Harmonic Analysis
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A Bilinear Strategy for Calderón’s Problem
(Revista Matemática Iberoamericana, 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 ... -
A note on generalized Poincaré-type inequalities with applications to weighted improved Poincaré-type inequalities
(2020)The main result of this paper supports a conjecture by C. P\'erez and E. Rela about the properties of the weight appearing in their recent self-improving result of generalized inequalities of Poincar\'e-type in the Euclidean ... -
A Decomposition of Calderón–Zygmund Type and Some Observations on Differentiation of Integrals on the Infinite-Dimensional Torus
(Potential Analysis, 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 ... -
Maximal estimates for a generalized spherical mean Radon transform acting on radial functions
(Annali de Matematica Pura et Applicata, 2020)We study a generalized spherical means operator, viz.\ generalized spherical mean Radon transform, acting on radial functions. As the main results, we find conditions for the associated maximal operator and its local ... -
Flow with $A_\infty(\mathbb R)$ density and transport equation in $\mathrm{BMO}(\mathbb R)$
(Forum of Mathematics, Sigma, 2019-11)We show that, if $b\in L^1(0,T;L^1_{\rm {loc}}(\mathbb R))$ has spatial derivative in the John-Nirenberg space ${\rm{BMO}}(\mathbb R)$, then it generates a unique flow $\phi(t,\cdot)$ which has an $A_\infty(\mathbb R)$ ... -
Bilinear Calderón--Zygmund theory on product spaces
(Journal des Math\'ematiques Pures et Appliqu\'ees, 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 ... -
Quantitative weighted estimates for Rubio de Francia's Littlewood--Paley square function
(Journal of Geometric Analysis, 2019-12)We consider the Rubio de Francia's Littlewood--Paley square function associated with an arbitrary family of intervals in $\mathbb{R}$ with finite overlapping. Quantitative weighted estimates are obtained for this operator. ... -
Análisis de Fourier en el toro infinito-dimensional
(2019-10-24)Se presentan algunos resultados originales de análisis armónico para funciones definidas en el toro infinito, que es el grupo topológico compacto consistente en el producto cartesiano de una familia numerable de toros ... -
Reconstruction of the Derivative of the Conductivity at the Boundary
(2019-08)We describe a method to reconstruct the conductivity and its normal derivative at the boundary from the knowledge of the potential and current measured at the boundary. This boundary determination implies the uniqueness ... -
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 ... -
$A_1$ theory of weights for rough homogeneous singular integrals and commutators
(Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, 2019)Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular integrals $T_{\Omega}$ and commutators of $\BMO$ symbols and $T_{\Omega}$ are obtained. In particular the following estimates are proved: \[ \|T_\Omega ... -
On the absolute divergence of Fourier series in the infinite dimensional torus
(Colloquium Mathematicum, 2019-03-22)In this note we present some simple counterexamples, based on quadratic forms in infinitely many variables, showing that the implication $f\in C^{(\infty}(\mathbb{T}^\omega)\Longrightarrow\sum_{\bar{p}\in\mathbb{Z}^\inf ... -
Improved fractional Poincaré type inequalities in John domains
(Arkiv för Matematik, 2019)We obtain improved fractional Poincaré inequalities in John domains of a metric space $(X, d)$ endowed with a doubling measure $\mu$ under some mild regularity conditions on the measure $\mu$. We also give sufficient ... -
On extension problem, trace hardy and Hardy’s inequalities for some fractional Laplacians
(Communications on pure and applied analysis, 2019-09)We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation ... -
Bloom type upper bounds in the product BMO setting
(Journal of Geometric Analysis, 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, ... -
Bloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutators
(International Mathematics Research Notices, 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 $T$ ... -
Sparse bounds for maximal rough singular integrals via the Fourier transform
(Annales de l'institut Fourier, 2019-03-12)We prove a quantified sparse bound for the maximal truncations of convolution-type singular integrals with suitable Fourier decay of the kernel. Our result extends the sparse domination principle by Conde-Alonso, Culiuc, ... -
Unique determination of the electric potential in the presence of a fixed magnetic potential in the plane
(2018-12)For electric and magnetic potentials with compact support, we consider the magnetic Schrödinger equation with fixed positive energy. Under a mild additional regularity hypothesis, and with fixed magnetic potential, we show ... -
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 ... -
Correlation imaging in inverse scattering is tomography on probability distributions
(Inverse Problems, 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 ...
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