We investigate the critical case of a weighted cylindrical Hardy inequality involving a logarithmic term and extend the classical result due to Edmunds and Triebel to this case: For functions in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7784_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="133" /> </InlineMediaObject> <EquationSource Format="TEX">${C}_{0}^{\infty }\left({\mathbb{R}}^{n}\backslash \left\{{x}{\prime}=0\right\}\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>C</mi> <mrow> <mn>0</mn> </mrow> <mi>∞</mi> </msubsup> <mfenced close=")" open="("> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mrow> <mo stretchy="true">\</mo> </mrow> <mfenced close="}" open="{"> <mi>x</mi> <mo>′</mo> <mo>=</mo> <mn>0</mn> </mfenced> </mfenced> </mrow> </math></EquationSource> </InlineEquation> we establish the sharp inequality<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7784_Article_IEq2.gif" Format="GIF" Height="37" Rendition="HTML" Resolution="72" Type="Linedraw" Width="366" /> </InlineMediaObject> <EquationSource Format="TEX">${\Vert \frac{f}{\left|x{^{\prime}}\right|N/2{\left(1+\left|\text{log}\left|x{^{\prime}}\right|\right|\right)}^{\beta /2}}\Vert }_{{L}^{2}}\le \left.\frac{2}{\left|1-\beta \right|}\right|{\Vert \frac{\left(x{^{\prime}}\bullet \nabla Nf\right)}{\left|x{^{\prime}}\right|N/2{\left(1+\left|\text{log}\left|x{^{\prime}}\right|\right|\right)}^{\beta /2}}\Vert }_{{L}^{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <mfrac> <mi>f</mi> <mrow> <mfenced close="|" open="|"> <mi>x</mi> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mfenced> <mi>N</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mmultiscripts> <mrow> <mfenced close=")" open="("> <mn>1</mn> <mo>+</mo> <mfenced close="|" open="|"> <mtext>log</mtext> <mfenced close="|" open="|"> <mi>x</mi> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mfenced> </mfenced> </mfenced> </mrow> <mrow /> <mrow> <mi>β</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </mmultiscripts> </mrow> </mfrac> <mrow> <mo stretchy="false">‖</mo> </mrow> </mrow> <msup> <mrow> <mi>L</mi> </mrow> <mn>2</mn> </msup> <mrow /> </mmultiscripts> <mo>≤</mo> <mfenced close="|"> <mfrac> <mn>2</mn> <mfenced close="|" open="|"> <mn>1</mn> <mo>-</mo> <mi>β</mi> </mfenced> </mfrac> </mfenced> <mmultiscripts> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <mfrac> <mfenced close=")" open="("> <mi>x</mi> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> <mo>∙</mo> <mi mathvariant="normal">∇</mi> <mi>N</mi> <mi>f</mi> </mfenced> <mrow> <mfenced close="|" open="|"> <mi>x</mi> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mfenced> <mi>N</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mmultiscripts> <mrow> <mfenced close=")" open="("> <mn>1</mn> <mo>+</mo> <mfenced close="|" open="|"> <mtext>log</mtext> <mfenced close="|" open="|"> <mi>x</mi> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mfenced> </mfenced> </mfenced> </mrow> <mrow /> <mrow> <mi>β</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </mmultiscripts> </mrow> </mfrac> <mrow> <mo stretchy="false">‖</mo> </mrow> </mrow> <msup> <mrow> <mi>L</mi> </mrow> <mn>2</mn> </msup> <mrow /> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation>where x = (x′, x′′) ∈ ℝN ×ℝn−N, 1 ≤ N ≤ n, and β ∈ℝ. We obtain improved versions of this inequality with remainder terms and discuss their nonattainability. As an application, we derive a family of Caffarelli–Kohn–Nirenberg type and uncertainty type inequalities.

Cylindrical Critical Hardy Type Inequalities and Identities

期刊论文
发表日期：

2025年6月7日

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Yerkin Shaimerdenov,
Nurgissa Yessirkegenov

摘要

We investigate the critical case of a weighted cylindrical Hardy inequality involving a logarithmic term and extend the classical result due to Edmunds and Triebel to this case: For functions in ${C}_{0}^{\infty }\left({\mathbb{R}}^{n}\backslash \left\{{x}{\prime}=0\right\}\right)$ $C_{0}^{\infty} (R^{n}, \, \{x' = 0\})$ we establish the sharp inequality

${\Vert \frac{f}{\left|x{^{\prime}}\right|N/2{\left(1+\left|\text{log}\left|x{^{\prime}}\right|\right|\right)}^{\beta /2}}\Vert }_{{L}^{2}}\le \left.\frac{2}{\left|1-\beta \right|}\right|{\Vert \frac{\left(x{^{\prime}}\bullet \nabla Nf\right)}{\left|x{^{\prime}}\right|N/2{\left(1+\left|\text{log}\left|x{^{\prime}}\right|\right|\right)}^{\beta /2}}\Vert }_{{L}^{2}}$ ${‖ \frac{f}{|x, ^{'}| N / 2 {(1 + |log, |x, ^{'}||)}^{β / 2}} ‖}_{L^{2}} \leq (\frac{2}{|1 - β|}| {‖ \frac{(x^{'} ∙ \nabla N f)}{|x, ^{'}| N / 2 {(1 + |log, |x, ^{'}||)}^{β / 2}} ‖}_{L^{2}}$

where x = (x′, x′′) ∈ ℝ^N ×ℝ^n−N, 1 ≤ N ≤ n, and β ∈ℝ. We obtain improved versions of this inequality with remainder terms and discuss their nonattainability. As an application, we derive a family of Caffarelli–Kohn–Nirenberg type and uncertainty type inequalities.