<p>We study the frequency function (introduced by Temur in [<CitationRef CitationID="CR17">17</CitationRef>]) in both the discrete and continuous settings. More precisely, we extend the definition of the frequency function to the higher-dimensional continuous setting and to the uncentered Hardy-Littlewood maximal function. We analyze the asymptotic behavior of the frequency function and the density of its small values for functions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\ell ^1(\mathbb {Z)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^1(\mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> answering some questions posed by Temur in [<CitationRef CitationID="CR17">17</CitationRef>]. Finally, we study the size of the frequency function for functions in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\ell ^p(\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, showing that this case differs significantly from the case <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the Frequency Function of Hardy-Littlewood Maximal Functions

期刊论文
发表日期：

2026年7月27日

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Carlos Garzón,
José Madrid

摘要

We study the frequency function (introduced by Temur in [17]) in both the discrete and continuous settings. More precisely, we extend the definition of the frequency function to the higher-dimensional continuous setting and to the uncentered Hardy-Littlewood maximal function. We analyze the asymptotic behavior of the frequency function and the density of its small values for functions in $\ell ^1(\mathbb {Z)}$ $ℓ^{1} (Z)$ and $L^1(\mathbb {R}^d)$ $L^{1} (R^{d})$ answering some questions posed by Temur in [17]. Finally, we study the size of the frequency function for functions in $\ell ^p(\mathbb {Z})$ $ℓ^{p} (Z)$ with $p>1$ $p > 1$ , showing that this case differs significantly from the case $p=1$ $p = 1$ .