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In an article by Cheeger, he presents an elementary method, which he refers to as quantitative differentiation, for obtaining a quantitative version of Rademacher's theorem. While these arguments use physical-space decompositions of functions, Cheeger comments in a talk that the argument can be phrased in terms of Littlewood-Paley theory, citing a short paper of Jones. Jones does in fact mention the "Littlewood-Paley inequality", however I don't see any semblance of a frequency-space decomposition used in the article.

My question is whether anyone can provide a rephrasing of these aforementioned results using frequency-space decompositions, as I have never seen any applications of Littlewood-Paley theory to these geometric measure theory-type results. In the case of Rademacher's theorem, I think this line of argument would also be nice in view of the translation-invariance of the statement.

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    $\begingroup$ While the modern incarnation of Littlewood-Paley theory is largely based around Fourier multipliers (or, in some cases, multipliers related to the heat kernel), classical Littlewood-Paley theory (of the type in Jones' paper) is instead based around multipliers arising from the Poisson kernel (harmonic extensions). See Stein's "Singular integrals and differentiability properties of functions" for a good introduction to the classical approach. It is instructive to rewrite these multipliers as Fourier multipliers to see the connection with the modern formalism. $\endgroup$ Commented Oct 16 at 22:21

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Here is a conversion table that might be useful for readers who wish to translate between different flavors of Littlewood-Paley theory. (Here $\Delta$ is the spatial Laplacian, with the negative semi-definite sign convention. Factors of $2\pi$ are ignored.)

Fourier-based Heat equation-based Harmonic extension-based
Time variable N/A $t$ $y$
Extension equation N/A $\partial_t v = \Delta v$ $\partial_{yy} u + \Delta u = 0$
Initial condition N/A $v(0,x)=f(x)$ $u(0,x)=f(x)$
In functional calculus N/A $v(t) = e^{t\Delta} f$ $u(y) = e^{-y|\nabla|} f$
Frequency scale $N$ $t^{-1/2}$ $y^{-1}$
Low-frequency projection $P_{\leq N} f$ $v(s) = e^{t\Delta}f$ $u(y) = e^{-y|\nabla|} f$
Low-frequency multiplier $\phi(\xi/N)$ $e^{t|\xi|^2}$ $e^{-y|\xi|}$
Low-frequency kernel Schwartz function Heat kernel Poisson kernel
Medium-frequency projection $P_N = P_{\leq N}-P_{\leq N/2}$ $-t\partial_t v = -t \Delta e^{t\Delta} f$ $-y\partial_y u = y|\nabla| e^{-y|\nabla|} f$
Medium-frequency multiplier $\psi(\xi/N)$ $t|\xi|^2 e^{t|\xi|^2}$ $y|\xi| e^{-y|\xi|}$
To sum over scales, use $\sum_{N \in 2^{\bf Z}}$ $\int_0^\infty \frac{dt}{t}$ $\int_0^\infty \frac{dy}{y}$
Resolution of the identity $f = \sum_N P_N f$ $f = \int_0^\infty (-t\partial_t v(t)) \frac{dt}{t}$ $f = \int_0^\infty (-y\partial_y u(y)) \frac{dy}{y}$
Littlewood-Paley square function $(\sum_N |P_N f|^2)^{1/2}$ $\int_0^\infty |-t\partial_t v(t)|^2\ \frac{dt}{t}$ $\int_0^\infty |-y\partial_y u(y)|^2 \frac{dy}{y}$
BMO condition $\int_Q |P_{\geq 1/\ell(Q)} f|^2\ dx \ll |Q|$ $\int_0^{\ell(Q)^2} \int_Q t |\partial_t v|^2\ dx dt \ll |Q|$ $\int_0^{\ell(Q)} \int_Q y |\nabla u|^2\ dx dy \ll |Q|$
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