Boundary multifractal behaviour for harmonic functions in the ball

It is well known that if $h$ is a nonnegative harmonic function in the ball of \RR^{d+1} or if $h$ is harmonic in the ball with integrable boundary values, then the radial limit of $h$ exists at almost every point of the boundary. In this paper, we are interested in the exceptional set of points of divergence and in the speed of divergence at these points. In particular, we prove that for generic harmonic functions and for any $\beta\in [0,d]$, the Hausdorff dimension of the set of points $\xi$ on the sphere such that $h(r\xi)$ looks like $(1-r)^{-\beta}$ is equal to $d-\beta$.Comment: 16 page

Multifractal analysis of the divergence of Fourier series: the extreme cases

International audienceWe study the size, in terms of the Hausdorff dimension, of the subsets of $\mathbb T$ such that the Fourier series of a generic function in L^1(\TT), L^p(\TT) or in $\mathcal C(\mathbb T)$ may behave badly. Genericity is related to the Baire category theorem or to the notion of prevalence

MULTIFRACTAL PHENOMENA AND PACKING DIMENSION

We undertake a general study of multifractal phenomena for functions. We show that the existence of several kinds of multifractal functions can be easily deduced from an abstract statement, leading to new results. This general approach does not work for Fourier or Dirichlet series. Using careful constructions, we extend our results to these cases