论文标题
通过$ m^{\ sharp} $ function的单面$ c_ {p} $估算
One-sided $C_{p}$ estimates via $M^{\sharp}$ function
论文作者
论文摘要
我们记得，如果存在$ \ varepsilon> 0 $和$ c> 0 $，则$ w \ in c_ {p}^{+} $，以便对于任何$ a <b <c $ a <b <c $带有$ c-b <b-a $以及任何可测量的套件$ e \ exeulset（a，b）$ c \ left（\ frac {| e |} {（c-b）} \ right）^{\ varepsilon} \ int _ {\ mathbb {r}} \ left（m^{+}χ_ {（+}χ_ {（a，c）}}} \ right）^p} \]这种情况是由Riveros和de la Torre引入的，是Muckenhoupt和Sawyer首先研究的$ C_ {P} $条件的单方面。在本文中，我们表明给定$ 1 <p <q <q <\ infty $如果$ w \ in c_ {q}^{+} $ then \ [\ | m^{+} f \ | _ {l^{l^{p}（p}（w）}（w）} \ simsim \ simsim \ | m^| m^| m^{\ sharp，+sharp，+sharp，+} p。如果这样的不平等，则在c_ {p}^{+}中$ w \ $ w \ $
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 \[ \int_{E}w\leq C\left(\frac{|E|}{(c-b)}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M^{+}χ_{(a,c)}\right)^{p}w<\infty. \] This condition was introduced by Riveros and de la Torre as a one-sided counterpart of the $C_{p}$ condition studied first by Muckenhoupt and Sawyer. 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 $w\in C_{p}^{+}.$
