Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions

Fabio Berra

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 4, page 1003-1017
  • ISSN: 0011-4642

Abstract

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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 p < . More precisely, given any measurable set E 0 , the estimate w ( { x n : M + , d ( 𝒳 E 0 ) ( x ) > t } ) C [ ( w , v ) ] A p + , d ( ) p t p v ( E 0 ) holds if and only if the pair ( w , v ) belongs to A p + , d ( ) , that is, | E | | Q | [ ( w , v ) ] A p + , d ( ) v ( E ) w ( Q ) 1 / p for every dyadic cube Q and every measurable set E Q + . The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the non-dyadic case in 2 by following the main ideas in L. Forzani, F. J. Martín-Reyes, S. Ombrosi (2011).

How to cite

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Berra, Fabio. "Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions." Czechoslovak Mathematical Journal 72.4 (2022): 1003-1017. <http://eudml.org/doc/298933>.

@article{Berra2022,
abstract = {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\le p<\infty $. More precisely, given any measurable set $E_0$, the estimate \[ w ( \lbrace x\in \mathbb \{R\}^n\colon M^\{+,d\}(\mathcal \{X\}\_\{E\_0\})(x)>t \rbrace )\le \frac\{C[(w,v)]\_\{A\_p^\{+,d\}(\mathcal \{R\})\}^p\}\{t^p\}v(E\_0) \] holds if and only if the pair $(w,v)$ belongs to $A_p^\{+,d\}(\mathcal \{R\})$, that is, \[ \frac\{|E|\}\{|Q|\}\le [(w,v)]\_\{A\_p^\{+,d\}(\mathcal \{R\})\} \Bigl (\frac\{v(E)\}\{w(Q)\}\Bigr )^\{ 1/p\} \] 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).},
author = {Berra, Fabio},
journal = {Czechoslovak Mathematical Journal},
keywords = {restricted weak type; one-sided maximal operator},
language = {eng},
number = {4},
pages = {1003-1017},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions},
url = {http://eudml.org/doc/298933},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Berra, Fabio
TI - Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 1003
EP - 1017
AB - 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\le p<\infty $. More precisely, given any measurable set $E_0$, the estimate \[ w ( \lbrace x\in \mathbb {R}^n\colon M^{+,d}(\mathcal {X}_{E_0})(x)>t \rbrace )\le \frac{C[(w,v)]_{A_p^{+,d}(\mathcal {R})}^p}{t^p}v(E_0) \] holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}(\mathcal {R})$, that is, \[ \frac{|E|}{|Q|}\le [(w,v)]_{A_p^{+,d}(\mathcal {R})} \Bigl (\frac{v(E)}{w(Q)}\Bigr )^{ 1/p} \] 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).
LA - eng
KW - restricted weak type; one-sided maximal operator
UR - http://eudml.org/doc/298933
ER -

References

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  1. Forzani, L., Martín-Reyes, F. J., Ombrosi, S., 10.1090/S0002-9947-2010-05343-7, Trans. Am. Math. Soc. 363 (2011), 1699-1719. (2011) Zbl1218.42008MR2746661DOI10.1090/S0002-9947-2010-05343-7
  2. Kinnunen, J., Saari, O., 10.1016/j.na.2015.07.014, Nonlinear Anal., Theory Methods Appl., Ser. A 131 (2016), 289-299. (2016) Zbl1341.42040MR3427982DOI10.1016/j.na.2015.07.014
  3. Kinnunen, J., Saari, O., 10.2140/apde.2016.9.1711, Anal. PDE 9 (2016), 1711-1736. (2016) Zbl1351.42023MR3570236DOI10.2140/apde.2016.9.1711
  4. Lerner, A. K., Ombrosi, S., 10.5565/PUBLMAT_54110_03, Publ. Mat., Barc. 54 (2010), 53-71. (2010) Zbl1183.42024MR2603588DOI10.5565/PUBLMAT_54110_03
  5. Martín-Reyes, F. J., 10.1090/S0002-9939-1993-1111435-2, Proc. Am. Math. Soc. 117 (1993), 691-698. (1993) Zbl0771.42011MR1111435DOI10.1090/S0002-9939-1993-1111435-2
  6. Martín-Reyes, F. J., Torre, A. de la, 10.1090/S0002-9939-1993-1110548-9, Proc. Am. Math. Soc. 117 (1993), 483-489. (1993) Zbl0769.42010MR1110548DOI10.1090/S0002-9939-1993-1110548-9
  7. Martín-Reyes, F. J., Salvador, P. Ortega, Torre, A. de la, 10.1090/S0002-9947-1990-0986694-9, Trans. Am. Math. Soc. 319 (1990), 517-534. (1990) Zbl0696.42013MR986694DOI10.1090/S0002-9947-1990-0986694-9
  8. Ombrosi, S., 10.1090/S0002-9939-05-07830-5, Proc. Am. Math. Soc. 133 (2005), 1769-1775. (2005) Zbl1063.42011MR2120277DOI10.1090/S0002-9939-05-07830-5
  9. Salvador, P. Ortega, 10.4064/sm-131-2-101-114, Stud. Math. 131 (1998), 101-114. (1998) Zbl0922.42012MR1636403DOI10.4064/sm-131-2-101-114
  10. Sawyer, E., 10.1090/S0002-9947-1986-0849466-0, Trans. Am. Math. Soc. 297 (1986), 53-61. (1986) Zbl0627.42009MR849466DOI10.1090/S0002-9947-1986-0849466-0

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