Related papers: Sobolev extremal polynomials with respect to mutua…

For finite-dimensional linear semigroups which leave a proper cone invariant it is shown that irreducibility with respect to the cone implies the existence of an extremal norm. In case the cone is simplicial a similar statement applies to…

We prove the non-degeneracy of the extremals of the Sobolev inequality $$\int\limits_{\mathbb R^N}|\nabla u|^pdx\ge \mathcal S_p\int\limits_{\mathbb R^N}|u|^{Np\over N-p}dx,\ u\in \mathcal D^{1,p}(\mathbb R^N)$$ when $1<p<N,$ as solutions…

We develop a theory of Sobolev orthogonal polynomials on the Sierpi\'nski gasket ($SG$). These orthogonal polynomials arise through the Gram-Schmidt orthogonalisation process applied on the set of monomials on $SG$ using several notions of…

We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollazek Polynomials as an example. The…

This paper is devoted to Markov's extremal problems of the form $M_{n,k}=\sup_{p\in\PP_n\setminus\{0\}}{{\|p^{(k)}\|}_X}/{{\|p\|}_X}$ $(1\le k\le n)$, where $\PP_n$ is the set of all algebraic polynomials of degree at most $n$ and $X$ is a…

The extremes of a stationary time series typically occur in clusters. A primary measure for this phenomenon is the extremal index, representing the reciprocal of the expected cluster size. Both a disjoint and a sliding blocks estimator for…

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class $A_p$…

In this paper, by utilizing a newly established variational principle on convex sets, we provide an existence and multiplicity result for a class of semilinear elliptic problems defined on the whole $\mathbb R^N$ with nonlinearities…

We study elliptic equations of order $2m$ with nonlocal boundary-value conditions in plane angles and in bounded domains, dealing with the case where the support of nonlocal terms intersects the boundary. We establish necessary and…

Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$ supporting a weak local $(1,p)$-Poincar\'e inequality. For each $\theta \in [0,p)$, we…

We consider the non-linear eigenvalue equations characterizing $L^p$ into $L^q$ Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the…

We initiate the study of fine $p$-(super)minimizers, associated with $p$-harmonic functions, on finely open sets in metric spaces, where $1 < p < \infty$. After having developed their basic theory, we obtain the $p$-fine continuity of the…

We study the perturbation by a critical term and a $(p-1)$-superlinear subcritical nonlinearity of a quasilinear elliptic equation containing a singular potential. By means of variational arguments and a version of the…

We prove new multiplicity results for some nonlocal critical growth elliptic equations in homogeneous fractional Sobolev spaces. The proofs are based on an abstract critical point theorem based on the ${\mathbb Z}_2$-cohomological index and…

In this paper we establish improved Sobolev inequalities on the quaternionic sphere under higher-order moment vanishing conditions with respect to the measure \(|u|^{p^*}\,d\xi\). As an application, we give a new proof of the existence of…

We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential…

Each extreme edge of the Newton diagram of a plurisubharmonic polynomial on $\mathbb{C}^2$ gives rise to a plurisubharmonic polynomial. It is tempting to believe that the union of the extreme edges or the convex hull of said union will do…

We study the supremum of some random Dirichlet polynomials and obtain sharp upper and lower bounds for supremum expectation that extend the optimal estimate of Hal\'asz-Queff\'elec and enable to cunstruct random polynomials with unusually…

We prove the Lp,q-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the…

In this paper we consider sequences of polynomials orthogonal with respect to certain discrete Laguerre-Sobolev inner product, with two perturbations (involving derivatives) located inside the oscillatory region for the classical Laguerre…