Title: Global solutions of the compressible Navier-Stokes equations Abstract : In this talk, I will talk about the existence of global weak solutions for the compressible Navier-Stokes equations, in particular, the viscosity coefficients depend on the density. Our main contribution is to further develop renormalized techniques so that the Mellet-Vasseur type inequality is not necessary for

Title: Bergman and Szego projections, Extremal Problems, and Square Functions Abstract: We study estimates for Hardy space norms of analytic projections. We first find a sufficient condition for the Bergman projection of a function in the unit disc to belong to the Hardy space $H^p$ for $1 < p < \infty$. We apply the result to prove

Title: The Fractional Porous Medium Equation on Manifolds with Conic Singularities Abstract: Due to the need to model long range diffusive interaction, during the last decade there has been a growing interest in considering diffusion equations involving non-local operators, e.g. the fractional powers of differential operators. In this talk, I will report some recent work

Title: Harmonic Conjugation in Variable Exponent Harmonic Bergman Spaces Abstract: I will talk about the harmonic conjugation in variable harmonic Bergman space. In the first part of the talk, I'll provide an overview of the main result for constant exponent spaces.  Then I'll illustrate our latest research on the boundedness of harmonic conjugation in variable harmonic Bergman

Title: Regularity Estimates for PDE with Data in Non-Standard Spaces Abstract: In this talk I present recent joint work with D. Cruz-Uribe. Given a weak super-solution $u\in W^{1,2}_0(\Omega)$ of the elliptic equation $$-\textrm{Div}\left(Q(x)\nabla u(x) \right) = f(x)$$ in a smooth domain $\Omega$ of $\mathbb{R}^n$ with $f$ in the Birnbaum-Orlicz space $L^A(\Omega)$ ($A(t) = t^{n/2}\log^\sigma(e+t)$ with