We characterize boundedness and compactness of the Toeplitz operator $T_{\mu}$, on the Bergman space $L_{a}^{1}(\Delta)$, where the symbols, $\mu$, are complex Borel measures on the unit disk of the complex plane, $\Delta$. The case of Toeplitz operators whose symbols are anti-analytic integrable functions is settled. Our results are related to the reproducing kernel thesis. We also study the case of symbols which are positive measures and the case of radial symbols. Moreover, we give a characterization of compactness for general bounded operators on $L^1_a.$

New estimates are obtained for the $\overline{\partial}$ on non-Stein domains in $\mathbb{C}^n$and the results are applied to the Corona problem in Carleman algebras on those domains.

In this work we study the following nonlinear elliptic boundary value problem, $b(u)-div \; a(x,\nabla u)=f \hbox{ in }\Omega$, $a(x,\nabla u).\eta=-\left|u\right|^{p(x)-2}u \hbox{ on }\partial \Omega$, where $\Omega$ is a smooth bounded open domain in $\mathbb{R}^{N}$, $N \geq 1$ with smooth boundary $\partial\Omega$. We prove the existence and uniqueness of a weak solution for $f \in L^{\infty}(\Omega)$, the existence and uniqueness of an entropy solution for $L^{1}$-data $f$. The functional setting involves Lebesgue and Sobolev spaces with variable exponent.

The definition of the coherence relations in non-abelian cohomology is a difficult problem studied by many authors. The purpose of this paper is to simplify the solution provided by the author which uses the notion of sequences of fibred categories and to apply the resulting theory to higher divisors and Chow theory.

We prove that in general surface bundle over the circle with pseudo-Anosov monodromy can have taut foliation which cannot be conjugated to the fibration or the models foliations. We also build smooth minimal foliations on surfaces bundles over the circle whose fibers are not compact.

In this article we consider the existence of positive solutions of a system of periodic neutral difference equations. The main tool employed is the Krasnosel'skii's fixed point theorem for the sum of a completely continuous operator and a contraction.

In this paper we survey research progress related to the existence of an adjoint for linear operators on Banach spaces. We introduce a new pair separable Banach spaces which are required for the general theory. We then discuss a number ways one can explicitly construct an adjoint and then prove that one always exists for bounded linear operators. However, this is not true for the class of closed densely defined linear operators. In this case, we can only show that one exists for operators of Baire class one. The existence of an adjoint allows us to construct the polar decomposition. As applications, we extend the Poincaré inequality and the Stone-von Neumann version of the spectral theorem to all operators of Baire class one on a separable Banach space. Our results even show that the spectral theorem is natural for Hilbert spaces (in a certain well-defined sense). As a final application, we provide the natural Banach space version of the Schatten class of compact operators.

Using the discrete fractional sum operator, we establish some inequalities of Chebyshev type.