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Differentiability of Lipschitz functions on metric measure spaces. (English) Zbl 0942.58018
The author extends to certain metric measure spaces the most essential part of calculus which depends upon first derivatives of functions. Among other results, a generalization is given of the theorem of Rademacher which states that a real valued Lipschitz function on \(\mathbb{R}^n\) is differentiable almost everywhere with respect to Lebesgue measure [H. Rademacher, Math. Ann. 79, 340-359 (1919; JFM 47.0243.01)]. This implies that in a suitably generalized sense, at almost all points, the blow ups of a real valued Lipschitz function converge to a unique linear function. Thus, one obtains that the underlying space possesses a degree of small scale and infinitesimal regularity.
This is an excellent paper, written by a master in the subject matter, and it will definitely be of essential use to both graduate students and mathematicians. It would be very nice if this paper could be extended and published as a research monograph.

MSC:
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
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