Moduli inequalities for W¹_n-1,loc-mappings with weighted bounded (q, p)-distortion

Abstract

We prove Poletskii-type moduli inequalities for the two-index scale of weighted bounded $(q,p)$-distortion under minimal regularity. This implies, in particular, a positive solution to a question formulated in a Tengval's paper on the validity of Poletskii-type moduli inequalities for nonspherical condensers, for mappings of Sobolev classes with the least possible summability exponent.

Keywords:

• Quasiconformal analysis
• Sobolev space
• Poletskii function
• modulus of a family of curves
• modulus estimate

AMS SUBJECT CLASSIFICATION:

• 30C65 (26B35 31B15 46E35)

Acknowledgments

I greatly appreciate the anonymous reviewers for critically reading and comments, which helped improve and clarify the initial manuscript. The author thanks also an anonymous reviewer for his recommendation to add papers [22,23,25–27] in the list of references.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In [2], there is also some improvement in this inequality in normal domains, see [2, Theorem 2]. The useful interpretation of the latter found by Väisälä [7, 3.1] is now called Väisälä's inequality in the literature. A little earlier, similar estimates were established for capacity, see [8–10].

2 Let $f:A\to {\mathbb{R}}^n$ where $A\subset {\mathbb{R}}^n$ is a measurable set. We say that f enjoys Luzin's $\mathcal{N}$-property if $E\subset A$, ${\mathcal{H}}^n\left(E\right)=0$, implies ${\mathcal{H}}^n\left(f\right(E\left)\right)=0$.

3 Here $J_m(x,f)=\sqrt{det\left(D{f}^{\ast }\right(x\left)Df\right(x\left)\right)}$ is the area factor [44, Section 3.2, Theorems 3, 4; Section 3.3, Theorem 1] of the Jacobi matrix $Df\left(x\right)$, $J_n(x,f)=J(x,f)=|detDf\left(x\right)|$ at m = n; and $\mathcal{N}(y,f,A\setminus E_f)=\mathrm{\#}\left\{f^{-1}\right(y)\cap A\setminus E_f\}$ is the Banach indicatrix.

4 Ahlfors and Beurling [50] introduced the concept of modulus of a family of curves on the plane in 1950. Fuglede [47] and Shabat [51] extended it to higher-dimensional spaces. The modulus of a family of curves has been applied not only to obtain an equivalent geometric description of quasiconformal mappings, but also to develop a method for studying the properties of this class of mappings.

5 The formulas below do the job at almost all points $t\in J_{2,l}$ provided conditions $t\in J_{2,l}$ ($x={\alpha }^{\ast }\left(t\right)\in E_l$) is a point of differentiability of ${\alpha }^{\ast }:J_{2,l}\to D$ ($f:E_l\to D^{\prime }$), $t\in J_{2,l}$ is a Lebesgue point for both integrals under consideration, and ${\dot{\alpha }}^{\ast }\left(t\right)\ne 0$. The derivative $\frac{\mathrm{d}}{\mathrm{d}t}(f\circ {\alpha }^{\ast })\left(t\right)=Df\left(x\right)\cdot \dot{\phantom{a}}{\alpha }^{\ast }\left(t\right)$ can be found by the chain rule similar to (45(46) $\begin{array}{rl}{\psi }_l(f\circ & {\alpha }^{\ast }\left(t\right))-{\psi }_l(f\circ {\alpha }^{\ast }\left(t_0\right))\\ & =D{\psi }_l\left(y_0\right)(f\circ {\alpha }^{\ast }(t)-f\circ {\alpha }^{\ast }(t_0\left)\right)+o\left(1\right)(f\circ {\alpha }^{\ast }(t)-f\circ {\alpha }^{\ast }(t_0\left)\right)\\ & =D{\psi }_l\left(y_0\right)\frac{\mathrm{d}}{\mathrm{d}t}(f\circ {\alpha }^{\ast })\left(t_0\right)(t-t_0)\\ & +D{\psi }_l\left(y_0\right)\tilde{o}\left(1\right)(t-t_0)+o\left(1\right)(\frac{\mathrm{d}}{\mathrm{d}t}(f\circ {\alpha }^{\ast })\left(t_0\right)+\tilde{o}\left(1\right))(t-t_0)\\ & =D{\psi }_l\left(y_0\right)\frac{\mathrm{d}}{\mathrm{d}t}(f\circ {\alpha }^{\ast })\left(t_0\right)(t-t_0)+o\left(1\right)(t-t_0)\end{array}$(46) ).

6 By the condition $n-1<q⩽p<n+\frac{1}{n-2}$ of Theorem 4.3 we conclude by [5, Theorem 26] that the given mapping $f:\mathrm{\Omega }\to {\mathbb{R}}^n$ is differentiable a.e. Thence from the very beginning the mapping $f:\mathrm{\Omega }\to {\mathbb{R}}^n$ is differentiable at all points of $x\in E_l$, $l\in N$, and its differential $Df\left(x\right)$ is non-degenerate. Therefore, f is locally homeomorphic around x and $i(x,f)=1$.

7 It is proved in [5, Corollary 30] that any mapping belonging to $\mathcal{I}\mathcal{D}({\mathbb{R}}^n;n,n;1,1)$ is a mapping with bounded distortion. It follows that its Jacobian is strictly positive a. e. (see details in [1])

Additional information

Funding

The work is supported by Mathematical Center in Akademgorodok under agreement with the Ministry of Science and Higher Education of the Russian Federation, No. 075-15-2019-1613.