We study conjugate-type phenomena for complex-valued harmonic quasiregular mappings in the unit disk across three function space families: <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Q(n,p,\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, F(p, q, s), and the non-derivative M(p, q, s). For a harmonic K-quasiregular mapping <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f=u+iv$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mi>u</mi> <mo>+</mo> <mi>i</mi> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation>, we first show that if the real part u belongs to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Q_h(1,p,\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Q</mi> <mi>h</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha >-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha +1<p<\alpha +2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>α</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>), the imaginary part v lies in the same space with a K-dependent quantitative bound. An analogous stability result is established for the harmonic F-scale, with sharp K-dependence. These results are extended to harmonic <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(K, K')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <msup> <mi>K</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-quasiregular mappings, yielding explicit estimates with an additional inhomogeneous term involving <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$K'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>K</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>. Finally, for normalized harmonic quasiconformal mappings, we derive membership criteria in the harmonic M- and F-scales, and obtain corresponding conclusions for their natural derivatives, with parameter ranges governed by the order <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha _K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>α</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> of the family of harmonic K-quasiconformal mappings.

Characterizations of Harmonic Quasiregular Mappings in Function Spaces

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发表日期：

2026年6月3日

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Jihua Sun,
Junming Liu,
Zhi-Gang Wang

摘要

We study conjugate-type phenomena for complex-valued harmonic quasiregular mappings in the unit disk across three function space families: $Q(n,p,\alpha )$ $Q (n, p, α)$ , F(p, q, s), and the non-derivative M(p, q, s). For a harmonic K-quasiregular mapping $f=u+iv$ $f = u + i v$ , we first show that if the real part u belongs to $Q_h(1,p,\alpha )$ $Q_{h} (1, p, α)$ (with $\alpha >-1$ $α > - 1$ and $\alpha +1<p<\alpha +2$ $α + 1 < p < α + 2$ ), the imaginary part v lies in the same space with a K-dependent quantitative bound. An analogous stability result is established for the harmonic F-scale, with sharp K-dependence. These results are extended to harmonic $(K, K')$ $(K, K^{'})$ -quasiregular mappings, yielding explicit estimates with an additional inhomogeneous term involving $K'$ $K^{'}$ . Finally, for normalized harmonic quasiconformal mappings, we derive membership criteria in the harmonic M- and F-scales, and obtain corresponding conclusions for their natural derivatives, with parameter ranges governed by the order $\alpha _K$ $α_{K}$ of the family of harmonic K-quasiconformal mappings.