<p>Recently, several authors have studied the problem of determining optimal constants in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-norm for Hardy-type inequalities. In this work, we continue this investigation by establishing sharp lower bounds for the norm of the difference between the Hardy operator and the identity when acting on various cones in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^p(\mathbb {R}^+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$1<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. This operator quantifies the oscillation of a function relative to its average. We also obtain optimal lower bounds for its adjoint. In the discrete setting, we conduct a similar analysis for the difference between the Cesàro and the identity, the difference between the Cesàro and the left-shift operators, as well as for its adjoint. As an application, we identify norms on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p(\mathbb {R}^+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\ell ^p(\mathbb {N})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> that are equivalent to their standard <i>p</i>-norm, and we provide the exact optimal constants for these equivalences.</p>

On sharp reverse Hardy-type inequalities for oscillation operators in the discrete and continuous settings

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发表日期：

2026年7月22日

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Achraf Ben Said,
Santiago Boza,
Javier Soria

摘要

Recently, several authors have studied the problem of determining optimal constants in $L^p$ $L^{p}$ -norm for Hardy-type inequalities. In this work, we continue this investigation by establishing sharp lower bounds for the norm of the difference between the Hardy operator and the identity when acting on various cones in $L^p(\mathbb {R}^+)$ $L^{p} (R^{+})$ , with $1<p<\infty $ $1 < p < \infty$ . This operator quantifies the oscillation of a function relative to its average. We also obtain optimal lower bounds for its adjoint. In the discrete setting, we conduct a similar analysis for the difference between the Cesàro and the identity, the difference between the Cesàro and the left-shift operators, as well as for its adjoint. As an application, we identify norms on $L^p(\mathbb {R}^+)$ $L^{p} (R^{+})$ and on $\ell ^p(\mathbb {N})$ $ℓ^{p} (N)$ that are equivalent to their standard p-norm, and we provide the exact optimal constants for these equivalences.