Composition operator into the space of function of bounded variation

Luděk Kleprlík

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arXiv · 2001.03459

Composition operator into the space of function of bounded variation

Luděk Kleprlík

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Abstract

Let $\Omega_1, \Omega_2\subset \mathbb R^n$ and $1\leq p <\infty$. We study the optimal conditions on a homeomorphism $f:\Omega_1$ onto $\Omega_2$ which guarantee that the composition $u\circ f$ belongs to the space $BV(\Omega_1)$ for every $u\in W^{1,p}(\Omega_2)$. We show that the sufficient and necessary condition is an existence of a function $K(y)\in L^{p'}(\Omega_2)$ such that $|Df|(f^{-1}(A))\leq \int_A K(y)\,dy$ for all Borel sets $A$.

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Luděk Kleprlík. 2020-01-08. Composition operator into the space of function of bounded variation. https://arxiv.org/abs/2001.03459

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