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Multiplicative square functions. (English) Zbl 1086.42010

L. Carleson [J. Anal. Math. 19, 1-13 (1967; Zbl 0186.13701)] characterized absolutely continuous doubling measures \(\mu\) on \(\mathbb{R}\) as those satisfying \[ \int_0^\infty \biggl(\sup_{| I| <t} \Bigl| \frac{2\mu(I)}{\mu(2I)}-1\Bigr| \biggr)^2\,\frac{dt}{t} <\infty.{\text{(C)}} \] The authors introduce a localized version of \(w(t)\) which they call a multiplicative square function, defined as \[ A^2(\mu) \, = \, \sum_{k=1}^\infty \max\left\{ \left| 1-\frac{2^N\mu(Q)}{\mu(Q_{k-1}(x))} \right| : Q\in F_k,\, Q\subset Q_{k-1}(x)\right\} \] in which \(F_k\) denotes the dyadic cubes in \(\mathbb{R}^N\) of sidelength \(\ell(Q)=2^{-k}\) and \(Q_{k}(x)\) is the unique element of \(F_k\) containing \(x\). Taking the finiteness of \(A^2(\mu)\) as a local version of finiteness of the integral in (C), the strengthening of Carleson’s result then takes the following form: If \(\mu\) is a doubling measure on a reference cube \(Q_0\) then the sets \[ \Bigl\{ x\in Q_0: \, \lim_{n\to\infty} \frac{\mu(Q_n(x))}{| Q_n(x)| }>0\Bigr\} \quad {\text{and} } \quad \Bigl\{ x\in Q_0: \, A^2(\mu)(x)<\infty \Bigr\} \] can differ by at most a set of Lebesgue measure zero. The authors also prove that there is a dimensional constant \(C\) such that \[ \frac{1}{C}< \frac{1}{A_n^2(\mu)(x)}\Bigl| \log \frac{\mu(Q_n(x))}{| Q_n(x)| }\Bigr| <C \] at almost every point \(x\in Q_0\) such that \(\mu(Q_n)(x)/| Q_n(x)| \to 0\) when \(n\) is large enough. Here \(A_n^2\) is the \(n\)-th partial sum of the series defining \(A^2\). This follows from a version of the law of the iterated logarithm for points in this set. Somewhat parallel results describing regularity properties of a measure in terms of a square function of its harmonic extension are also established.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions

Citations:

Zbl 0186.13701
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