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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

We prove that for certain positive operators T, such as the Hardy-Littlewood maximal function and fractional integrals, there is a constant D>1, depending only on the dimension n, such that the two weight norm inequality ...

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 ...

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 ...