Related papers: Three circles theorems for harmonic functions

We prove that subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some asymptotically small sets on spheres, are bounded from above everywhere. It follows that subharmonic functions of…

We prove an abstract result ensuring that one-sided geometric control yields two-sided estimates for functions satisfying general conditions. Our findings resonate in the context of nonlinear elliptic problems, including supersolutions to…

While discrete harmonic functions have been objects of interest for quite some time, this is not the case for discrete polyharmonic functions, as appear for instance in the asymptotics of path counting problems. In this article, a novel…

Raimi's classical theorem establishes a partition of the natural numbers with a remarkable unavoidability property: for every finite coloring of $\mathbb{N}$, there is a color class whose translate meets both parts of the partition in…

It is well-known that for any inner function $\theta$ defined in the unit disk $D$ the following two conditons: $(i)$ there exists a sequence of polynomials $\{p_n\}_n$ such that $\lim_{n \to \infty} \theta(z) p_n(z) = 1$ for all $z \in D$,…

The ring of cyclic quasi-symmetric functions and its non-Escher subring are introduced in this paper. A natural basis consists of fundamental cyclic quasi-symmetric functions; for the non-Escher subring they arise as toric $P$-partition…

Let $Z$ be a quadratic harmonic cone in $\mathbb{R}^{3}$. We consider the family $\mathcal{H}(Z)$ of all harmonic functions vanishing on $Z$. Is $\mathcal{H}(Z)$ finite or infinite dimensional? Some aspects of this question go back to as…

For an infinite penny graph, we study the finite-dimensional property for the space of harmonic functions, or ancient solutions of the heat equation, of polynomial growth. We prove the asymptotically sharp dimensional estimate for the above…

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set…

We characterize the set of positive harmonic functions with Dirichlet boundary conditions in unbounded domains which are union of several different chambers. We analyze the asymptotic behavior of the solutions in connection with the changes…

We prove an $\epsilon$-regularity theorem for vector-valued p-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere. This does not seem to follow from…

We consider upper bounds on the growth of $L^p$ norms of restrictions of eigenfunctions and quasimodes to geodesic segments in a nonpositively curved manifold in the high frequency limit. This sharpens results of Chen and Sogge as well as…

The Three Gap Theorem, also known as the Steinhaus Conjecture, is a classical result on the combinatorics of the fractional part function, and has since been generalized in many ways. In this paper, we pose a new problem related to these…

We study a certain truncation of the ring of arithmetical functions with unitary convolution, consisting of functions vanishing on arguments >n. The truncations are artinian monomial quotients of a polynomial ring in finitely many…

For a proper immersed minimal disk in $\bf{R}^N$ with quadratic area growth, we show that any harmonic function whose negative part grows at a slow sub-linear rate is constant. This leads to a higher codimensional Bernstein theorem for…

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl…

In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic function whose analytic part is convex, is univalent and close-to-convex. In this paper, certain cases are discussed under which the conclusion of this result can be…

Some years ago, the harmonic polynomial was introduced in order to understand better the harmonic topological index; for instance, it allows to obtain bounds of the harmonic index of the main products of graphs. Here, we obtain several…

The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The barrier classification of regular…

Let $u$ and $v$ be harmonic in $ \Omega \subset \mathbb{R}^n$ functions with the same zero set $Z$. We show that the ratio $f$ of such functions is always well-defined and is real analytic. Moreover it satisfies the maximum and minimum…