Class Of Integral Operators Research Articles

Regularity of Integral Operators

In this paper we consider a class of integral operators whose kernels satisfy certain generalizations of Hormander condition and establish their \( L^{p}\longrightarrow I_{\alpha}(L^{p}) \) regularity results.

Starlikeness of functions in the range of a class of integral operators

For β < 1, let 𝒫β denote the class of normalized analytic functions whose derivatives have real part greater than β. Recently, Fournier and Ruscheweyh [5] developed a method for finding the best (least) bound β such that a certain integral operator V λ maps the class 𝒫β into the familiar class 𝒮* of starlike functions in the unit disk. The main object of this paper is to show how their method can be made more readily available for several other classes of integral operators. *E-mail: jllimath@nwpu.edu.cn

Numerical approximation of entropy solutions for hyperbolic integro-differential equations

Scalar hyperbolic integro-differential equations arise as models for e.g. radiating or self-gravitating fluid flow. We present finite volume schemes on unstructured grids applied to the Cauchy problem for such equations. For a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem. For a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme. Under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h1/4|ln(h)| where h denotes the grid parameter. The convergence proofs rely on appropriate variants of the classical Kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator.

A Generalization of the Weighted Hardy Inequality for One Class of Integral Operators

We consider the problem of finding necessary and sufficient conditions for validity of an estimate for a function by some differential operation containing a weight function rather than the ordinary derivative. This operation is referred to as the ρ-weighted derivative.

Fixed points of an integral operator

A characterization of the functions fixed by a class of integral operators associated with hyperbolically-harmonic functions is given.

Estimates for the Approximation Numbers of One Class of Integral Operators. I

Asymptotic estimates are obtained for the approximation numbers of the Hardy integral operator with variable limits of integration.

On a Class of Hilbert-Schmidt Operators

We consider a class of integral operators with L 2 kernel which are often encountered in the resolution of boundary value problems and in scattering theory. We prove that, under certain conditions, these operators are contrac- tions; applications of this result to a nonlinear differential equation at resonance and to an integral equation of inverse scattering theory are discussed.

Convergence and Rate of Approximation in BVΦ for a Class of Integral Operators

We obtain estimates and convergence results with respect to ϕ-variation in spaces BVΦ for a class of linear integral operators whose kernels satisfy a general homogeneity condition. Rates of approximation are also obtained. As applications, we apply our general theory to the case of Mellin convolution operators, to that one of moment operators and finally to a class of operators of fractional order.

Riesz Summability of Spectral Expansions for a Class of Integral Operators

Riesz Summability of Spectral Expansions for a Class of Integral Operators

Generalizations of the Forelli–Rudin Estimates

We give elementary proofs of the Forelli–Rudin estimates and generalizations of these estimates. As an application we obtain Lp-boundedness results for classes of integral operators that include Bergman projections and Hankel operators.

Weighted inequality for some classical integral operators: 0 &lt; p &lt; 1

Weighted inequality for some classical integral operators: 0 &lt; p &lt; 1

Moser's second one-dimensional inequality for a class of integral operators

Suppose that \(1 < p < \infty \), \(q=p/(p-1)\), and for non-negative \(f\in L^p(-\infty\! ,\infty )\) and any real x we let \(F(x)-F(0)=\int _0^xf(t)\ dt\); suppose in addition that \(\int\limits _{-\infty }^\infty F(t)\exp (-|t|)\ dt=0\). Moser's second one-dimensional inequality states that there is a constant \(C_p\), such that \(\int\limits _{-\infty }^\infty \exp [a |F(x)|^q-|x|] \ dx\le C_p\) for each f with \(||f||_p\le 1\) and every \(a\le 1\). Moreover the value a = 1 is sharp. We replace the operation connecting f with F by a more general integral operation; specifically we consider non-negative kernels K(t,x) with the property that xK(t,x) is homogeneous of degree 0 in t, x. We state an analogue of the inequality above for this situation, discuss some applications and consider the sharpness of the constant which replaces a.

Multi-resolution approximation to the stochastic inverse problem

The linear inverse problem of estimating the input random field in a first-kind stochastic integral equation relating two random fields is considered. For a wide class of integral operators, which includes the positive rational functions of a self-adjoint elliptic differential operator on L2(ℝd), the ill-posed nature of the problem disappears when such operators are defined between appropriate fractional Sobolev spaces. In this paper, we exploit this fact to reconstruct the input random field from the orthogonal expansion (i.e. with uncorrelated coefficients) derived for the output random field in terms of wavelet bases, transformed by a linear operator factorizing the output covariance operator. More specifically, conditions under which the direct orthogonal expansion of the output random field coincides with the integral transformation of the orthogonal expansion derived for the input random field, in terms of an orthonormal wavelet basis, are studied.

Multi-resolution approximation to the stochastic inverse problem

The linear inverse problem of estimating the input random field in a first-kind stochastic integral equation relating two random fields is considered. For a wide class of integral operators, which includes the positive rational functions of a self-adjoint elliptic differential operator on L 2(ℝ d ), the ill-posed nature of the problem disappears when such operators are defined between appropriate fractional Sobolev spaces. In this paper, we exploit this fact to reconstruct the input random field from the orthogonal expansion (i.e. with uncorrelated coefficients) derived for the output random field in terms of wavelet bases, transformed by a linear operator factorizing the output covariance operator. More specifically, conditions under which the direct orthogonal expansion of the output random field coincides with the integral transformation of the orthogonal expansion derived for the input random field, in terms of an orthonormal wavelet basis, are studied.

Spectral boundary value problems for the Helmholtz equation with spectral parameter in boundary conditions on a non-smooth surface

The spectral properties of four problems for the Helmholtz equation with spectral parameter in boundary or transmission conditions on a closed Lipschitz surface S are studied. These problems are related to the classical integral operators of potential type on S for the Helmholtz equation. They have been studied before in the case when S is infinitely smooth. It is shown that the most important properties of eigenvalues and root functions hold also for Lipschitz surfaces S. The machinery of potential theory in Lipschitz domains and of spectral theory is used in the proofs.

Monogenic hardy spaces on lipschitz domains and compensated compactness

In this paper we establish a boundary behavior theory for H p spaces of Clifford valued functions monogenic in a Lipschitz domain in the case ρ>1 The action of a natural class of integral operators with monogenic kernels is studied on these spaces and several applications to PDE's in non-smooth domains are presented.

One-dimensional Hardy-type inequalities in many dimensions

Weighted inequalities for certain Hardy-type averaging operators in are shown to be equivalent to weighted inequalities for one-dimensional operators. Known results for the one-dimensional operators are applied to give weight characterisations, with best constants in some cases, in the higher-dimensional setting. Operators considered include averages over all dilations of very general starshaped regions as well as averages over all balls touching the origin. As a consequence, simple weight conditions are given which imply weighted norm inequalities for a class of integral operators with monotone kernels.

Fredholm determinants and the mKdV/Sinh-Gordon hierarchies

For a particular class of integral operators $K$ we show that the quantity \[\ph:=\log \det (I+K)-\log \det (I-K)\] satisfies both the integrated mKdV hierarchy and the sinh-Gordon hierarchy. This proves a conjecture of Zamolodchikov.

On the Singular Numbers of Certain Volterra Integral Operators

A criterion for a certain class of integral operators to belong to Schatten-von Neumann symmetric normed ideals is given. In particular, when 2 ≤ p < ∞, it is shown that the Schatten p-norm of such an operator can be estimated by constant multiples of an integral expression which is a natural extension of the well-known formula for the Hilbert-Schmidt norm.

The starlikeness of a certain class of integral operators

The object of the present paper is to investigate the starlikeness of certain integral operators which are defined for analytic functions. The main theorem proved here provides a generalization of a recent result due to Y. C. Kim, K. S. Lee, and H. M. Srivastava [2] Some other interesting consequences of the main result are also considered.