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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.09351 |
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| _version_ | 1866929313717157888 |
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| author | Perdices, Eduard Roure |
| author_facet | Perdices, Eduard Roure |
| contents | We present a multi-variable extension of Rubio de Francia's restricted weak-type extrapolation theory that does not involve Rubio de Francia's iteration algorithm; instead, we rely on the following Sawyer-type inequality for the weighted Hardy-Littlewood maximal operator $M_u$:
$$ \left \Vert \frac{M_u (fv)}{v} \right \Vert_{L^{1,\infty}(uv)} \leq C_{u,v} \Vert f \Vert_{L^1(uv)}, \quad u, \, uv \in A_{\infty}. $$
Our approach can be adapted to recover weak-type $A_{\vec P}$ extrapolation schemes, including an endpoint result that falls outside the classical theory.
Among the applications of our work, we highlight extending outside the Banach range the well-known equivalence between restricted weak-type and weak-type for characteristic functions, and obtaining mixed and restricted weak-type bounds with $A_{p}^{\mathcal R}$ weights for relevant families of multi-variable operators, addressing the lack in the literature of these types of estimates. We also reveal several standalone properties of the class $A_{p}^{\mathcal R}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extrapolation via Sawyer-type inequalities Perdices, Eduard Roure Functional Analysis Classical Analysis and ODEs 42B25 (Primary), 46E30 (Secondary) We present a multi-variable extension of Rubio de Francia's restricted weak-type extrapolation theory that does not involve Rubio de Francia's iteration algorithm; instead, we rely on the following Sawyer-type inequality for the weighted Hardy-Littlewood maximal operator $M_u$: $$ \left \Vert \frac{M_u (fv)}{v} \right \Vert_{L^{1,\infty}(uv)} \leq C_{u,v} \Vert f \Vert_{L^1(uv)}, \quad u, \, uv \in A_{\infty}. $$ Our approach can be adapted to recover weak-type $A_{\vec P}$ extrapolation schemes, including an endpoint result that falls outside the classical theory. Among the applications of our work, we highlight extending outside the Banach range the well-known equivalence between restricted weak-type and weak-type for characteristic functions, and obtaining mixed and restricted weak-type bounds with $A_{p}^{\mathcal R}$ weights for relevant families of multi-variable operators, addressing the lack in the literature of these types of estimates. We also reveal several standalone properties of the class $A_{p}^{\mathcal R}$. |
| title | Extrapolation via Sawyer-type inequalities |
| topic | Functional Analysis Classical Analysis and ODEs 42B25 (Primary), 46E30 (Secondary) |
| url | https://arxiv.org/abs/2404.09351 |