On functions of bounded β-dimensional mean oscillation
You-Wei Chen, Daniel Spector
Abstract
Abstract In this paper, we define a notion of β-dimensional mean oscillation of functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>:</m:mo> <m:mrow> <m:msub> <m:mi>Q</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> <m:mo>→</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> </m:math> {u:Q_{0}\subset\mathbb{R}^{d}\to\mathbb{R}} which are integrable on β-dimensional subsets of the cube <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Q</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {Q_{0}} : <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mi>u</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mi>BMO</m:mi> <m:mi>β</m:mi> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>Q</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mo>:=</m:mo> <m:mrow> <m:munder> <m:mo movablelimits="false">sup</m:mo> <m:mrow> <m:mi>Q</m:mi> <m:mo>⊂</m:mo> <m:msub> <m:mi>Q</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:munder> <m:mo></m:mo> <m:mrow> <m:munder> <m:mo movablelimits="false">inf</m:mo> <m:mrow> <m:mi>c</m:mi> <m:mo>∈</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:munder> <m:mo></m:mo> <m:mrow> <m:mstyle displaystyle="true"> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mi>l</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>Q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>β</m:mi> </m:msup> </m:mrow> </m:mfrac> </m:mstyle> <m:mo></m:mo> <m:mrow> <m:mstyle displaystyle="true"> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi>Q</m:mi> </m:msub> </m:mstyle> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>-</m:mo> <m:mi>c</m:mi> </m:mrow> <m:mo rspace="4.2pt" stretchy="false">|</m:mo> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:msubsup> <m:mi mathvariant="script">ℋ</m:mi> <m:mi mathvariant="normal">∞</m:mi> <m:mi>β</m:mi> </m:msubsup> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> \displaystyle\|u\|_{\mathrm{BMO}^{\beta}(Q_{0})}\vcentcolon=\sup_{Q\subset Q_{% 0}}\inf_{c\in\mathbb{R}}\frac{1}{l(Q)^{\beta}}\int_{Q}|u-c|\,d\mathcal{H}^{% \beta}_{\infty}, where the supremum is taken over all finite subcubes Q parallel to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>Q</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {Q_{0}} , <jats:inline-formula id="