^{d};V)$ with a Hilbert space $V$ into $L^{q}((a,b)\times \mathbb{R}^{d})$. Secondly, we prove that the sharp function of the Littlewood-Paley $g$-function is bounded by some maximal function. Finally, by applying Fefferman-Stein theorem and Hardy-Littlewood maximal theorem, we prove the Littlewood-Paley type inequality for evolution systems associated with pseudo-differential operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15221 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Generalization of Littlewood-Paley Type Inequality for Evolution Systems Associated with Pseudo Differential Operators Ji, Un Cig Kim, Jae Hun Analysis of PDEs 42B25, 42B37, 47G30 In this paper, we first prove that the Littlewood-Paley $g$-function, related to the convolution corresponding to the composition of pseudo-differential operator and evolution system associated with pseudo-differential operators, is a bounded operator from $L^{q}((a,b)\times \mathbb{R}^{d};V)$ with a Hilbert space $V$ into $L^{q}((a,b)\times \mathbb{R}^{d})$. Secondly, we prove that the sharp function of the Littlewood-Paley $g$-function is bounded by some maximal function. Finally, by applying Fefferman-Stein theorem and Hardy-Littlewood maximal theorem, we prove the Littlewood-Paley type inequality for evolution systems associated with pseudo-differential operators. |
| title | A Generalization of Littlewood-Paley Type Inequality for Evolution Systems Associated with Pseudo Differential Operators |
| topic | Analysis of PDEs 42B25, 42B37, 47G30 |
| url | https://arxiv.org/abs/2502.15221 |