Browsing Analysis of Partial Differential Equations (APDE) by Title
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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 ... 
Bilinear CalderónZygmund theory on product spaces
(201910)We develop a wide general theory of bilinear biparameter singular integrals $T$. This includes general Calder\'onZygmund type principles in the bilinear biparameter setting: easier bounds, like estimates in the Banach ... 
Bilinear identities involving the $k$plane transform and Fourier extension operators
(2019)We prove certain $L^2(\mathbb{R}^n)$ bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the $k$plane transform. As the estimates are $L^2$based, they follow from ... 
Bilinear identities involving the kplane transform and Fourier extension operators
(20191130)We prove certain L2pRnq bilinear estimates for Fourier extension operators associ ated to spheres and hyperboloids under the action of the kplane transform. As the estimates are L2based, they follow from bilinear ... 
Bilinear representation theorem
(20180101)We represent a general bilinear CalderónZygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so ... 
Bilinear Spherical Maximal Functions of Product Type
(20210812)In this paper we introduce and study a bilinear spherical maximal function of product type in the spirit of bilinear Calderón–Zygmund theory. This operator is different from the bilinear spherical maximal function considered ... 
A Bilinear Strategy for Calderón's Problem
(201908)Electrical Impedance Imaging would suffer a serious obstruction if for two different conductivities the potential and current measured at the boundary were the same. The Calder\'on's problem is to decide whether the ... 
A Bilinear Strategy for Calderón’s Problem
(202005)Electrical Impedance Imaging would suffer a serious obstruction if two different conductivities yielded the same measurements of potential and current at the boundary. The Calderón’s problem is to decide whether the ... 
Bloom type inequality for biparameter singular integrals: efficient proof and iterated commutators
(20190314)Utilising some recent ideas from our bilinear biparameter theory, we give an efficient proof of a twoweight Bloom type inequality for iterated commutators of linear biparameter singular integrals. We prove that if $T$ ... 
Bloom type upper bounds in the product BMO setting
(20190408)We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral $T_n$ in $\mathbb R^n$ and a bounded singular integral $T_m$ in $\mathbb R^m$ we prove that $$ \ [T_n^1, ... 
Borderline Weighted Estimates for Commutators of Singular Integrals
(20160701)In this paper we establish the following estimate \[ w\left(\left\{ x\in\mathbb{R}^{n}\,:\,\left[b,T]f(x)\right > \lambda\right\} \right)\leq \frac{c_{T}}{\varepsilon^{2}}\int_{\mathbb{R}^{n}}\Phi\left(\b\_{BMO}\f ... 
Boundary Triples for the Dirac Operator with CoulombType Spherically Symmetric Perturbations
(201902)We determine explicitly a boundary triple for the Dirac operator $H:=i\alpha\cdot \nabla + m\beta + \mathbb V(x)$ in $\mathbb R^3$, for $m\in\mathbb R$ and $\mathbb V(x)= x^{1} ( \nu \mathbb{I}_4 +\mu \beta i \lambda ... 
The Calderón problem with corrupted data
(201701)We consider the inverse Calderón problem consisting of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the DirichlettoNeumann map and, therefore, ... 
Carleman type inequalities for fractional relativistic operators
(20190922)In this paper, we derive Carleman estimates for the fractional relativistic operator. Firstly, we consider changingsign solutions to the heat equation for such operators. We prove monotonicity inequalities and convexity ... 
A characterization of two weight norm inequality for LittlewoodPaley $g_{\lambda}^{*}$function
(2017)Let $n\ge 2$ and $g_{\lambda}^{*}$ be the wellknown high dimensional LittlewoodPaley function which was defined and studied by E. M. Stein, $$g_{\lambda}^{*}(f)(x)=\bigg(\iint_{\mathbb R^{n+1}_{+}} \Big(\frac{t}{t+xy ... 
A comparison principle for vector valued minimizers of semilinear elliptic energy, with application to dead cores
(2021)We establish a comparison principle providing accurate upper bounds for the modulus of vector valued minimizers of an energy functional, associated when the potential is smooth, to elliptic gradient systems. Our assumptions ... 
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 ... 
Convex Integration Arising in the Modelling of ShapeMemory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations
(20190330)We study convex integration solutions in the context of the modelling of shapememory alloys. The purpose of the article is twofold, treating both rigidity and flexibility prop erties: Firstly, we relate the maximal ... 
Correlation imaging in inverse scattering is tomography on probability distributions
(20181204)Scattering from a nonsmooth random field on the time domain is studied for plane waves that propagate simultaneously through the potential in variable angles. We first derive sufficient conditions for stochastic moments ... 
A Decomposition of Calderón–Zygmund Type and Some Observations on Differentiation of Integrals on the InfiniteDimensional Torus
(20200213)In this note we will show a Calder\'onZygmund decomposition associated with a function $f\in L^1(\mathbb{T}^{\omega})$. The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting ...