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

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 -