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Let $(X,d,m)$ and $(Y,\rho,n)$ be metric measure spaces and let $f:X\rightarrow Y$ be a Borel-measurable function for which there is some $y_0$ and some $p\geq 0$ such that $$ \int_{x\in X}\,d(y_0,f(x))^p\,dn(x) <\infty. $$

Are there known, non-trivial, conditions under which $f$ factors in the following manner:

  • There exists some Borel measurable $f_1:X\rightarrow \mathbb{R}^n$ whose components belong to the Hajłasz-Sobolev space $W_1^s(X)$.
  • There is a Lipschitz function $f_2:f_1(X)\rightarrow Y$ (which need not be Lipschitz on the entire $\mathbb{R}^n$; here this is interesting if $f_1$ is not continuous and $X$ is non-compact) $$ f(x) = f_2\circ f_1(x)\qquad (\forall x \in X)? $$

I would also be interested in the ``simplest case'' where $X$ and $Y$ are Riemannian manifolds with $X$ a closed manifold, and $m$ and $n$ are the measures induced by their volume forms.

Bonus Question: If yes, then can we also relate the Lipschitz constant of $f_2$ to $f$'s regularity?

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  • $\begingroup$ Is $dn(u)$ meant to be $dn(y)$? And what does $f$ have to do with this condition? $\endgroup$
    – Dirk Werner
    Jan 27, 2022 at 14:07
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    $\begingroup$ And is $\int_{y\in Y}$ meant to be $\int_X$? $\endgroup$
    – Dirk Werner
    Jan 27, 2022 at 18:57
  • $\begingroup$ @DirkWerner Any leads? $\endgroup$
    – ABIM
    Jan 31, 2022 at 12:31
  • $\begingroup$ I feel that the follow-up and converse question may be easier: mathoverflow.net/questions/416157/stability-of-sobolev-class $\endgroup$
    – ABIM
    Feb 14, 2022 at 23:30

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