Related papers: On the complex \L ojasiewicz inequality with param…

We prove a generalization of Dunham Jackson's famous approximation inequality to the case of compact sets in the complex plane admitting both upper and lower bounds for their Green's functions, i.e. the well known Holder Continuity Property…

We give a strong version of a classic inequality of \L ojasiewicz; one which collapses to the usual inequality in the complex analytic case. We show that this inequality for a pair, quadruple, or octuple of real analytic functions allows us…

Let $F(x) := (f_{ij}(x))_{i=1,\ldots,p; j=1,\ldots,q},$ be a ($p\times q$)-real polynomial matrix and let $f(x)$ be the smallest singular value function of $F(x).$ In this paper, we first give the following {\em nonsmooth} version of \L…

In this paper, we give a short proof of a theorem by Koll\'{a}r on hereditarily rational functions. This is an answer to his appeal to find an elementary proof which does not rely so much on resolution of singularities. Our approach does…

In this note, we study the behaviour of the Lojasiewicz exponent under hyperplane sections and its relation to the order of tangency.

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay…

We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and…

In this article we proved an interesting property of the class of continuous convex functions. This leads to the form of pre-Hermite-Hadamard inequality which in turn admits a generalization of the famous Hermite-Hadamard inequality. Some…

In this paper, we study polar quotients and \L ojasiewicz exponents of plane curve singularities, which are {\em not necessarily reduced}. We first show that the polar quotients is a topological invariant. We next prove that the \L…

We study asymptotics of integrals of certain rational functions that depend on parameters in a field $K$ of characteristic zero. We use formal power series to represent the integral and prove certain identities about its coefficients…

We consider the most general class of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of an arbitrary order whose solutions and right-hand sides belong to appropriate Sobolev spaces. For…

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable…

We establish a unique continuation property for solutions of the differential inequality $|\nabla u|\leq V|u|$, where $V$ is locally $L^n$ integrable on a domain in $\mathbb R^n$. A stronger uniqueness result is obtained if in addition the…

\noindent Let $I$ be an ideal of the ring of formal power series $\bK[[x,y]]$ with coefficients in an algebraically closed field $\bK$ of arbitrary characteristic. Let $\Phi$ denote the set of all parametrizations…

For displacement convex functionals in the probability space equip\-ped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type \L oja\-sie\-wicz inequalities. \chg{We also discuss the more…

We consider an abstract space of measurable linear cocycles and we assume the availability in this space of some appropriate uniform large deviation type estimates. Under these hypotheses we establish the continuity of the Oseledets…

For a continuous function, we prove that the function is pluriharmonic if and only if the equality part of the optimal Ohsawa--Takegoshi $L^2$-extension theorem is satisfied with respect to the metric having the function as a weight. This…

We investigate several possibilities of obtaining a {\L}ojasiewicz inequality for definable multifunctions and give some examples of applications thereof. In particular, we prove that the Hausdorff distance and its extension to closed sets…

For graphs with non-negative Ollivier curvature, we prove the Liouville property, i.e., every bounded harmonic function is constant. Moreover, we improve Ollivier's results on concentration of the measure under positive Ollivier curvature.

The article is devoted to the study of mappings that distort the modulus of families of paths by the Poletsky inequality type. At boundary points of a domain, we have obtained the H\"{o}lder inequality for such mappings, provided that their…