Related papers: Green's function asymptotics and sharp point-wise …

Some results on the approximation of functions from the Sobolev spaces on metric graphs by step functions are obtained. The estimates are uniform with respect to all graphs of a given finite length, and the constant factors in the…

Sharp asymptotic for norms of Fourier projections on two-point homogeneous manifolds (the real sphere, the real, complex and quaternionic projective spaces and the Cayley elliptic plain) are established. These results extend sharp…

We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show…

A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in…

A rather complete investigation of anisotropic Bessel potential, Besov, and H\"older spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends'…

We prove certain $L^p$ Sobolev-type and Poincar\'e-type inequalities for functions on real and complex manifolds for the gradient operator $\nabla$, the Laplace operator $\Delta$, and the operator $\bar\partial$. Integral representations…

Asymptotically sharp Bernstein- and Markov-type inequalities are established for rational functions on $C^2$ smooth Jordan curves and arcs. The results are formulated in terms of the normal derivatives of certain Green's functions with…

In [Comm. Anal. Geom., 13(5):845-885, 2005.], Bartnik described the phase space for the Einstein equations, modelled on weighted Sobolev spaces with local regularity $(g,\pi)\in H^2\times H^1$. In particular, it was established that the…

We consider the Sobolev (Bessel potential) spaces H^ell(R^d, C), and their standard norms || ||_ell (with ell integer or noninteger). We are interested in the unknown sharp constant K_{ell m n d} in the inequality || f g ||_{ell} \leqs…

We develop a general method to calculate entropy numbers of standard Sobolev's classes on an arbitrary compact homogeneous Riemannian manifold. Our method is essentially based on a detailed study of geometric characteristics of norms…

In this note we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first such inequality is a variation on the classical Schwarz Lemma from complex analysis,…

In the first part of this paper, we prove local interior and boundary gradient estimates for p-harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an…

We establish P\'olya-Szeg\H{o}-type inequalities (PSIs) for Sobolev-functions defined on a regular $n$-dimensional submanifold $\Sigma$ (possibly with boundary) of a $(n+m)$-dimensional Euclidean space, under explicit upper bounds of the…

We estimate the linear isoperimetric constants of an n-dimensional ellipse. Using these estimates and a technique of Gromov, we estimate the Hopf and linking invariants of Lipschitz maps from ellipses to round spheres. Using these…

We prove Michael-Simon type Sobolev inequalities for $n$-dimensional submanifolds in $(n+m)$-dimensional Riemannian manifolds with nonnegative $k$-th intermediate Ricci curvature by using the Alexandrov-Bakelman-Pucci method. Here…

We investigate the non-existence and existence of positive solutions to biharmonic elliptic inequalities on manifolds. Using Green function and volume growth conditions, we establish the critical exponent for biharmonic problem.

Two sharp comparison results are derived for three-dimensional complete noncompact manifolds with scalar curvature bounded from below. The first one concerns the Green's function. When the scalar curvature is nonnegative, it states that the…

We obtain sharp estimates for heat kernels and Green's functions on complete noncompact Riemannian manifolds with Euclidean volume growth and nonnegative Ricci curvature. We will then apply these estimates to obtain sharp Moser-Trudinger…

We give a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence of and the asymptotic expansions for the extremals, best constants, various remainder terms, etc. Most attention is…

Analogous to Weil-Petersson quasicircles, we investigate infinite circle patterns in the Euclidean plane parameterized by discrete harmonic functions of finite Dirichlet energy. The space of such circle patterns forms an…