Characterizations of harmonic quasiregular mappings in function spaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908756635287552 |
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| author | Sun, Jihua Liu, Junming Wang, Zhi-Gang |
| author_facet | Sun, Jihua Liu, Junming Wang, Zhi-Gang |
| contents | We study conjugate-type phenomena for complex-valued harmonic quasiregular mappings in the unit disk across three function space families: $Q(n,p,α)$, $F(p,q,s)$, and the non-derivative $M(p,q,s)$. For a harmonic $K$-quasiregular mapping $f=u+iv$, we first show that if the real part $u$ belongs to $Q_h(1,p,α)$ (with $α>-1$ and $α+1<p<α+2$), 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 $(K, K')$-quasiregular mappings, yielding explicit estimates with an additional inhomogeneous term involving $K'$. Finally, for normalized harmonic quasiconformal mappings, %$f\in\mathcal S_H(K)$, 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 $α_K$ of the family of harmonic quasiconformal mappings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01017 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterizations of harmonic quasiregular mappings in function spaces Sun, Jihua Liu, Junming Wang, Zhi-Gang Complex Variables 31A05, 30H05, 30C62 We study conjugate-type phenomena for complex-valued harmonic quasiregular mappings in the unit disk across three function space families: $Q(n,p,α)$, $F(p,q,s)$, and the non-derivative $M(p,q,s)$. For a harmonic $K$-quasiregular mapping $f=u+iv$, we first show that if the real part $u$ belongs to $Q_h(1,p,α)$ (with $α>-1$ and $α+1<p<α+2$), 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 $(K, K')$-quasiregular mappings, yielding explicit estimates with an additional inhomogeneous term involving $K'$. Finally, for normalized harmonic quasiconformal mappings, %$f\in\mathcal S_H(K)$, 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 $α_K$ of the family of harmonic quasiconformal mappings. |
| title | Characterizations of harmonic quasiregular mappings in function spaces |
| topic | Complex Variables 31A05, 30H05, 30C62 |
| url | https://arxiv.org/abs/2601.01017 |