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<p>In this paper we address the following question: given a holomorphic function with prescribed <InlineEquation ID="IEq1"> <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> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^q(\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>q</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> norm (with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1\le p,q \le \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>) along two parallel lines in the complex plane, what is the largest value that the function can attain at a prescribed point between these lines? Here we show that this problem is well-posed in suitable Hardy-like spaces on the strip. Moreover, in this setting we completely solve this problem by providing not only an explicit formula for the optimizers but also for the optimal values. In addition, we briefly discuss some applications of these results to interpolation theory and to Lieb-Thirring inequalities.</p>

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A Generalized Three Lines Lemma in Hardy-like Spaces

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  • 发表日期：

    2026年7月16日

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  • Thiago Carvalho Corso

摘要

In this paper we address the following question: given a holomorphic function with prescribed \(L^p(\mathbb {R})\) L p ( R ) and \(L^q(\mathbb {R})\) L q ( R ) norm (with \(1\le p,q \le \infty \) 1 ≤ p , q ≤ ∞ ) along two parallel lines in the complex plane, what is the largest value that the function can attain at a prescribed point between these lines? Here we show that this problem is well-posed in suitable Hardy-like spaces on the strip. Moreover, in this setting we completely solve this problem by providing not only an explicit formula for the optimizers but also for the optimal values. In addition, we briefly discuss some applications of these results to interpolation theory and to Lieb-Thirring inequalities.