We provide examples of Hölder equivalence between two unit spheres of Banach spaces. Specifically, we prove the following: (i) If X has a normalized 1-unconditional basis <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {E}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">E</mi> </math></EquationSource> </InlineEquation>, then unit spheres <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$S(X^{(p)})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S(X^{(q)})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left( \min \{\frac{p}{q},1\},\min \{\frac{q}{p},1\}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mo movablelimits="true">min</mo> <mrow> <mo stretchy="false">{</mo> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> <mo movablelimits="true">min</mo> <mrow> <mo stretchy="false">{</mo> <mfrac> <mi>q</mi> <mi>p</mi> </mfrac> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </mfenced> </math></EquationSource> </InlineEquation>-Hölder homeomorphic for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$1\le p,q<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. (ii) If two Banach spaces X and Y are locally <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\alpha ,\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Hölder homeomorphic for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$0<\alpha ,\beta \le {1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, then unit spheres <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$S\left( (X\oplus \mathbb {R})_{\infty }\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mfenced close=")" open="("> <msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>⊕</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> </msub> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$S\left( (Y\oplus \mathbb {R})_{\infty }\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mfenced close=")" open="("> <msub> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>⊕</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> </msub> </mfenced> </mrow> </math></EquationSource> </InlineEquation> are <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$(\alpha ,\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Hölder homeomorphic. In addition, some stability results for the equivalence of Banach spaces and their unit spheres are given.

Hölder equivalence of unit spheres in Banach spaces

期刊论文
发表日期：

2026年4月13日

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Junxi Chen,
Chunyan Luo,
Jianjian Wang,
Bo Xiang

摘要

We provide examples of Hölder equivalence between two unit spheres of Banach spaces. Specifically, we prove the following: (i) If X has a normalized 1-unconditional basis $\mathcal {E}$ $E$ , then unit spheres $S(X^{(p)})$ $S (X^{(p)})$ and $S(X^{(q)})$ $S (X^{(q)})$ are $\left( \min \{\frac{p}{q},1\},\min \{\frac{q}{p},1\}\right) $ $(min {\frac{p}{q}, 1}, min {\frac{q}{p}, 1})$ -Hölder homeomorphic for all $1\le p,q<\infty $ $1 \leq p, q < \infty$ . (ii) If two Banach spaces X and Y are locally $(\alpha ,\beta )$ $(α, β)$ -Hölder homeomorphic for $0<\alpha ,\beta \le {1}$ $0 < α, β \leq 1$ , then unit spheres $S\left( (X\oplus \mathbb {R})_{\infty }\right) $ $S ({(X \oplus R)}_{\infty})$ and $S\left( (Y\oplus \mathbb {R})_{\infty }\right) $ $S ({(Y \oplus R)}_{\infty})$ are $(\alpha ,\beta )$ $(α, β)$ -Hölder homeomorphic. In addition, some stability results for the equivalence of Banach spaces and their unit spheres are given.