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Our goal in this work is to present some function spaces on the complex plane $\mathbb{C},\,X(\mathbb{C})$ , for which the quasiregular solutions of the Beltrami equation, $\bar{\partial }f(z)\,=\,\mu (z)\partial f(z)$ , have first derivatives locally in $X(\mathbb{C})$ , provided that the Beltrami coefficient $\mu $ belongs to $X(\mathbb{C})$ .

Using the BMO-H1 duality (among other things), D. R. Adams proved in [1] the strong type inequality

∫Mf(x)dHα(x)[les ]C∫[mid ]f(x)[mid ]dHα(x), 0<α<n,(1)

where C is some positive constant independent of f. Here M is the Hardy–Littlewood maximal operator in ℝn, Hα is the α-dimensional Hausdorff content, and the integrals are taken in the Choquet sense. The Choquet integral of ϕ[ges ]0 with respect to a set function C is defined by

formula here

Precise definitions of M and Hα will be given below. For an application of (1) to the Sobolev space W1, 1 (ℝn), see [1, p. 114].

The purpose of this note is to provide a self-contained, direct proof of a result more general than (1).

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Beltrami Equation with Coefficient in Sobolev and Besov Spaces

CHOQUET INTEGRALS, HAUSDORFF CONTENT AND THE HARDY–LITTLEWOOD MAXIMAL OPERATOR

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Beltrami Equation with Coefficient in Sobolev and Besov Spaces

CHOQUET INTEGRALS, HAUSDORFF CONTENT AND THE HARDY–LITTLEWOOD MAXIMAL OPERATOR