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The Regularity of Inverses to Sobolev Mappings and the Theory of \( \mathcal{Q}_{q,p} \)-Homeomorphisms

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Abstract

We prove that each homeomorphism \( \varphi:D\to D^{\prime} \) of Euclidean domains in \( 𝕉^{n} \), \( n\geq 2 \), belonging to the Sobolev class \( W^{1}_{p,\operatorname{loc}}(D) \), where \( p\in[1,\infty) \), and having finite distortion induces a bounded composition operator from the weighted Sobolev space \( L^{1}_{p}(D^{\prime};\omega) \) into \( L^{1}_{p}(D) \) for some weight function \( \omega:D^{\prime}\to(0,\infty) \). This implies that in the cases \( p>n-1 \) and \( n\geq 3 \) as well as \( p\geq 1 \) and \( n\geq 2 \) the inverse \( \varphi^{-1}:D^{\prime}\to D \) belongs to the Sobolev class \( W^{1}_{1,\operatorname{loc}}(D^{\prime}) \), has finite distortion, and is differentiable \( {\mathcal{H}}^{n} \)-almost everywhere in \( D^{\prime} \). We apply this result to \( \mathcal{Q}_{q,p} \)-homeomorphisms; the method of proof also works for homeomorphisms of Carnot groups. Moreover, we prove that the class of \( \mathcal{Q}_{q,p} \)-homeomorphisms is completely determined by the controlled variation of the capacity of cubical condensers whose shells are concentric cubes.

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Notes

  1. For our description of an annular condenser, see Definition 5.

  2. The definition of generalized derivatives assumes that \( \frac{\partial u}{dy_{j}}\in L_{1,\operatorname{loc}}(D^{\prime}) \).

  3. Recall that the norm \( |x|_{p} \) of a vector \( x=(x_{1},x_{2},\dots,x_{n})\in 𝕉^{n} \) is defined as \( |x|_{p}=\bigl{(}\sum\nolimits_{k=1}^{n}|x_{k}|^{p}\bigr{)}^{\frac{1}{p}} \) for \( p\in[1,\infty) \) and \( |x|_{\infty}=\max\nolimits_{k=1,\dots,n}|x_{k}| \). Each ball of the norm \( |x|_{2} \) or \( |x|_{\infty} \) is a Euclidean ball or cube respectively.

  4. Instead of balls, we can use cubes as elementary sets.

  5. Henceforth \( B_{\delta} \) is an arbitrary ball \( B(z,\delta)\subset D^{\prime} \) containing \( y \).

  6. Here the Lebesgue Differentiation Theorem is applied in the form

    $$ \lim\limits_{r\to 0}\frac{1}{\mathcal{H}^{n-1}(B_{j}(z,r))}\int\limits_{B_{j}(z,r)}\bigg{|}\int\limits_{\Delta_{i}}\omega(w,y_{j})\,dy_{j}-\int\limits_{\Delta_{i}}\omega(z,y_{j})\,dy_{j}\bigg{|}\,dw=0 $$

    for \( \mathcal{H}^{n-1} \)-almost all \( z\in P_{j} \).

  7. As [51, § 1.1.3, Theorems 1 and 2] shows, a locally summable function \( u:\Omega\to 𝕉 \) belongs to \( L_{p}^{1}(\Omega) \) with \( p\geq 1 \) if and only if \( u \) can be changed on a set of \( \mathcal{H}^{n} \)-measure zero so that the modified function is absolutely continuous on \( \mathcal{H}^{n-1} \)-almost all straight lines parallel to each coordinate axis, and has ordinary partial derivatives belonging to \( L_{p}(\Omega) \). Furthermore, the weak gradient \( \nabla u \) of \( u \) in the sense of generalized functions coincides \( {\mathcal{H}}^{n} \)-almost everywhere with the ordinary gradient.

  8. The arguments in the indicated fragments of [12] are applicable to every homeomorphism \( \varphi:D\to D^{\prime} \) from \( W^{1}_{1,\operatorname{loc}}(D) \) having finite distortion and satisfying (2).

  9. If \( dx \) is a volume form on \( 𝔾 \) of degree \( N \) then \( i(X_{1j}) \) is a form of degree \( N-1 \), which at the smooth vector fields \( Y_{1},Y_{2},\dots,Y_{N-1} \) on \( 𝔾 \) takes the value \( i(X_{1j})(Y_{1},Y_{2},\dots,Y_{N-1})=dx(X_{1j},Y_{1},Y_{2},\dots,Y_{N-1}) \).

  10. The value of \( M \) is governed by the necessity to ensure (65)–(67), while the exact value of \( M \) is not used in the proof.

  11. In the context of this article \( \mathcal{H}^{\nu-1}(B_{\tilde{\rho}}(z,r)) \) is equivalent to \( r^{\nu-1} \) on \( Q_{0} \) if and only if there exist positive reals \( \zeta_{1} \) and \( \zeta_{2} \) such that \( \zeta_{1}r^{\nu-1}\leq\mathcal{H}^{\nu-1}(B_{\tilde{\rho}}(z,r))\leq\zeta_{2}r^{\nu-1} \) for all \( z\in Q_{0} \) and all \( r \) with \( B_{\tilde{\rho}}(z,r)\subset Q_{0} \).

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Acknowledgment

The author is grateful to the referee for critical consideration of the manuscript and comments, which helped improve the first version of this article.

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The author was supported by the Mathematical Center in Akademgorodok and the Ministry of Science and Higher Education of the Russian Federation (Contract 075–15–2019–1613).

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Vodopyanov, S.K. The Regularity of Inverses to Sobolev Mappings and the Theory of \( \mathcal{Q}_{q,p} \)-Homeomorphisms. Sib Math J 61, 1002–1038 (2020). https://doi.org/10.1134/S0037446620060051

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