Isocapacity estimates for Hessian operators

Through a new powerful potential-theoretic analysis, this paper is devoted to discovering the geometrically equivalent isocapacity forms of Chou–Wang's Sobolev type inequality and Tian–Wang's Moser–Trudinger type inequality for the fully nonlinear 1≤k≤n/2$1\le k\le n/2$ Hessian operators.     

Hessian estimates for convex solutions to quadratic Hessian equation

We derive Hessian estimates for convex solutions to quadratic Hessian equation by compactness argument.     

Isoperimetric estimates in products

In a product M ₁ × M ₂ of Riemannian manifolds, the least perimeter required to enclose given volume among general regions is at least 1/√ 2 times that among regions of product form, assuming that the isoperimetric profiles of M ₁ and M ₂ are concave. This result sharpens earlier work of Grigor'yan, generalizes results of Bollobás and Leader and of Barthe, and yields a lower bound on the Cheeger isoperimetric constant of a product.     

Identities for eigenfunctions of ordinary differential operators

The paper contains formulae for regularized k-sums of residues of the Green's function for an ordinary differential operator with regular boundary conditions.Translated from Trudy Seminara imeni I. G. Petrovskogo, Vol. 10, pp. 107–117, 1984.The author would like to express his gratitude to Victor Antonovich Sadovnichii for his interest in this work.     

Essential spectrum and exponential estimates of eigenfunctions of lattice Schrödinger and Dirac operators

The main aim of the paper is the investigation of a relation between the essential spectrum and the exponential decay at infinity of eigenfunctions of the lattice analogs of Schrödinger and Dirac operators.     

Isoperimetric estimates in low-dimensional Riemannian products

Annals of Global Analysis and Geometry - Let $$(T^k,h_k)=(S_{r_1}^1\times S_{r_2}^1 \times \cdots \times S_{r_k}^1, dt_1^2+dt_2^2+\cdots +dt_k^2)$$ be flat tori, $$r_k\ge \cdots \ge r_2\ge...     

Expansions in generalized eigenfunctions of selfadjoint operators

Mathematische Zeitschrift -     

Anisotropic estimates for sub-elliptic operators

In the 1970's,Folland and Stein studied a family of subelliptic scalar operators L_λwhich arise naturally in the(?)_b-complex.They introduced weighted Sobolev spaces as the natural spaces for this complex,and then obtained sharp estimates for(?)b in these spaces using integral kernels and approximate inverses.In the 1990's,Rumin introduced a differential complex for compact contact manifolds,showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator,and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper,we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.     

Hardy-type estimates for Dirac operators

We prove some Hardy type inequalities related to the Dirac operator by elementary methods, for a large class of potentials, which even includes measure valued potentials. Optimality is achieved by the Coulomb potential. When potentials are smooth enough, our estimates provide some spectral information on the operator.     

An integral representation for eigenfunctions of linear ordinary differential operators

This paper generalizes an integral representation formula for eigenfunctions of Sturm-Liouville operators, known as the Volterra transformation operator in the theory of the inverse scattering problem, to higher-order differential operators. A specific fourth-order initial value problem is considered: $\text{Lφ = k}{}^4\text{φ, L =}\text{d}{}^4\text{dx}{}^4\text{+}\text{d}\text{dx}\text{(q}\text{d}\text{dx}\text{) + r}$φ(0) = 1, φ′(0) = 0, φ″(0) = ?k², φ? = 0 The solution for complex k is expressed as an inverse-Laplace-Borel transform. Jump formulae are obtained relating the representing kernel directly to the coefficients of L. The result admits obvious generalization to operators of arbitrary order.     

Computable bounds for eigenvalues and eigenfunctions of elliptic differential operators

Summary We are concerned with bounds for the error between given approximations and the exact eigenvalues and eigenfunctions of self-adjoint operators in Hilbert spaces. The case is included where the approximations of the eigenfunctions don't belong to the domain of definition of the operator. For the eigenvalue problem with symmetric elliptic differential operators these bounds cover the case where the trial functions don't satisfy the boundary conditions of the problem. The error bounds suggest a certain defectminization method for solving the eigenvalue problems. The method is applied to the membrane problem.     

Triebel-lizorkin space estimates for multilinear operators of sublinear operators

In this paper, we obtain the continuity for some multilinear operators related to certain non-convolution operators on the Triebel-Lizorkin space. The operators include Littlewood-Paley operator and Marcinkiewicz operator.     

On asymptotics of polynomial eigenfunctions for exactly solvable differential operators

In this paper we study the class of differential operators with polynomial coefficients Q_j in one complex variable satisfying the condition degQ_jj with equality for at least one j. We show that if degQ_k<k then the root with the largest modulus of the nth degree eigenpolynomial p_n of T tends to infinity when n→∞, as opposed to the case when degQ_k=k, which we have treated previously in [T. Bergkvist, H. RullgÅrd, On polynomial eigenfunctions for a class of differential operators, Math. Res. Lett. 9 (2002) 153–171]. Moreover, we present an explicit conjecture and partial results on the growth of the largest modulus of the roots of p_n. Based on this conjecture we deduce the algebraic equation satisfied by the Cauchy transform of the asymptotic root measure of the appropriately scaled eigenpolynomials, for which the union of all roots is conjecturally contained in a compact set.     

Weak first-order interior estimates and Hessian equations

Development of the modern theory of fully nonlinear, second-order partial differential equations has amazingly enriched the classical collection of ideas and methods. In this paper, we construct first-order interior a priori estimates of new type for solutions of Hessian equations. These estmates are applied to present the most transparent version of Krylov’s method and its tight connection with fully nonlinear equations. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 55–66.