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Abstract <p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(({{x}_{n}})\)</EquationSource> <!--RusMath2570045Demir-m1--> </InlineEquation> be a sequence and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho \geqslant 1\)</EquationSource> <!--RusMath2570045Demir-m2--> </InlineEquation>. For two fixed sequences <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="139" /> </InlineMediaObject> <EquationSource Format="TEX">\({{n}_{1}} &lt; {{n}_{2}} &lt; {{n}_{3}} &lt; \ldots \)</EquationSource> <!--RusMath2570045Demir-m3--> </InlineEquation>, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\)</EquationSource> <!--RusMath2570045Demir-m4--> </InlineEquation> define the oscillation operator <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq5.gif" Format="GIF" Height="79" Rendition="HTML" Resolution="72" Type="Linedraw" Width="306" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{O}}_{\rho }}({{x}_{n}}) = {{\left( {\sum\limits_{k = 1}^\infty \,\mathop {\sup }\limits_{\substack{ {{n}_{k}} \leqslant m &lt; {{n}_{{k + 1}}} \\ m \in M } } {{{\left| {{{x}_{m}} - {{x}_{{{{n}_{k}}}}}} \right|}}^{\rho }}} \right)}^{{1/\rho }}}.\)</EquationSource> <!--RusMath2570045Demir-m5--> </InlineEquation>Let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">\((X,\mathcal{B},\mu ,\tau )\)</EquationSource> <!--RusMath2570045Demir-m6--> </InlineEquation> be a dynamical system with <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\((X,\mathcal{B},\mu )\)</EquationSource> <!--RusMath2570045Demir-m7--> </InlineEquation> a probability space and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau \)</EquationSource> <!--RusMath2570045Demir-m8--> </InlineEquation> a measurable, invertible, measure preserving point transformation from <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource> <!--RusMath2570045Demir-m9--> </InlineEquation> to itself. Suppose that the sequences <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(({{n}_{k}})\)</EquationSource> <!--RusMath2570045Demir-m10--> </InlineEquation> is a lacunary, and <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\)</EquationSource> <!--RusMath2570045Demir-m11--> </InlineEquation> is any sequence of positive real numbers such that there exists an <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq12.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ell \in \mathbb{R}\)</EquationSource> <!--RusMath2570045Demir-m12--> </InlineEquation> satisfying <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="242" /> </InlineMediaObject> <EquationSource Format="TEX">\(\# \{ m \in M:{{n}_{k}} \leqslant m &lt; {{n}_{{k + 1}}}\} \leqslant \ell \)</EquationSource> <!--RusMath2570045Demir-m13--> </InlineEquation> for all <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq14.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(k \in \mathbb{N}\)</EquationSource> <!--RusMath2570045Demir-m14--> </InlineEquation> to obtain the above mentioned results, where <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq15.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\# \)</EquationSource> <!--RusMath2570045Demir-m15--> </InlineEquation> denotes cardinality. Then we prove the following results for <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq16.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho \geqslant 2\)</EquationSource> <!--RusMath2570045Demir-m16--> </InlineEquation>: (i) Define <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq17.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="129" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\phi }_{n}}(x) = \frac{1}{n}{{\chi }_{{[0,n]}}}(x)\)</EquationSource> <!--RusMath2570045Demir-m17--> </InlineEquation> on <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq18.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{R}\)</EquationSource> <!--RusMath2570045Demir-m18--> </InlineEquation>. Then there exists a constant <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq19.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(C &gt; 0\)</EquationSource> <!--RusMath2570045Demir-m19--> </InlineEquation> such that <InlineEquation ID="IEq20"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq20.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="223" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\left\| {{{\mathcal{O}}_{\rho }}({{\phi }_{n}} * f)} \right\|}_{{{{L}^{1}}(\mathbb{R})}}} \leqslant C{{\left\| f \right\|}_{{{{H}^{1}}(\mathbb{R})}}}\)</EquationSource> <!--RusMath2570045Demir-m20--> </InlineEquation>for all <InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq21.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(f \in {{H}^{1}}(\mathbb{R})\)</EquationSource> <!--RusMath2570045Demir-m21--> </InlineEquation>.(ii) Let <InlineEquation ID="IEq22"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq22.gif" Format="GIF" Height="39" Rendition="HTML" Resolution="72" Type="Linedraw" Width="163" /> </InlineMediaObject> <EquationSource Format="TEX">\({{A}_{n}}f(x) = \frac{1}{n}\sum\limits_{k = 1}^n \,f({{\tau }^{k}}x)\)</EquationSource> <!--RusMath2570045Demir-m22--> </InlineEquation>be the usual ergodic averages in ergodic theory. Then <InlineEquation ID="IEq23"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq23.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="216" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\left\| {{{\mathcal{O}}_{\rho }}({{A}_{n}}f)} \right\|}_{{{{L}^{1}}(X)}}} \leqslant C{{\left\| f \right\|}_{{{{H}^{1}}(X)}}}\)</EquationSource> <!--RusMath2570045Demir-m23--> </InlineEquation>for all <InlineEquation ID="IEq24"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq24.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(f \in {{H}^{1}}(X)\)</EquationSource> <!--RusMath2570045Demir-m24--> </InlineEquation>. (iii) If <InlineEquation ID="IEq25"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq25.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">\({{[f(x)\log (x)]}^{ + }}\)</EquationSource> <!--RusMath2570045Demir-m25--> </InlineEquation> is integrable, then <InlineEquation ID="IEq26"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq26.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathcal{O}}_{\rho }}({{A}_{n}}f)\)</EquationSource> <!--RusMath2570045Demir-m26--> </InlineEquation> is integrable. In the author’s previously published article titled “Oscillation inequalities on real and ergodic <InlineEquation ID="IEq27"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq27.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\({{H}^{1}}\)</EquationSource> <!--RusMath2570045Demir-m27--> </InlineEquation> spaces” the above results have been obtained when both <InlineEquation ID="IEq28"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(({{n}_{k}})\)</EquationSource> <!--RusMath2570045Demir-m28--> </InlineEquation> and <InlineEquation ID="IEq29"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\)</EquationSource> <!--RusMath2570045Demir-m29--> </InlineEquation> are lacunary. Thus the results of this work extents those results to a nonlacunary sequence <InlineEquation ID="IEq30"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10501_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\)</EquationSource> <!--RusMath2570045Demir-m30--> </InlineEquation> with a more general growth condition.</p>

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Oscillation Inequalities on Real and Ergodic H1 Spaces. II

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • Sakin Demir

摘要

Abstract

Let \(({{x}_{n}})\) be a sequence and \(\rho \geqslant 1\) . For two fixed sequences \({{n}_{1}} < {{n}_{2}} < {{n}_{3}} < \ldots \) , and \(M\) define the oscillation operator \({{\mathcal{O}}_{\rho }}({{x}_{n}}) = {{\left( {\sum\limits_{k = 1}^\infty \,\mathop {\sup }\limits_{\substack{ {{n}_{k}} \leqslant m < {{n}_{{k + 1}}} \\ m \in M } } {{{\left| {{{x}_{m}} - {{x}_{{{{n}_{k}}}}}} \right|}}^{\rho }}} \right)}^{{1/\rho }}}.\) Let \((X,\mathcal{B},\mu ,\tau )\) be a dynamical system with \((X,\mathcal{B},\mu )\) a probability space and \(\tau \) a measurable, invertible, measure preserving point transformation from \(X\) to itself. Suppose that the sequences \(({{n}_{k}})\) is a lacunary, and \(M\) is any sequence of positive real numbers such that there exists an \(\ell \in \mathbb{R}\) satisfying \(\# \{ m \in M:{{n}_{k}} \leqslant m < {{n}_{{k + 1}}}\} \leqslant \ell \) for all \(k \in \mathbb{N}\) to obtain the above mentioned results, where \(\# \) denotes cardinality. Then we prove the following results for \(\rho \geqslant 2\) : (i) Define \({{\phi }_{n}}(x) = \frac{1}{n}{{\chi }_{{[0,n]}}}(x)\) on \(\mathbb{R}\) . Then there exists a constant \(C > 0\) such that \({{\left\| {{{\mathcal{O}}_{\rho }}({{\phi }_{n}} * f)} \right\|}_{{{{L}^{1}}(\mathbb{R})}}} \leqslant C{{\left\| f \right\|}_{{{{H}^{1}}(\mathbb{R})}}}\) for all \(f \in {{H}^{1}}(\mathbb{R})\) .(ii) Let \({{A}_{n}}f(x) = \frac{1}{n}\sum\limits_{k = 1}^n \,f({{\tau }^{k}}x)\) be the usual ergodic averages in ergodic theory. Then \({{\left\| {{{\mathcal{O}}_{\rho }}({{A}_{n}}f)} \right\|}_{{{{L}^{1}}(X)}}} \leqslant C{{\left\| f \right\|}_{{{{H}^{1}}(X)}}}\) for all \(f \in {{H}^{1}}(X)\) . (iii) If \({{[f(x)\log (x)]}^{ + }}\) is integrable, then \({{\mathcal{O}}_{\rho }}({{A}_{n}}f)\) is integrable. In the author’s previously published article titled “Oscillation inequalities on real and ergodic \({{H}^{1}}\) spaces” the above results have been obtained when both \(({{n}_{k}})\) and \(M\) are lacunary. Thus the results of this work extents those results to a nonlacunary sequence \(M\) with a more general growth condition.