Restricted weighted weak boundedness for product type operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913557316108288 |
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| author | Carro, María Jesús Ombrosi, Sheldy |
| author_facet | Carro, María Jesús Ombrosi, Sheldy |
| contents | Given a bilinear (or sub-bilinear) operator $B$, we prove restricted weighted weak type inequalities of the form $$ ||B(f_1, f_2)||_{L^{p, \infty}(w_1^{p/p_1}w_2^{p/p_2})}\lesssim ||f_1||_{L^{p_1, 1}(w_1)}||f_2||_{L^{p_2, 1}(w_2)}, $$ whenever $B(f_1, f_2)= (T_1f_1) (T_2 f_2)$ is the product of two singular integral operators satisfying Dini conditions. Additionally, we also establish, as an application, the boundedness of a certain class of bounded variation bilinear Fourier multipliers solving a question posted in [Bilinear Fourier multipliers of bounded variation; Int. Math. Res. Not. (2023), no.24, 21943--21975 by Baena-Miret, Carro, Luque and Sanchez-Pascuala]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_15759 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Restricted weighted weak boundedness for product type operators Carro, María Jesús Ombrosi, Sheldy Classical Analysis and ODEs Functional Analysis 47B90, 42B15 Given a bilinear (or sub-bilinear) operator $B$, we prove restricted weighted weak type inequalities of the form $$ ||B(f_1, f_2)||_{L^{p, \infty}(w_1^{p/p_1}w_2^{p/p_2})}\lesssim ||f_1||_{L^{p_1, 1}(w_1)}||f_2||_{L^{p_2, 1}(w_2)}, $$ whenever $B(f_1, f_2)= (T_1f_1) (T_2 f_2)$ is the product of two singular integral operators satisfying Dini conditions. Additionally, we also establish, as an application, the boundedness of a certain class of bounded variation bilinear Fourier multipliers solving a question posted in [Bilinear Fourier multipliers of bounded variation; Int. Math. Res. Not. (2023), no.24, 21943--21975 by Baena-Miret, Carro, Luque and Sanchez-Pascuala]. |
| title | Restricted weighted weak boundedness for product type operators |
| topic | Classical Analysis and ODEs Functional Analysis 47B90, 42B15 |
| url | https://arxiv.org/abs/2410.15759 |