For superposition operators we prove rigidity for mappings from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^q$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Z_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> and into <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Q_{\log ,p>0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mrow> <mo>log</mo> <mo>,</mo> <mi>p</mi> <mo>></mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>, obtain sharp integral characterizations for the actions from <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {B}_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">B</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Q_{\log ,p>0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mrow> <mo>log</mo> <mo>,</mo> <mi>p</mi> <mo>></mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> and to F(2, q, s), and rule out the mappings <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$M_p\!\rightarrow \! H^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>p</mi> </msub> <mspace width="-0.166667em" /> <mo stretchy="false">→</mo> <mspace width="-0.166667em" /> <msup> <mi>H</mi> <mi>∞</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$M_p\!\rightarrow \!\mathcal {B}_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>p</mi> </msub> <mspace width="-0.166667em" /> <mo stretchy="false">→</mo> <mspace width="-0.166667em" /> <msub> <mi mathvariant="script">B</mi> <mi>α</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for nonconstant symbols. On the quadratic side we establish a Korenblum-type domination theorem in M(2, q, s) for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$f(z)=z^2+a$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>a</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$g(z)=z(1+\bar{a} z^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>z</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mover accent="true"> <mrow> <mi>a</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with an explicit range for |a|. The arguments rely on automorphism normalization, finite-area test maps, the Hardy mean identity with beta-integral weights, and a monotonicity lemma.

Superposition operators on several function spaces

期刊论文
发表日期：

2026年4月6日

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Wenxuan Zhang,
Jihua Sun,
Junming Liu

摘要

For superposition operators we prove rigidity for mappings from $H^q$ $H^{q}$ and $H^\infty $ $H^{\infty}$ into $Z_p$ $Z_{p}$ and into $Q_{\log ,p>0}$ $Q_{log, p > 0}$ , obtain sharp integral characterizations for the actions from $\mathcal {B}_\alpha $ $B_{α}$ to $Q_{\log ,p>0}$ $Q_{log, p > 0}$ and to F(2, q, s), and rule out the mappings $M_p\!\rightarrow \! H^\infty $ $M_{p} \to H^{\infty}$ and $M_p\!\rightarrow \!\mathcal {B}_\alpha $ $M_{p} \to B_{α}$ for nonconstant symbols. On the quadratic side we establish a Korenblum-type domination theorem in M(2, q, s) for $f(z)=z^2+a$ $f (z) = z^{2} + a$ and $g(z)=z(1+\bar{a} z^2)$ $g (z) = z (1 + \bar{a} z^{2})$ with an explicit range for |a|. The arguments rely on automorphism normalization, finite-area test maps, the Hardy mean identity with beta-integral weights, and a monotonicity lemma.