1 results
One-sided $C_{p}$
estimates via $M^{\sharp }$
function
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 6 / December 2023
- Published online by Cambridge University Press:
- 27 December 2022, pp. 2079-2105
- Print publication:
- December 2023
-
- Article
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We recall that $w\in C_{p}^+$
if there exist $\varepsilon >0$
and $C>0$
such that for any $a< b< c$
with $c-b< b-a$
and any measurable set $E\subset (a,b)$
, the following holds\[ \int_{E}w\leq C\left(\frac{|E|}{(c-b)}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M^+\chi_{(a,c)}\right)^{p}w<\infty. \]This condition was introduced by Riveros and de la Torre [33] as a one-sided counterpart of the $C_{p}$
condition studied first by Muckenhoupt and Sawyer [30, 34]. In this paper we show that given $1< p< q<\infty$
if $w\in C_{q}^+$
then\[ \|M^+f\|_{L^{p}(w)}\lesssim\|M^{\sharp,+}f\|_{L^{p}(w)} \]and conversely if such an inequality holds, then $w\in C_{p}^+$
. This result is the one-sided counterpart of Yabuta's main result in [37]. Combining this estimate with known pointwise estimates for $M^{\sharp,+}$
in the literature we recover and extend the result for maximal one-sided singular integrals due to Riveros and de la Torre [33] obtaining counterparts a number of operators.
