Given an open subset U of a complex Banach space E, a weight v on U, and a complex Banach space F, let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathscr {H}_v^{\infty }(U,F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">H</mi> <mi>v</mi> <mi>∞</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the Banach space of all weighted holomorphic mappings <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f:U\rightarrow F$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation>, under the weighted supremum norm <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Vert f\Vert _v:=\sup \hspace{0.55542pt}\{v(x)\Vert f(x)\Vert \,{:}\, x\in U\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mi>v</mi> </msub> <mo>:</mo> <mo>=</mo> <mo movablelimits="true">sup</mo> <mspace width="0.55542pt" /> <mrow> <mo stretchy="false">{</mo> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">‖</mo> <mspace width="0.166667em" /> <mo>:</mo> <mspace width="0.166667em" /> <mi>x</mi> <mo>∈</mo> <mi>U</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we introduce and study the class <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Pi _p^{\mathscr {H}_v^{\infty }}(U,F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Π</mi> <mi>p</mi> <msubsup> <mi mathvariant="script">H</mi> <mi>v</mi> <mi>∞</mi> </msubsup> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of p-summing weighted holomorphic mappings. We prove that it is an injective Banach ideal of weighted holomorphic mappings. Variants for weighted holomorphic mappings of Pietsch Domination Theorem, Pietsch Factorization Theorem and Maurey Extrapolation Theorem are presented. We also identify the spaces of p-summing weighted holomorphic mappings from U into <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$F^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>F</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> under the norm <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\pi ^{\mathscr {H}_v^{\infty }}_p$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>π</mi> <mi>p</mi> <msubsup> <mi mathvariant="script">H</mi> <mi>v</mi> <mi>∞</mi> </msubsup> </msubsup> </math></EquationSource> </InlineEquation> with the duals of F-valued <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathscr {H}_v^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="script">H</mi> <mi>v</mi> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation>-molecules on U under a suitable version <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$d^{\mathscr {H}_v^{\infty }}_{p^}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>d</mi> <msup> <mi>p</mi> <mo>∗</mo> </msup> <msubsup> <mi mathvariant="script">H</mi> <mi>v</mi> <mi>∞</mi> </msubsup> </msubsup> </math></EquationSource> </InlineEquation> of the Chevet–Saphar tensor norms.

On p-summability in weighted Banach spaces of holomorphic functions

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2026年5月3日

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María G. Cabrera-Padilla,
Antonio Jiménez-Vargas,
Ayşegül Keten Çopur

摘要

Given an open subset U of a complex Banach space E, a weight v on U, and a complex Banach space F, let $\mathscr {H}_v^{\infty }(U,F)$ $H_{v}^{\infty} (U, F)$ denote the Banach space of all weighted holomorphic mappings $f:U\rightarrow F$ $f : U \to F$ , under the weighted supremum norm $\Vert f\Vert _v:=\sup \hspace{0.55542pt}\{v(x)\Vert f(x)\Vert \,{:}\, x\in U\}$ ${‖ f ‖}_{v} : = sup {v (x) ‖ f (x) ‖ : x \in U}$ . In this paper, we introduce and study the class $\Pi _p^{\mathscr {H}_v^{\infty }}(U,F)$ $Π_{p}^{H_{v}^{\infty}} (U, F)$ of p-summing weighted holomorphic mappings. We prove that it is an injective Banach ideal of weighted holomorphic mappings. Variants for weighted holomorphic mappings of Pietsch Domination Theorem, Pietsch Factorization Theorem and Maurey Extrapolation Theorem are presented. We also identify the spaces of p-summing weighted holomorphic mappings from U into $F^*$ $F^{*}$ under the norm $\pi ^{\mathscr {H}_v^{\infty }}_p$ $π_{p}^{H_{v}^{\infty}}$ with the duals of F-valued $\mathscr {H}_v^{\infty }$ $H_{v}^{\infty}$ -molecules on U under a suitable version $d^{\mathscr {H}_v^{\infty }}_{p^*}$ $d_{p^{*}}^{H_{v}^{\infty}}$ of the Chevet–Saphar tensor norms.