相关论文: On extremal properties of Jacobian elliptic functi…
A random ensemble of cusp forms for the full modular group is introduced. For a weight-$k$ cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be $\asymp…
In dimensions $n\ge 2$ we obtain $L^{p_1}(\mathbb R^n) \times\dots\times L^{p_m}(\mathbb R^n)$ to $L^p(\mathbb R^n)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide…
We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function, in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the…
Given a bounded open connected Lipschitz set $\Omega \subset \mathbb R^2$, we show that the relaxed Cartesian area functional $\overline{\mathcal A}(u,\Omega)$ of a map $u\in W^{1,1}(\Omega;\mathbb S^1)$ is finite, and provide a useful…
The elliptic integral and its various generalizations are playing very important and basic role in different branches of modern mathematics. It is well known that they cannot be represented by the elementary transcendental functions.…
El Soufi-Ilias' theorem establishes a connection between minimal submanifolds of spheres and extremal metrics for eigenvalues of the Laplace-Beltrami operator. Recently, this connection was used to provide several explicit examples of…
We prove that for a homogeneous linear partial differential operator $\mathcal A$ of order $k \le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation…
The main purpose of this paper is to introduce and study the notion of plurifinely-maximal plurifinely plurisubharmonic functions, which extends the notion of maximal plurisubharmonic functions on a Euclidean domain to a plurifine domain of…
We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have…
We consider Bergman spaces and variations of them in one or several complex variables. For some domains we show that in these spaces the generic function is totally unbounded and hence non - extendable. We also show that the generic…
Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for the Macdonald function $K_{i\tau}(x)$. The results can be applied, for instance, to study the summability of the divergent…
The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among simplexes for the regular simplex (the regular tetrahedron, in three dimensions), maximal among parallelepipeds for the hypercube, and maximal among…
We show that for a positive proportion of Laplace eigenvalues $\lambda_j$ the associated Hecke-Maass $L$-functions $L(s,u_j)$ approximate with arbitrary precision any target function $f(s)$ on a closed disc with center in $3/4$ and radius…
Symmetric submodular functions are an important family of submodular functions capturing many interesting cases including cut functions of graphs and hypergraphs. Maximization of such functions subject to various constraints receives little…
The century old extremal problem, solved by Carath\'eodory and Fej\'er, concerns a nonnegative trigonometric polynomial normalized by a0 = 1, and the quantity to be maximized is the coefficient a1. In the complex exponential form, the…
We show how localization and smoothing techniques can be used to establish universality at the edge of the spectrum for a fixed positive measure on [-1,1]. Assume that the measure is a regular measure, and is absolutely continuous in some…
In 1962 J. Siciak introduced polynomial extremal function. The Siciak extremal function is one of the most important tool to investigate derivatives of polynomials. W. Ple\'sniak in the circle of papers (most important joint with W.…
We study the integrality properties of the coefficients of the mirror map attached to the generalized hypergeometric function $_{n}F_{n-1}$ with rational parameters and with a maximal unipotent monodromy. We present a conjecture on the…
The optimal $L^p \to L^q$ mapping properties for the (local) helical maximal function are obtained, except for endpoints. The proof relies on tools from multilinear harmonic analysis and, in particular, a localised version of the…
The problem of maximizing non-negative submodular functions has been studied extensively in the last few years. However, most papers consider submodular set functions. Recently, several advances have been made for the more general case of…