<p>This paper concerns the variation bounds and continuity of the maximal function <Equation ID="Equ1"> <EquationSource Format="TEX">$M_\Phi f(x)=\underset{s+t>0}{\underset{s,t\ge 0}{\sup }}\Phi (s+t)\int _{x-s}^{x+t}|f(y)|dy,\ \ \ x\in \mathbb {R}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>M</mi> <mi mathvariant="normal">Φ</mi> </msub> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <munder> <mo movablelimits="false">sup</mo> <mrow> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mrow> <mi>s</mi> <mo>+</mo> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mi mathvariant="normal">Φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msubsup> <mo>∫</mo> <mrow> <mi>x</mi> <mo>-</mo> <mi>s</mi> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mi>t</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">|</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mspace width="4pt" /> <mspace width="4pt" /> <mspace width="4pt" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Phi :(0,\infty )\rightarrow (0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a measurable function, which contains some classic maximal operators as its prototypical examples. The variation boundedness and continuity of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$M_\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi mathvariant="normal">Φ</mi> </msub> </math></EquationSource> </InlineEquation> is established under a more restrictive condition on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>. The main results extend some known ones and provide a large class of maximal operators which possess the variation boundedness and continuity.</p>

The Variation of Maximal Function Associated to a Function

期刊论文
发表日期：

2026年2月16日

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Ting Chen,
Feng Liu,
Huoxiong Wu,
Qingying Xue,
Kôzô Yabuta

摘要

This paper concerns the variation bounds and continuity of the maximal function $M_\Phi f(x)=\underset{s+t>0}{\underset{s,t\ge 0}{\sup }}\Phi (s+t)\int _{x-s}^{x+t}|f(y)|dy,\ \ \ x\in \mathbb {R}, $ $M_{Φ} f (x) = \underset{s + t > 0}{sup_{s, t \geq 0}} Φ (s + t) \int_{x - s}^{x + t} | f (y) | d y, x \in R,$ where $\Phi :(0,\infty )\rightarrow (0,\infty )$ $Φ : (0, \infty) \to (0, \infty)$ is a measurable function, which contains some classic maximal operators as its prototypical examples. The variation boundedness and continuity of $M_\Phi $ $M_{Φ}$ is established under a more restrictive condition on $\Phi $ $Φ$ . The main results extend some known ones and provide a large class of maximal operators which possess the variation boundedness and continuity.