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Keywords:
restricted weak type; one-sided maximal operator
restricted weak type; one-sided maximal operator
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).
References:
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