Related papers: Nikolskii inequality for lacunary spherical polyno…

The dependence of the smoothness of variational solutions to the first boundary value problems for second order elliptic operators are studied. The results use Sobolev-Slobodetskii and Nikolskii-Besov spaces and their properties. Methods…

Oberchkoff's inequality says that a polynomial of degree d with non-negative coefficients has at most 2ad/pi zeros in the angle {|arg z|<a}. We improve and generalize this inequality, and study the case of equality.

Let $\Pi_n$ be the class of algebraic polynomials $P$ of degree $n$, all of whose zeros lie on the segment $[-1,1]$. In 1995, S.P. Zhou has proved the following Tur\'{a}n type reverse Markov-Nikol'skii inequality: $\|P'\|_{L_p[-1,1]}>c\,…

This paper studies the inverse Steklov spectral problem for curvilinear polygons. For generic curvilinear polygons with angles less than $\pi$, we prove that the asymptotics of Steklov eigenvalues obtained in arXiv:1908.06455 determines, in…

A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the $h$-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial $d$-polytopes,…

It was proved that the fundamental group of the space of harmonic polynomials of degree $n(n \geq 2)$, with the same Gaussian curvature is not trivial. Furthermore, we give an example of topologically nonequivalent conjugate harmonic…

Let $\h_n$ be the $(2n+1)$-dimensional Heisenberg group. and let ${\cal L}_\alpha$ be the sublaplacian of the Lie algebra of $\h_n$ A new spherical harmonics with its orthogonal polynomial properties is presented for the group.

It is shown that if $p \ge 3$ and $u \in W^{1,p}(\Omega,\mathbb{R}^N)$ solves the inhomogenous $p$-Laplace system \[ \operatorname{div} (|\nabla u|^{p-2} \nabla u) = f, \qquad f \in W^{1,p'}(\Omega,\mathbb{R}^N), \] then locally the…

Nikolskii type inequalities for entire functions of exponential type on Rn for the Lorentz Zygmund spaces are obtained. Some new limiting cases are examined. Application to Besov type spaces of logarithmic smoothness is given.

In this article, we prove that for a broad class of second order elliptic PDEs, including the Laplacian, the zero sets of solutions to the Dirichlet problem are smooth for "generic" $L^2$ data. When the zero set of a solution (e.g. a…

In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff-Backelman-Pucci estimate corresponding to the full class $\cS^{\fL_0}$ of…

In this article, we consider the reciprocal antisymmetric polynomial \[P(z) = \sum_{j = 0}^{s}(-1)^j\gamma_j\left(z^j - z^{N + s + 1 - j}\right), \ \gamma_0 = 1.\] It is shown that if all the zeros of $P(z)$ are located on the unit circle,…

Assume $n\geq 2$. Consider the elementary symmetric polynomials $e_k(y_1,y_2,\ldots, y_n)$ and denote by $E_0,E_1,\ldots,E_{n-1}$ the elementary symmetric polynomials in reverse order \begin{align*}…

Let $d\geq 2$ be an integer, $S^d\subset {\mathbb R}^{d+1}$ the unit sphere and $\sigma$ a finite signed measure whose positive and negative parts are supported on $S^d$ with finite energy. In this paper, we derive an error estimate for the…

We obtain the following version of Lidskii theorem. Let L, M, N be p-dimensional subspaces in R^n. Let \psi_j be the angles between L and M, let \phi_j be the angles between M and N, and let \theta_j be the angles between L and N. Consider…

We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the $n$-sphere involving an operator of order $2s> n$. In this case the Sobolev exponent is negative. Our results…

In a previous work, both the constants of motion of a classical system and the symmetries of the corresponding quantum version have been computed with the help of factorizations. As their expressions were not polynomial, in this paper the…

The aim of this paper is to give an overview of some inequalities about $L^p$-norms ($p= 1$ or $p= 2$) of harmonic (periodic) and non-harmonic trigonometric polynomials. Among the material covered, we mention Ingham's Inequality about 2…

We consider the inhomogeneous biharmonic nonlinear Schr\"odinger equation $$ i u_t +\Delta^2 u+\lambda|x|^{-b}|u|^\alpha u = 0, $$ where $\lambda=\pm 1$ and $\alpha$, $b>0$. In the subctritical case, we improve the global well-posedness…

The primary objective of this paper is to establish several sharp versions of improved Bohr inequalities, refined Bohr inequalities, and Bohr-Rogosinski inequalities for the class of $K$-quasiconformal sense-preserving harmonic mappings…