Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 1003-1017, 2022
Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions
Fabio Berra
Received August 20, 2021. Published online May 5, 2022.
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 $1\leq p<\infty$. 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. Martin-Reyes, S. Ombrosi (2011).
Affiliations: Fabio Berra, CONICET and Universidad Nacional del Litoral, Bv. Pellegrini 2750, (3000) Santa Fe, Argentina e-mail: fberra@santafe-conicet.gov.ar
