Article
| Title: | Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions (English) |
| Author: | Berra, Fabio |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 1003-1017 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality for $1\leq p<\infty $. More precisely, given any measurable set $E_0$, the estimate $$ w ( \{x\in \mathbb {R}^n\colon M^{+,d}(\mathcal {X}_{E_0})(x)>t \})\leq \frac {C[(w,v)]_{A_p^{+,d}(\mathcal {R})}^p}{t^p}v(E_0) $$ holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}(\mathcal {R})$, that is, $$ \frac {|E|}{|Q|}\leq [(w,v)]_{A_p^{+,d}(\mathcal {R})} \Bigl (\frac {v(E)}{w(Q)}\Bigr )^{ 1/p} $$ for every dyadic cube $Q$ and every measurable set $E\subset Q^+$. The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the non-dyadic case in $\mathbb {R}^2$ by following the main ideas in L. Forzani, F. J. Martín-Reyes, S. Ombrosi (2011). (English) |
| Keyword: | restricted weak type |
| Keyword: | one-sided maximal operator |
| MSC: | 28B99 |
| MSC: | 42B25 |
| idZBL: | Zbl 07655777 |
| idMR: | MR4517590 |
| DOI: | 10.21136/CMJ.2022.0296-21 |
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| Date available: | 2022-11-28T11:34:20Z |
| Last updated: | 2023-04-11 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151124 |
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| Reference: | [1] Forzani, L., Martín-Reyes, F. J., Ombrosi, S.: Weighted inequalities for the two-dimensional one-sided Hardy-Littlewood maximal function.Trans. Am. Math. Soc. 363 (2011), 1699-1719. Zbl 1218.42008, MR 2746661, 10.1090/S0002-9947-2010-05343-7 |
| Reference: | [2] Kinnunen, J., Saari, O.: On weights satisfying parabolic Muckenhoupt conditions.Nonlinear Anal., Theory Methods Appl., Ser. A 131 (2016), 289-299. Zbl 1341.42040, MR 3427982, 10.1016/j.na.2015.07.014 |
| Reference: | [3] Kinnunen, J., Saari, O.: Parabolic weighted norm inequalities and partial differential equations.Anal. PDE 9 (2016), 1711-1736. Zbl 1351.42023, MR 3570236, 10.2140/apde.2016.9.1711 |
| Reference: | [4] Lerner, A. K., Ombrosi, S.: A boundedness criterion for general maximal operators.Publ. Mat., Barc. 54 (2010), 53-71. Zbl 1183.42024, MR 2603588, 10.5565/PUBLMAT_54110_03 |
| Reference: | [5] Martín-Reyes, F. J.: New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions.Proc. Am. Math. Soc. 117 (1993), 691-698. Zbl 0771.42011, MR 1111435, 10.1090/S0002-9939-1993-1111435-2 |
| Reference: | [6] Martín-Reyes, F. J., Torre, A. de la: Two weight norm inequalities for fractional one-sided maximal operators.Proc. Am. Math. Soc. 117 (1993), 483-489. Zbl 0769.42010, MR 1110548, 10.1090/S0002-9939-1993-1110548-9 |
| Reference: | [7] Martín-Reyes, F. J., Salvador, P. Ortega, Torre, A. de la: Weighted inequalities for one- sided maximal functions.Trans. Am. Math. Soc. 319 (1990), 517-534. Zbl 0696.42013, MR 986694, 10.1090/S0002-9947-1990-0986694-9 |
| Reference: | [8] Ombrosi, S.: Weak weighted inequalities for a dyadic one-sided maximal function in ${\mathbb R}^n$.Proc. Am. Math. Soc. 133 (2005), 1769-1775. Zbl 1063.42011, MR 2120277, 10.1090/S0002-9939-05-07830-5 |
| Reference: | [9] Salvador, P. Ortega: Weighted inequalities for one-sided maximal functions in Orlicz spaces.Stud. Math. 131 (1998), 101-114. Zbl 0922.42012, MR 1636403, 10.4064/sm-131-2-101-114 |
| Reference: | [10] Sawyer, E.: Weighted inequalities for the one-sided Hardy-Littlewood maximal functions.Trans. Am. Math. Soc. 297 (1986), 53-61. Zbl 0627.42009, MR 849466, 10.1090/S0002-9947-1986-0849466-0 |
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