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The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure, which only satisfies an upper control on the size…

The manuscript is devoted to the boundary behavior of mappings with bounded and finite distortion, which has been actively studied recently. We consider mappings of domains of the Euclidean space that satisfy the inverse Poletsky inequality…

In this paper we will deal with problems in approximation theory of bounded analytic functions on the unit disc and their boundary behavior on the unit circle. We will attempt to unify two known such theorems to create a stronger theorem.…

It is shown that the quadrature derivatives in some analytical gradients of energies evaluated with a multi-centre radial-angular grid do not vanish even in the limit of an infinitely dense grid, causing severe errors when neglected. The…

The thesis comprises three chapters. Chapter 1 investigates generalizations of the theorem of Fatou for convolution type integral operators with general approximate identities. It is introduced $\lambda(r)$-convergence, which is a…

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $\phi=b/a$, where $a$ is the Pythagorean mate of $b$. It is known that, if $f$ is a function holomorphic on a…

We study the hot spots conjecture for domains in the Gaussian space $(\mathbb{R}^n, (2\pi)^{-n/2} e^{-|x|^2/2} dx)$ for $n \ge 2$. Given a bounded domain $\Omega$ with a piecewise smooth boundary, we consider the first nontrivial…

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…

We show that the complex saddle points of the no-boundary wave function with a positive cosmological constant and a positive scalar potential have a representation in which the geometry consists of a regular Euclidean AdS domain wall that…

The $k$-Cauchy-Fueter complex, $k=0,1,\ldots$, in quaternionic analysis are the counterpart of the Dolbeault complex in the theory of several complex variables. In this paper, we construct explicitly boundary complexes of these complexes on…

A recently developed viewpoint on the fundamentals of density-functional theory for finite interacting spin-lattice systems that centers around the notion of degeneracy regions is presented. It allows for an entirely geometrical description…

We investigate the 1D Riemann-Liouville fractional derivative focusing on the connections with fractional Sobolev spaces, the space $BV$ of functions of bounded variation, whose derivatives are not functions but measures and the space…

The Langlands correspondence for complex curves is traditionally formulated in terms of sheaves rather than functions. Recently, Langlands asked whether it is possible to construct a function-theoretic version. In this paper we use the…

In this paper we extend the standard differential geometric theory of Hamiltonian dynamics to noncommutative spaces, beginning with symplectic forms. Derivations on the algebra are used instead of vector fields, and interior products and…

Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…

Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\mathcal{T}(S)$ its Teichm{\"u}ller space. For any isotopy class of closed curves $\gamma$, we compute the first three derivatives of the length function…

We obtain some H\"older regularity estimates for an Hamilton-Jacobi with fractional time derivative of order $\alpha \in (0,1)$ cast by a Caputo derivative. The H\"older seminorms are independent of time, which allows to investigate the…

Let $M$ be a complete simply connected manifold which is in addition Gromov hyperbolic, coercive and roughly starlike. For a given harmonic function on $M$, a local Fatou Theorem and a pointwise criteria of non-tangential convergence coming…

We build on work of Muro-Raptis in [Ann. K-Theory 2 (2017), no. 2, 303-340] and Cisinski-Neeman in [Adv. Math. 217 (2008), no. 4, 1381-1475] to prove that the additivity of derivator K-theory holds for a large class of derivators that we…

We first propose a conformal geometry for Connes-Landi noncommutative manifolds and study the associated scalar curvature. The new scalar curvature contains its Riemannian counterpart as the commutative limit. Similar to the results on…