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TWO-WEIGHTED COMPOSITION OPERATORS ON SOBOLEV SPACES AND QUASICONFORMAL ANALYSIS

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Abstract

We establish some equivalent conditions for a homeomorphism \(\varphi : D\rightarrow D'\) of Euclidean domains in \(\mathbb R^n\), \(n\ge 2\), to induce a bounded composition operator \(\varphi ^*: {L}^1_p(D';\omega ) \cap \text {Lip}_l(D')\rightarrow L^1_q(D;\theta )\), where \(1< q \le p<\infty\), by the composition rule: \(\varphi ^*(f)=f\circ \varphi\). Here \(\omega :D'\rightarrow (0,\infty )\) is an arbitrary weight function on the domain \(D'\), and \(\theta :D\rightarrow (0,\infty )\) is some weight function in Muckenhoupt’s \(A_{q}\)-class on the domain D. Moreover, we prove that the class of homeomorphisms under consideration is completely determined by the controlled variation of the weighted capacity of cubical condensers whose shells are concentric cubes.

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Notes

  1. The definition of generalized derivatives assumes that \(\frac{\partial u}{dy_j}\in L_{1,\text {loc}}(D')\).

  2. For our description of a cubical condenser, see Definition 3.

  3. Recall that the norm \(|x|_p\) of a vector \(x=(x_1,x_2,\dots ,x_n)\in \mathbb R^n\) 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 an Euclidean ball B or cube Q respectively.

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

  5. We understand (9) in the sense that any function \(u\in {L}^1_p(W;\omega ) \cap \overset{\circ }{\text {Lip}}_l(W)\) extended by zero beyond W belongs to \({L}^1_p(D';\omega ) \cap \text {Lip}_l(D')\).

  6. Henceforth, \(B_\delta\) is an arbitrary ball \(B(z,\delta )\subset D'\) containing y.

  7. It is not required that \(u_i\in \mathcal R(U_i)\) and \(v_i=\varphi ^*u_i\).

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  1. Sobolev Institute of Mathematics, Acad. Koptyug av.4, 630090, Novosibirsk, Russia

    S. K. Vodopyanov

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The paper is dedicated to the anniversary of Professor S. Samko.

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Vodopyanov, S.K. TWO-WEIGHTED COMPOSITION OPERATORS ON SOBOLEV SPACES AND QUASICONFORMAL ANALYSIS. J Math Sci 266, 491–509 (2022). https://doi.org/10.1007/s10958-022-05903-y

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