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<p>In this paper, we develop a theory of symmetrization on the one dimensional integer lattice. More precisely, we associate a radially decreasing function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(u^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> with a function <i>u</i> defined on the integers and prove the corresponding Polya-Szegő inequality. Along the way we also prove the weighted Polya-Szegő inequality for the decreasing rearrangement on the non-negative integers (<i>half-line</i>). As a consequence, we prove the discrete weighted Hardy’s inequality with the weight <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n^\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>n</mi> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1 &lt; \alpha \le 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>&lt;</mo> <mi>α</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Symmetrization Inequalities on One-Dimensional Integer Lattice

  • 期刊论文

  • 发表日期：

    2026年1月7日

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  • Shubham Gupta

摘要

In this paper, we develop a theory of symmetrization on the one dimensional integer lattice. More precisely, we associate a radially decreasing function \(u^*\) u ∗ with a function u defined on the integers and prove the corresponding Polya-Szegő inequality. Along the way we also prove the weighted Polya-Szegő inequality for the decreasing rearrangement on the non-negative integers (half-line). As a consequence, we prove the discrete weighted Hardy’s inequality with the weight \(n^\alpha \) n α for \(1 < \alpha \le 2\) 1 < α ≤ 2 .