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

Zbl 0186.13701
Full Text: