Analysis of Partial Differential Equations (APDE)
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RESTRICTED TESTING FOR POSITIVE OPERATORS
(2020)We prove that for certain positive operators T, such as the HardyLittlewood maximal function and fractional integrals, there is a constant D>1, depending only on the dimension n, such that the two weight norm inequality ... 
Static and Dynamical, Fractional Uncertainty Principles
(202103)We study the process of dispersion of lowregularity solutions to the Schrödinger equation using fractional weights (observables). We give another proof of the uncertainty principle for fractional weights and use it to get ... 
Extensions of the JohnNirenberg theorem and applications
(2021)The John–Nirenberg theorem states that functions of bounded mean oscillation are exponentially integrable. In this article we give two extensions of this theorem. The first one relates the dyadic maximal function to the ... 
Convergence over fractals for the Schrödinger equation
(202101)We consider a fractal refinement of the Carleson problem for the Schrödinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with ... 
Multilinear operatorvalued calderónzygmund theory
(2020)We develop a general theory of multilinear singular integrals with operator valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the Rboundedness ... 
Invariant measures for the dnls equation
(20201002)We describe invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schr\"odinger equation (DNLS) constructed in \cite{MR3518561, Genovese2018}. The construction works for small $L^2$ ... 
Endpoint estimates, extrapolation for multilinear muckenhoupt classes, and applications
(2019)In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the socalled multilinear Muckenhoupt classes. ... 
Magnetic domaintwin boundary interactions in NiMnGa
(202004)The stress required for the propagation of twin boundaries in a sample with fine twins increases monotonically with ongoing deformation. In contrast, for samples with a single twin boundary, the stress exhibits a plateau ... 
Sensitivity of twin boundary movement to sample orientation and magnetic field direction in NiMnGa
(2019)When applying a magnetic field parallel or perpendicular to the long edge of a parallelepiped Ni MnGa stick, twin boundaries move instantaneously or gradullay through the sample. We evaluate the sample shape dependence ... 
Generalized PoincaréSobolev inequalities
(202012)PoincaréSobolev inequalities are very powerful tools in mathematical analysis which have been extensively used for the study of differential equations and their validity is intimately related with the geometry of the ... 
Sparse and weighted estimates for generalized Hörmander operators and commutators
(2019)In this paper a pointwise sparse domination for generalized Ho ̈rmander and also for iterated commutators with those operators is provided generalizing the sparse domination result in [24]. Relying upon that sparse domination ... 
The Well Order Reconstruction Solution for ThreeDimensional Wells, in the Landaude Gennes theory.
(2019)We study nematic equilibria on threedimensional square wells, with emphasis on Well Order Reconstruction Solu tions (WORS) as a function of the well size, characterized by λ, and the well height denoted by ε. The WORS ... 
A sharp lorentzinvariant strichartz norm expansion for the cubic wave equation in \mathbb{R}^{1+3}
(2020)We provide an asymptotic formula for the maximal Stri chartz norm of small solutions to the cubic wave equation in Minkowski space. The leading coefficient is given by Foschi’s sharp constant for the linear Strichartz ... 
Multilinear singular integrals on noncommutative lp spaces
(2019)We prove Lp bounds for the extensions of standard multilinear Calderón Zygmund operators to tuples of UMD spaces tied by a natural product structure. The product can, for instance, mean the pointwise product in UMD ... 
Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory
(2019)Abstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − Lσ,μ[φ(u)] = f(x,t) in RN × (0,T), where Lσ,μ is a general ... 
Bayesian approach to inverse scattering with topological priors
(2020)We propose a Bayesian inference framework to estimate uncertainties in inverse scattering problems. Given the observed data, the forward model and their uncertainties, we find the posterior distribution over a finite ... 
The Schrödinger equation and Uncertainty Principles
(202009)The main task of this thesis is the analysis of the initial data u0 of Schrödinger’s initial value problem in order to determine certain properties of its dynamical evolution. First we consider the elliptic Schrödinger ... 
The observational limit of wave packets with noisy measurements
(2019)The authors consider the problem of recovering an observable from certain measurements containing random errors. The observable is given by a pseudodifferential operator while the random errors are generated by a Gaussian ... 
Scattering with criticallysingular and δshell potentials
(2019)The authors consider a scattering problem for electric potentials that have a component which is critically singular in the sense of Lebesgue spaces, and a component given by a measure supported on a compact Lipschitz ... 
Topics in Harmonic Analysis; commutators and directional singular integrals
(20200301)This dissertation focuses on two main topics: commutators and maximal directional operators. Our first topic will also distinguish between two cases: commutators of singular integral operators and BMO functions and ...