Inequalities with angular integrability and applications

We prove an extension of the Stein-Weiss weighted estimates for fractional integrals, in the context of Lp spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can be unified with their improved radial versions. A number of consequences are obtained: in particular we deduce precised versions of weighted Sobolev embeddings, Caffarelli-Kohn-Nirenberg estimates, and Strichartz estimates for the wave equation, which extend the radial improvements to the case of arbitrary functions. Then we apply this technology in order to give new a priori assumptions on weak solutions of the Navier-Stokes equation so as to be able to conclude that they are smooth. The regularity criteria are given in terms of mixed radial-angular weighted Lebesgue space norms.Comment: Phd Thesis at University Sapienza, advisor professor Piero D'Ancon

Random Walk on Lattice with an Antisymmetric Perturbation in One Point

We study an homogeneous irreducible markovian random walk in a square lattice of arbitrary dimension, with an antisymmetric perturbation acting only in one point. We compute exactly spatial correction to the diffusive behaviour in the asympotics of probability, in the spirit of local limit theorems for random walks.Comment: This paper has been withdrawn by the author due to a error in the proo

Singular integrals with angular integrability

In this note we prove a class of sharp inequalities for singular integral operators in weighted Lebesgue spaces with angular integrability.Comment: 5 pages - updated bibliograph

Convergence over fractals for the periodic Schr\"odinger equation

We consider a fractal refinement of the Carleson problem for pointwise convergence of solutions to the periodic Schr\"odinger equation to their initial datum. For $\alpha \in (0,d]$ and $s < \frac{d}{2(d+1)} (d + 1 - \alpha)$, we find a function in $H^s(\mathbb{T}^d)$ whose corresponding solution diverges in the limit $t \to 0$ on a set with strictly positive $\alpha$-Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that $s > \frac{d}{2(d+2)}\left( d+2-\alpha \right)$ is sufficient for the solution corresponding to every datum in $H^s(\mathbb T^d)$ to converge $\alpha$-almost everywhere