Related papers: On zero sets in the Dirichlet space

We show that the boundedness of the Hardy-Littlewood maximal operator on a K\"othe function space ${\mathbb{X}}$ and on its K\"othe dual ${\mathbb{X}}'$ is equivalent to the well-posedness of the $\mathbb{X}$-Dirichlet and…

We study how small is the set of critical values of the distance function from a compact (resp. closed) set in the plane or in a connected complete two-dimensional Riemannian manifold. We show that for a compact set, the set of critical…

We consider the Dirichlet series associated to the number of representations of an integer as the sum of primes. Assuming the Riemann hypothesis on the distribution of the zeros of the Riemann zeta function we obtain the domain of…

The zeros of the size-$n$ partition functions for a statistical mechanical model can be used to help understand the critical behaviour of the model as $n\to\infty$. Here we use weighted Dyck paths as a simple model of two-dimensional…

Nakao's stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. Ito's formula in terms of the extended stochastic…

We study properties of $A^p_\alpha$ spaces in the Dirichlet range, recently defined by Brevig, Kulikov, Seip and Zlotnikov as the set of all holomorphic functions on the unit disc $\mathbb{D}$ such that \[ \int_{\mathbb{D}} |f(z)|^{p-2}…

In this note we present a new proof of the Carleson Embedding Theorem on the unit disc and unit ball. The only technical tool used in the proof of this fact is Green's formula. The starting point is that every Carleson measure gives rise to…

We introduce new functional spaces that generalize the weighted Bergman and Dirichlet spaces on the disk D(0,R) in the complex plane and the Bargmann-Fock spaces on the whole complex plane. We give a complete description of the considered…

We prove two compactness results for function spaces with finite Dirichlet energy of half-space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic…

B.\,Ya.\,Levin has proved that zero set of a sine type function can be presented as a union of a finite number of separated sets, that is an important result in the theory of exponential Riesz bases. In the present paper we extend Levin's…

We study the closure of the convex hull of a compact set in a complete CAT(0) space. First we give characterization results in terms of compact sets and the closure of their convex hulls for locally compact CAT(0) spaces that are either…

Inspired by a recent article on Fr\'echet spaces of ordinary Dirichlet series $\sum a_n n^{-s}$ due to J.~Bonet, we study topological and geometrical properties of certain scales of Fr\'echet spaces of general Dirichlet spaces $\sum a_n…

We study the asymptotics of zeros for entire functions of the form \sin z + \int_{-1}^1 f(t)e^{izt}dt with f belonging to a space X \hookrightarrow L_1(-1,1) possessing some minimal regularity properties.

We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods.

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,\alpha)$-summable to the function in the norm of the space, provided that $\alpha>1/2$. We…

Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function…

We improve existing explicit bounds of Vinogradov-Korobov type for zero-free regions of the Riemann zeta function, both for large height t and for every t. A primary input is an explicit bound of the author (Proc. London Math. Soc. 85…

We consider the Dirichlet problem for the Beltrami equation in some simply connected domain. We consider the class of all homeomorphic solutions of such a problem with a normalization condition and set-theoretic constraints on their complex…

For any $s \in \mathbb{C}$ with $\Re(s)>0$, denote by $\eta_{n-1}(s)$ the $(n-1)^{th}$ partial sum of the Dirichlet series for the eta function $\eta(s)=1-2^{-s}+3^{-s}-\cdots \;$, and by $R_n(s)$ the corresponding remainder. Denoting by…

In this paper, we prove a general theorem concerning the analyticity of the closure of a subspace defined by a family of variations of mixed Hodge structures, which includes the analyticity of the zero loci of degenerating normal functions.…