Hardy spaces and quasiregular mappings

Adamowicz, Tomasz; González, María

doi:10.1090/tran/9446

Hardy spaces and quasiregular mappings
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by Tomasz Adamowicz and María J. González;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9446

Published electronically: June 10, 2025

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Abstract:

We study Hardy spaces $\mathcal {H}^p$, $0<p<\infty$ for quasiregular mappings on the unit ball $B$ in ${\mathbb R}^n$ which satisfy appropriate growth and multiplicity conditions. Under these conditions we recover several classical results for analytic functions and quasiconformal mappings in $\mathcal {H}^p$. In particular, we characterize $\mathcal {H}^p$ in terms of non-tangential limit functions and non-tangential maximal functions of quasiregular mappings. Among applications we show that every quasiregular map in our class belongs to $\mathcal {H}^p$ for some $p=p(n,K)$. Moreover, we provide characterization of Carleson measures on $B$ via integral inequalities for quasiregular mappings on $B$. We also discuss the Bergman spaces of quasiregular mappings and their relations to $\mathcal {H}^p$ spaces and analyze correspondence between results for $\mathcal {H}^p$ spaces and $\mathcal {A}$-harmonic functions.

A key difference between the previously known results for quasiconformal mappings in ${\mathbb R}^n$ and our setting is the role of multiplicity conditions and the growth of mappings that need not be injective.

Our paper extends results by Astala and Koskela, Jerison and Weitsman, Jones, Nolder, and Zinsmeister.

References

Tomasz Adamowicz and Katrin Fässler, Hardy spaces and quasiconformal maps in the Heisenberg group, J. Funct. Anal. 284 (2023), no. 6, Paper No. 109832, 68. MR 4530903, DOI 10.1016/j.jfa.2022.109832
Tomasz Adamowicz and María J. González, Hardy spaces for quasiregular mappings and composition operators, J. Geom. Anal. 31 (2021), no. 11, 11417–11427. MR 4310177, DOI 10.1007/s12220-021-00687-0
Lars V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics, University of Minnesota, School of Mathematics, Minneapolis, MN, 1981. MR 725161
Kari Astala, Tadeusz Iwaniec, and Gaven Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, vol. 48, Princeton University Press, Princeton, NJ, 2009. MR 2472875
Kari Astala and Pekka Koskela, $H^p$-theory for quasiconformal mappings, Pure Appl. Math. Q. 7 (2011), no. 1, 19–50. MR 2900163, DOI 10.4310/PAMQ.2011.v7.n1.a3
Tuomo Äkkinen, Radial limits of mappings of bounded and finite distortion, J. Geom. Anal. 24 (2014), no. 3, 1298–1322. MR 3223554, DOI 10.1007/s12220-012-9373-6
B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $\textbf {R}^{n}$, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 2, 257–324. MR 731786, DOI 10.5186/aasfm.1983.0806
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 268655
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), no. 1, 81–116. MR 1555303, DOI 10.1007/BF02547518
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
V. M. Gol′dšteĭn, The behavior of mappings with bounded distortion when the distortion coefficient is close to one, Sibirsk. Mat. Ž. 12 (1971), 1250–1258 (Russian). MR 294634
Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Dover Publications, Inc., Mineola, NY, 2006. Unabridged republication of the 1993 original. MR 2305115
T. Iwaniec and C. A. Nolder, Hardy-Littlewood inequality for quasiregular mappings in certain domains in $\textbf {R}^n$, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 267–282. MR 802488, DOI 10.5186/aasfm.1985.1030
David Jerison and Allen Weitsman, On the means of quasiregular and quasiconformal mappings, Proc. Amer. Math. Soc. 83 (1981), no. 2, 304–306. MR 624919, DOI 10.1090/S0002-9939-1981-0624919-X
Peter W. Jones, Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980), no. 1, 41–66. MR 554817, DOI 10.1512/iumj.1980.29.29005
Pekka Koskela, Juan J. Manfredi, and Enrique Villamor, Regularity theory and traces of ${\scr A}$-harmonic functions, Trans. Amer. Math. Soc. 348 (1996), no. 2, 755–766. MR 1311911, DOI 10.1090/S0002-9947-96-01430-4
Pekka Koskela and Vesna Manojlović, Quasi-nearly subharmonic functions and quasiconformal mappings, Potential Anal. 37 (2012), no. 2, 187–196. MR 2944066, DOI 10.1007/s11118-011-9252-y
G. R. MacLane, Holomorphic functions, of arbitrarily slow growth, without radial limits, Michigan Math. J. 9 (1962), 21–24. MR 136742
Juan J. Manfredi, $p$-harmonic functions in the plane, Proc. Amer. Math. Soc. 103 (1988), no. 2, 473–479. MR 943069, DOI 10.1090/S0002-9939-1988-0943069-2
O. Martio and S. Rickman, Boundary behavior of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 507 (1972), 17. MR 379846
O. Martio, S. Rickman, and J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 488 (1971), 31. MR 299782
O. Martio and U. Srebro, Locally injective automorphic mappings in $\mathbf R^n$, Math. Scand. 85 (1999), no. 1, 49–70. MR 1707745, DOI 10.7146/math.scand.a-13884
Ruth Miniowitz, Distortion theorems for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 1, 63–74. MR 538089, DOI 10.5186/aasfm.1978-79.0415
Craig A. Nolder, The $H^p$-norm of a quasiconformal mapping, J. Math. Anal. Appl. 275 (2002), no. 2, 557–561. MR 1943765, DOI 10.1016/S0022-247X(02)00236-6
Kiyoshi Noshiro, Cluster sets, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 28, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 133464
K. Rajala, The local homeomorphism property of spatial quasiregular mappings with distortion close to one, Geom. Funct. Anal. 15 (2005), no. 5, 1100–1127. MR 2221159, DOI 10.1007/s00039-005-0530-y
Kai Rajala, Radial limits of quasiregular local homeomorphisms, Amer. J. Math. 130 (2008), no. 1, 269–289. MR 2382149, DOI 10.1353/ajm.2008.0001
Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 994644, DOI 10.1090/mmono/073
Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229. Springer–Verlag, Berlin–New York, 1971.
Matti Vuorinen, On the boundary behavior of locally $K$-quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 79–95. MR 595179, DOI 10.5186/aasfm.1980.0532
Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174, DOI 10.1007/BFb0077904
Michel Zinsmeister, A distortion theorem for quasiconformal mappings, Bull. Soc. Math. France 114 (1986), no. 1, 123–133 (English, with French summary). MR 860655

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Hardy spaces and quasiregular mappings
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Abstract: