99 research outputs found
Hardy spaces and quasiconformal maps in the Heisenberg group
We define Hardy spaces , , for quasiconformal mappings on
the Kor\'{a}nyi unit ball in the first Heisenberg group . Our
definition is stated in terms of the Heisenberg polar coordinates introduced by
Kor\'{a}nyi and Reimann, and Balogh and Tyson. First, we prove the existence of
such that every -quasiconformal map belongs to for all . Second, we give two
equivalent conditions for the membership of a quasiconformal map , one
in terms of the radial limits of , and one using a nontangential maximal
function of . As an application, we characterize Carleson measures on
via integral inequalities for quasiconformal mappings on and their radial
limits. Our paper thus extends results by Astala and Koskela, Jerison and
Weitsman, Nolder, and Zinsmeister, from to . A
crucial difference between the proofs in and is
caused by the nonisotropic nature of the Kor\'{a}nyi unit sphere with its two
characteristic points.Comment: 51 p
Isoperimetric inequalities and regularity of -harmonic functions on surfaces
We investigate the logarithmic and power-type convexity of the length of the
level curves for -harmonic functions on smooth surfaces and related
isoperimetric inequalities. In particular, our analysis covers the -harmonic
and the minimal surface equations. As an auxiliary result, we obtain higher
Sobolev regularity properties of the solutions, including the
regularity.
The results are complemented by a number of estimates for the derivatives
and of the length of the level curve function , as well as by
examples illustrating the presentation.
Our work generalizes results due to Alessandrini, Longinetti, Talenti and
Lewis in the Euclidean setting, as well as a recent article of ours devoted to
the harmonic case on surfaces.Comment: 27 p
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