Mini-cours (I)

The study  of $L_p$-convergence for  Fourier series is a classical topic in harmonic analysis. In dimension one, the Fourier series $S_n(f)$ of a periodic function is given by a convolution which can also be seen as a Fourier multiplier by the characteristic function of the interval $[-n,n]$. In bigger dimension, it is a diffcult task to determine sets whose characteristic functions are Fourier multipliers on $L_p({\mathbb R}^n)$. A celebrated result of Fefferman asserts that this surprisingly not true for the ball in ${\mathbb R}^n$ ($n>1$). Kakaya sets of directions are important geometric tools to deal with this question and many other related to Littlewood-Paley square function inequalities or Bochner Riesz summability...

Loosely speaking, a Kakeya set in the plane is a small measurable set including a unit segment in every possible direction. A closely related notion has been recently introduced for infinite sets $Z$ of directions in the unit circle. On the other hand, 2D lacunarity is nothing but a fast accumulation phenomenon for sets of directions. To some sense, the lacunarity order of a 2D set of directions $Z$ establishes how far is that set from being of Kakeya type.

During this course we shall review some classical results involving these notions. We shall also explore new higher dimensional analogues and their consequences in Euclidean and noncommutative harmonic analysis. If time permits, we shall also discuss a few open problems.