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I understand the basic definition of a metric measure space to be the following:

A metric measure space is a triple of a space $X$, metric $d$, and measure $m$: $(X,d,m)$ in the sense that the metric induces a topology and the measure is the borel measure arising from the sigma field induced by the metric.

I frequently hear/read that, that analysis on these spaces is, informally, analysis in spaces with no a priori smooth structure.

In what sense, formal or informal, is analysis on these spaces "non-smooth"? Does it have to do with the fact the measure isn't complete or something?

Also feel free to change the tags on this question, I'm not sure where it goes.

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    $\begingroup$ It simply has to do with the fact there is no notion of a smooth or differentiable function between metric measure spaces. $\endgroup$
    – Wojowu
    Jan 28 '19 at 14:28
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    $\begingroup$ One example that comes to mind is the class of Alexandrov spaces, which can be thought of as limits of smooth manifolds. For example the distance function is never smooth on a compact smooth manifold, I think. $\endgroup$ Jan 28 '19 at 14:39
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    $\begingroup$ Another example is analysis on Cantor sets. No smooth structures. Since you are at Stony Brook you should talk to Raanan Schul or Silvia Ghinassi. They would provide you many interesting examples. $\endgroup$ Jan 28 '19 at 14:55
  • $\begingroup$ I would compare to the analysis on the "smoothest" spaces, i.e. real vector spaces with an inner product. $\endgroup$
    – Dirk
    Jan 28 '19 at 19:59
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    $\begingroup$ @Wojowu Actually in many situation there are "differentiable" mappings between metric spaces. In particular Cheeger proved existence of a measurable differentiable structure on spaces supporting Poincare inequalities. $\endgroup$ Mar 5 '20 at 18:05
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I highly recommend the survey article "Nonsmooth Calculus" by Juha Heinonen, available here.

The beginning of the introduction reads:

"The word nonsmooth in the title refers both to functions and spaces. Calculus is a field of study where infinitesimal data yields global information. Mathematicians have been practicing calculus with nonsmooth functions for over a century, but only recently in spaces that are not smooth in the traditional sense. In this article, we first survey calculus with nonsmooth functions and then move on to discuss current advances involving singular spaces."

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Nonsmooth functions have been studied in analysis since the nineteenth century. This led to the theory of Sobolev spaces, which turned out to be the right tool to study nonlinear partial differential equations and calculus of variations. While the functions were not smooth, they were defined on smooth objects: domains in the Euclidean space or, more generally, Riemannian manifolds. By the end of the 1970s it had been discovered that a substantial part of harmonic analysis could be generalized to spaces that are not smooth, namely to spaces of homogeneous type which are metric spaces equipped with a so-called doubling measure. This included the study of maximal functions, Hardy spaces and BMO, but it was only the zeroth-order analysis in the sense that no derivatives were involved. The study of first-order analysis with suitable generalizations of derivatives, a fundamental theorem of calculus, and Sobolev spaces, in the setting of spaces of homogeneous type, has been developed since the 1990s. This area is growing and plays an important role in many areas of contemporary mathematics. It is known as analysis on metric spaces. As a sign of recognition, analysis on metric spaces has been included in the 2010 MSC classification as a category (30L: Analysis on metric spaces). You can find more information about the scope of applications of analysis on metric spaces in a recent brief survey paper that has some of the most important references to books and articles in the subject.

M. Bonk, L. Capogna, P. Hajłasz, N. Shanmugalingam, J. Tyson, Analysis in metric spaces. Notices Amer. Math. Soc. 67 (2020), no. 2, 253–256. https://www.ams.org/journals/notices/202002/rnoti-p253.pdf.

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