Tohoku Mathematical Journal
2025
December
SECOND SERIES VOL. 77, NO. 4

Tohoku Math. J.
77 (2025), 473-497

Title QUASICONFORMAL AND SOBOLEV MAPPINGS IN NON-AHLFORS REGULAR METRIC SPACES

Author Panu Lahti and Xiaodan Zhou

(Received May 12, 2023, revised December 18, 2023)
Abstract. We show that a mapping $f: X\to Y$ satisfying the metric condition of quasiconformality outside suitable exceptional sets is in the Newton-Sobolev class $N_{\mathrm{loc}}^{1,1}(X;Y)$. Unlike previous works, we only assume an asymptotic version of Ahlfors-regularity on $X,Y$. This allows many non-Ahlfors regular spaces, such as weighted spaces to be included in the theory. Unexpectedly, already in the classical setting of Euclidean spaces, our theory detects Sobolev mappings that are not recognized by previous results.

Mathematics Subject Classification. Primary 30L10; Secondary 30L15, 46E36.
Key words and phrases. Quasiconformal mapping, Newton-Sobolev mapping, Ahlfors regular space, weighted space, equi-integrability.

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