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#### Generalized Poincaré-Sobolev inequalities

(2020-12)

Poincaré-Sobolev 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 ...

#### 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 ...

#### Topics in Harmonic Analysis; commutators and directional singular integrals

(2020-03-01)

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 ...

#### 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 ...

#### 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 ...

#### Sharp reverse Hölder inequality for Cp weights and applications

(2020)

We prove an appropriate sharp quantitative reverse Hölder inequality for the $C_p$ class of
weights fromwhich we obtain as a limiting case the sharp reverse Hölder inequality for
the $A_\infty$ class of weights (Hytönen ...

#### Multilinear operator-valued calderón-zygmund 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 R-boundedness ...

#### Maximal estimates for a generalized spherical mean Radon transform acting on radial functions

(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 ...