The development of the theory of martingale Hardy spaces has been strongly influenced by the classical theory. Because of this it is inevitable to compare results of this theory to those on classical Hardy spaces and during the decades this new direction was developing in this way and many similarities between these theories have already been found. But nowadays a lot of new results were obtained in the theory of martingale Hardy spaces which are new in classical case.

This lecture is devoted to review theory of martingale Hardy spaces. In particular, we give atomic decomposition of these spaces and show how this theorem simplify the proofs of boundedness of any sub-linear operators on these spaces. For the illustration of bounded operators on the martingale Hardy spaces we define Vilenkin groups and their characters which are called Vilenkin functions. We also define Fourier series with respect to Vilenkin system and maximal operators related to these operators. Moreover, we define modulus of continuity in martingale Hardy spaces and derive necessary and sufficient conditions in the terms of modulus of continuity such that partial sums with respect to one Vilenkin-Fourier series converge in $H^p$ norm. We also investigate classical Hardy spaces, modulus of continuity in these spaces and present one to one analogues of these results for partial sums in this classical case.

At the end of this presentation we state some open problem how to generate these two similar results for Hardy spaces on some locally compact Abelian groups and partial sums with respect to character functions of these groups. Moreover, we also state similar open problems for variable martingale Hardy spaces. Finally, we will choose opposite approach and will try to state and investigate results for partial sums of the two-dimensional Fourier series in martingale Hardy spaces for classical case of Hardy spaces.

Let us consider two notions of concentration for homogeneous polynomials in d complex variables on the unit sphere: a local notion measuring the fraction of the $L^2$-norm supported on a measurable subset and a global notion given by the generalized Wehrl entropy. Lieb and Solovej proved that the extremizers in both cases are monomials up to a unitary rotation. Their result generalizes the one by Lieb in 1978 on the Wehrl entropy conjecture for coherent states in representations of the Heisenberg group to symmetric representations of the groups $\operatorname{SU}(d)$.

In this talk, we will focus on the stability of the previous inequalities. Namely, if the concentration is close to the optimal one, we will quantify how close the polynomial is to the extremizers. This is obtained in full generality in the case $d=2$, while in the case of higher dimensions restrictions on the size of the subset or on the degree of the polynomials arise. We will finally recover analogous stability results in the Bargmann–Fock space.

This is a joint work with Joaquim Ortega-Cerdà (UB-CRM).