Two-sided Estimates Research Articles

Two-sided Lieb–Thirring bounds

We prove upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. As a corollary, we obtain two-sided estimates for the sum of the negative eigenvalues of atomic Hamiltonians with Kato potentials. Instead of being in terms of the potential itself, as in the usual Lieb–Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of (−Δ + V + M)uM = 1 in Rd; here M∈R is chosen so that the operator is positive. We further prove that the infimum of (uM−1−M) is a lower bound for the ground state energy E0 and derive a simple iteration scheme converging to E0.

On dihedral angle sums and number of facets for product polytopes

In this paper we present a method for computing the dihedral angle sums (and their two-sided estimates) of cartesian and skew product polytopes provided the sums of dihedral angles (or their estimates) are known for the factors. In addition, a formula for computing the number of facets of such product polytopes is derived. The method proposed is very universal and illustrated by several examples. The estimates are in a full agreement with known results for prisms and hexahedra.

Heat Kernel Estimates of Fractional Schrödinger Operators with Hardy Potential on Half-line

AbstractWe provide sharp two-sided estimates of the heat kernel of the Dirichlet fractional Laplacian on the half-line perturbed by a Hardy potential.

Two-sided Estimates for the Sum of Probabilities of Errors in the Multiple Hypothesis Testing Problem with Finite Number of Hypotheses on a Nonhomogeneous Sample

Two-sided Estimates for the Sum of Probabilities of Errors in the Multiple Hypothesis Testing Problem with Finite Number of Hypotheses on a Nonhomogeneous Sample

Two-Sided Estimates of Solutions with a Blow-Up Mode for a Nonlinear Heat Equation with a Quadratic Source

Two-Sided Estimates of Solutions with a Blow-Up Mode for a Nonlinear Heat Equation with a Quadratic Source

Two-sided heat kernel estimates for Schrödinger operators with unbounded potentials

Two-sided heat kernel estimates for Schrödinger operators with unbounded potentials

Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit

We study a d-dimensional non-symmetric strictly α-stable Lévy process X, whose spherical density is bounded and bounded away from the origin. First, we give sharp two-sided estimates on the transition density of X killed when leaving an arbitrary κ-fat set. We apply these results to get the existence of the Yaglom limit for arbitrary κ-fat cone. In the meantime we also obtain the spacial asymptotics of the survival probability at the vertex of the cone expressed by means of the Martin kernel for Γ and its homogeneity exponent. Our results hold for the dual process X̂, too.

Two-sided estimates of Lebesgue constants for spaces of functions of generalized bounded variation and Hölder spaces

A proof of two-sided estimates of Lebesgue constants for spaces Hω⋂BV(X), where Hω is the Hölder space and BV(X) is the space of functions of bounded variation constructed from the symmetric space of sequences X. Choosing different spaces of sequences as X, we obtained new criteria for the convergence of classical Fourier series.

On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations

UDC 517.9 We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid L 2 -norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order.

POLYNOMIALS LEAST DEVIATING FROM ZERO IN \(L^p(-1;1)\), \(0 \le p \le \infty \), WITH A CONSTRAINT ON THE LOCATION OF THEIR ROOTS

We study Chebyshev's problem on polynomials that deviate least from zero with respect to \(L^p\)-means on the interval \([-1;1]\) with a constraint on the location of roots of polynomials. More precisely, we consider the problem on the set \(\mathcal{P}_n(D_R)\) of polynomials of degree \(n\) that have unit leading coefficient and do not vanish in an open disk of radius \(R \ge 1\). An exact solution is obtained for the geometric mean (for \(p=0\)) for all \(R \ge 1\); and for \(0<p<\infty\) for all \(R \ge 1\) in the case of polynomials of even degree. For \(0<p<\infty\) and \(R\ge 1\), we obtain two-sided estimates of the value of the least deviation.

APPROXIMATION OF DIFFERENTIATION OPERATORS BY BOUNDED LINEAR OPERATORS IN LEBESGUE SPACES ON THE AXIS AND RELATED PROBLEMS IN THE SPACES OF \((p,q)\)-MULTIPLIERS AND THEIR PREDUAL SPACES

We consider a variant \(E_{n,k}(N;r,r;p,p)\) of the four-parameter Stechkin problem \(E_{n,k}(N;r,s;p,q)\) on the best approximation of differentiation operators of order \(k\) on the class of \(n\) times differentiable functions \((0<k<n)\) in Lebesgue spaces on the real axis. We discuss the state of research in this problem and related problems in the spaces of multipliers of Lebesgue spaces and their predual spaces. We give two-sided estimates for \(E_{n,k}(N;r,r;p,p)\). The paper is based on the author's talk at the S.B. Stechkin's International Workshop-Conference on Function Theory (Kyshtym, Chelyabinsk region, August 1–10, 2023).

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

Estimates for the Approximation and Eigenvalues of the Resolvent of a Class of Singular Operators of Parabolic Type

In this paper, we study a differential operator of parabolic type with a variable and unbounded coefficient, defined on an infinite strip. Sufficient conditions for the existence and compactness of the resolvent are established, and an estimate for the maximum regularity of solutions of the equation Lu=f∈L2(Ω) is obtained. Two-sided estimates for the distribution function of approximation numbers are obtained. As is known, estimates of approximation numbers show the rate of best approximation of the resolvent of an operator by finite-dimensional operators. The paper proves the assertion about the existence of positive eigenvalues among the eigenvalues of the given operator and finds two-sided estimates for them.

On global extrema of power Takagi functions

By construction, power Takagi functions Sp are similar to Takagi's continuous nowhere differentiable function described in 1903. These real-valued functions Sp(x) have one real parameter p>0 . They are defined on the real axis R by the series Sp(x)=∑∞n=0(S0(2nx)/2n)p , where S0(x) is the distance from real number x to the nearest integer number. We show that for every p>0 , the functions Sp are everywhere continuous, but nowhere differentiable on R . Next, we derive functional equations for Takagi power functions. With these, it is possible, in particular, to calculate the values Sp(x) at rational points x . In addition, for all values of the parameter p from the interval (0;1) , we find the global extrema of the functions Sp , as well as the points where they are reached. It turns out that the global maximum of Sp equals to 2p/(3p(2p−1)) and is reached only at points q+1/3 and q+2/3 , where q is an arbitrary integer. The global minimum of the functions Sp equals to 0 and is reached only at integer points. Using the results on global extremes, we obtain two-sided estimates for the functions Sp and find the points at which these estimates are reached.

Two-sided estimate of effective thermal conductivity coefficients of a textured composite with anisotropic ellipsoidal inclusions

Two-sided estimate of effective thermal conductivity coefficients of a textured composite with anisotropic ellipsoidal inclusions

A Posteriori Error Control of Approximate Solutions to Boundary Value Problems Found by Neural Networks

The paper discusses how to verify the quality of approximate solutions to partial differential equations constructed by deep neural networks. A posterior error estimates of the functional type, that have been developed for a wide range of boundary value problems, are used to solve this problem. It is shown, that they allow one to construct guaranteed two-sided estimates of global errors and get distribution of local errors over the domain. Results of numerical experiments are presented for elliptic boundary value problems. They show that the estimates provide much more reliable information on the quality of approximate solutions generated by networks than the loss function, which is used as a quality criterion in the Deep Galerkin method.

Stability of estimates for fundamental solutions under Feynman–Kac perturbations for symmetric Markov processes

In this paper, when a given symmetric Markov process X satisfies the stability of global heat kernel two-sided (upper) estimates by Markov perturbations (see Definition 1.2), we give a necessary and sufficient condition on the stability of global two-sided (upper) estimates for fundamental solution of Feynman–Kac semigroup of X. As a corollary, under the same assumptions, a weak type of global two-sided (upper) estimates holds for the fundamental solution of Feynman–Kac semigroup with (extended) Kato class conditions for measures. This generalizes all known results on the stability of global integral kernel estimates by symmetric Feynman–Kac perturbations with Kato class conditions in the framework of symmetric Markov processes.

The four-dimensional Yang–Mills partition function in the vicinity of the vacuum

The partition function of four-dimensional Euclidean, non-supersymmetric textrm{SU}(2) Yang–Mills theory is calculated in the perturbative and weak coupling regime, i.e., in a small open ball about the flat connection and when the gauge coupling constant acquires a small but finite value. The computation is based on various known inequalities, valid only in four dimensions, providing two-sided estimates for the exponentiated Yang–Mills action in terms of the L^2-norm of the derivative of the gauge potential only; these estimates then give rise to Gaußian-like infinite- dimensional formal integrals involving the Laplacian and hence can be computed via zeta-function and heat kernel techniques. It then turns out that these formal integrals give a sharp value for the partition function in the aforementioned perturbative and weak coupling regime of the theory. In the resulting expression for the partition function, the original classical value of the coupling constant is shifted to a smaller one which can be interpreted as the manifestation, in this approach, of a non-trivial beta -function and asymptotic freedom in pure non-Abelian gauge theories.

Brownian motion on the golden ratio Sierpinski gasket

We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without a finitely ramified cell structure, via a study on the trace of an electrical network on an infinite graph. The Dirichlet form is the unique one that is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. The proof is based on a fixed point problem of a renormalization map, inspired by Sabot's celebrated work for finitely ramified fractals. Lastly, the Hunt process associated with the Dirichlet form satisfies a two-sided sub-Gaussian heat kernel estimate.

Heat kernels for time-dependent non-symmetric mixed Lévy-type operators

In this paper we establish the existence, uniqueness and regularity of heat kernels to a large class of time-inhomogeneous non-symmetric nonlocal operators with Dini's continuous kernels. Moreover, quantitative estimates including two-sided estimates, gradient and fractional derivative estimates of the heat kernels are obtained.