Browsing Harmonic Analysis by Title
Now showing items 5-24 of 100
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Bilinear Calderón--Zygmund theory on product spaces
(2019-10)We develop a wide general theory of bilinear bi-parameter singular integrals $T$. This includes general Calder\'on--Zygmund type principles in the bilinear bi-parameter setting: easier bounds, like estimates in the Banach ... -
Bilinear representation theorem
(2018-01-01)We represent a general bilinear Calderón--Zygmund 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
(2021-08-12)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
(2019-08)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
(2020-05)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 bi-parameter singular integrals: efficient proof and iterated commutators
(2019-03-14)Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom type inequality for iterated commutators of linear bi-parameter singular integrals. We prove that if $T$ ... -
Bloom type upper bounds in the product BMO setting
(2019-04-08)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
(2016-07-01)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 ... -
Boundedness properties of maximal operators on Lorentz spaces
(2023)We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal M$ acting on Lorentz spaces. Given $p \in (1,\infty)$ and a metric measure space $\mathcal X = (X, \rho, \mu)$ we let $\Omega^p_{\rm ... -
The Calderón problem with corrupted data
(2017-01)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 Dirichlet-to-Neumann map and, therefore, ... -
A characterization of two weight norm inequality for Littlewood-Paley $g_{\lambda}^{*}$-function
(2017)Let $n\ge 2$ and $g_{\lambda}^{*}$ be the well-known high dimensional Littlewood-Paley function which was defined and studied by E. M. Stein, $$g_{\lambda}^{*}(f)(x)=\bigg(\iint_{\mathbb R^{n+1}_{+}} \Big(\frac{t}{t+|x-y ... -
Cones with convoluted geometry that always scatter or radiate
(2021)We investigate fixed energy scattering from conical potentials having an irregular cross-section. The incident wave can be an arbitrary non-trivial Herglotz wave. We show that a large number of such local conical scatterers ... -
Convergence over fractals for the Schrödinger equation
(2021-01)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 ... -
Correlation imaging in inverse scattering is tomography on probability distributions
(2018-12-04)Scattering from a non-smooth 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 ... -
Corrigendum to: An extension problem and trace Hardy inequality for the sublaplacian on H-type groups
(2021-03-10)Recently we have found a couple of errors in our paper entitled An extension problem and trace Hardy inequality for the sub-Laplacian on $H$-type groups, Int. Math. Res. Not. IMRN (2020), no. 14, 4238--4294. They concern ... -
Counterexamples for the fractal Schrödinger convergence problem with an Intermediate Space Trick
(2021-12-09)We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of Du--Kim--Wang--Zhang. We confirm that the same ... -
A Decomposition of Calderón–Zygmund Type and Some Observations on Differentiation of Integrals on the Infinite-Dimensional Torus
(2020-02-13)In this note we will show a Calder\'on--Zygmund 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 ... -
Degenerate Poincare-Sobolev inequalities
(2021)Abstract. We study weighted Poincar ́e and Poincar ́e-Sobolev type in- equalities with an explicit analysis on the dependence on the Ap con- stants of the involved weights. We obtain inequalities of the form with different ... -
Determination of convection terms and quasi-linearities appearing in diffusion equations
(2018-12)We consider the highly nonlinear and ill posed inverse problem of determining some general expression appearing in the a diffusion equation from measurements of solutions on the lateral boundary. We consider both linear ... -
Discrete Carleman estimates and three balls inequalities
(2021-10-16)We prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schrödinger operators. These quantitatively connect the discrete setting in which the unique continuation property fails and the ...