<p>We extend the subrepresentation formula <Equation ID="Equ19"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="44007_2025_151_Article_Equ19.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} |f(x)|\le c_n\,I_1(|\nabla f|)(x) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <msub> <mi>c</mi> <mi>n</mi> </msub> <mspace width="0.166667em" /> <msub> <mi>I</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in several ways. First, we consider more general <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44007_2025_151_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$A_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-potential operators on the right-hand side and prove local and global pointwise inequalities for these operators. Second, we show that we can improve the right-hand side using fractional derivatives. Finally, we extend our results to rough singular integral operators, similar to the main result in [HMP1].</p>

A New Look at the Subrepresentation Formulas

期刊论文
发表日期：

2025年4月28日

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Cong Hoang,
Kabe Moen,
Carlos Pérez

摘要

We extend the subrepresentation formula $\begin{aligned} |f(x)|\le c_n\,I_1(|\nabla f|)(x) \end{aligned}$ $\begin{matrix} | f (x) | \leq c_{n} I_{1} (| \nabla f |) (x) \end{matrix}$ in several ways. First, we consider more general $A_1$ $A_{1}$ -potential operators on the right-hand side and prove local and global pointwise inequalities for these operators. Second, we show that we can improve the right-hand side using fractional derivatives. Finally, we extend our results to rough singular integral operators, similar to the main result in [HMP1].