Speaker
Description
The space of measures of bounded total variation and $L_1$ quite often lack good properties. Usually this is related to the unboundedness of the maximal function on $L_1$. The interest in these spaces is justified not only by the simplicity of their norms, but also by direct connections with the geometric measure theory, and, thus, geometry. In harmonic analysis,
we sometimes replace $L_1$ with a narrower space $H_1$. Here $H_1$ is the real Hardy class, it behaves ‘better’. However, we lose the relationship with the geometric measure theory when passing to $H_1$: by ’the Riesz brothers’ theorem’ there are no analogs of the space of measures whose norm resembles that of $H_1$. We will try to suggest a scale of spaces that interpolates $L_1$ and $H_1$. These intermediate spaces do contain singular measures of fractional dimension and also posses some properties of the Hardy class; for example, some ’trace’ inequalities for fractional integration operators are valid for measures in these spaces. We do not have complete proofs yet and are also slightly unsure whether our definitions are best possible. A larger part of work is done in a martingale model that simplifies the classical setting of Euclidean spaces. Nevertheless, the story seems worth telling. For example, the aforementioned scale extends the definition of lower Hausdorff dimension to arbitrary distributions.
Work in progress with Daniel Spector, Taiwan
