Integral operators with homogeneous kernels in grand Lebesgue spaces

Sufficient conditions on the kernel and the grandizer that ensure the boundedness of integral operators with homogeneous kernels in grand Lebesgue spaces on ? ⁿ as well as an upper bound for their norms are obtained. For some classes of grandizers, necessary conditions and lower bounds for the norm of these operators are also obtained. In the case of a radial kernel, stronger estimates are established in terms of one-dimensional grand norms of spherical means of the function. A sufficient condition for the boundedness of the operator with homogeneous kernel in classical Lebesgue spaces with arbitrary radial weight is obtained. As an application, boundedness in grand spaces of the one-dimensional operator of fractional Riemann–Liouville integration and of a multidimensional Hilbert-type operator is studied.     

An Integral Formula for Weighted Bergman Reproducing Kernels

An integral formula is obtained for reproducing kernels in weighted Bergman spaces with radial and logarithmically subharmonic weights in the unit disk. We deduce from it that these reproducing kernels have a special structure leading to the contractive divisor property of extermal functions.     

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

We obtain estimates and convergence results with respect to ?-variation in spaces BV_Φ 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.     

Weighted norm inequalities for L'-valued integral operators and applications

In this paper we prove weighted norm estimates for vector valued integral operators with positive kernels. In addition weighted norm inequalities for certain general vector valued singular integral operators are obtained. Applications of these results include a generalized Sobolev Theorem for Lizorkin-Triebel spaces and estimates of various Littlewood-Paley operators.     

变量核分数次积分在Hardy空间的有界性

§ 1　 Introduction and main resultsLet Sn- 1 be the unitsphere in Rn(n≥ 2 ) equipped with normalized Lebesgue measure dσ= dσ(z′) .We say that a functionΩ(x,z) defined on Rn× Rnbelongs to L∞ (Rn)× Lr(Sn- 1 )(r≥ 1 ) ,ifΩ(x,z) satisfies the following two conditions,(i) for any x,z∈Rnandλ>0 ,there hasΩ(x,λz) =Ω(x,z) ;(ii)‖Ω‖L∞(Rn)× Lr(Sn- 1) :=supx∈ Rn∫Sn- 1|Ω(x,z′) | rdσ(z′) 1 / r<∞ .For 0 <α相似文献   

Hardy type estimates for commutators of singular integrals

In this paper we study the Hardy type estimates for commutators Tb of standard Calderon-Zygmund singular integral operators T with a Lipschitz function b. The corresponding results are also obtained on the commutators Sb generated by b with singular integral operators S with variable kernels.     

On Compactness of Regular Integral Operators in the Space <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>

In this paper we obtain a sufficient condition for quite continuity of Fredholm type integral operators in the space L₁(a, b). Uniform approximations by operators with degenerate kernels of horizontally striped structures are constructed. A quantitative error estimate is obtained. We point out the possibility of application of the obtained results to second kind integral equations, including convolution equations on a finite interval, equations with polar kernels, one-dimensional equations with potential type kernels, and some transport equations in non-homogeneous layers.     

Multilinear commutators of BMO functions and multilinear singular integrals with non-smooth kernels

This paper is concerned with certain multilinear commutators of BMO functions and multilinear singular integral operators with non-smooth kernels. By the sharp maximal functions estimates, the weighted norm inequalities for this kind of commutators are established.     

L p bounds for singular integrals with rough kernels on product domains

This paper is concerned with singular integral operators on product domains with rough kernels both along radial direction and on spherical surface.Some rather weaker size conditions,which imply the Lp-boundedness of such operators for certain fixed p(1 p ∞),are given.     

On the algebra of multidimensional integral operators with homogeneous SO(n)-invariant kernels and weakly radially oscillating coefficients

In this paper, we study the Banach algebra B generated by multidimensional integral operators whose kernels are homogeneous functions of degree (?n) invariant with respect to the rotation group SO(n) and by the operators of multiplication by radial weakly oscillating functions. A symbolic calculus is developed for the algebra 25. The Fredholm property and the formula for calculating the index are described in terms of this calculus.     

Extrapolation results in grand Lebesgue spaces defined on product sets

Extrapolation results in weighted grand Lebesgue spaces defined with respect to product measure \(\mu \times \nu \) on \(X\times Y\), where \((X, d, \mu )\) and \((Y, \rho , \nu )\) are spaces of homogeneous type, are obtained. As applications of the derived results we prove new one-weight estimates for multiple integral operators such as strong maximal, Calderón–Zygmund and fractional integral operators with product kernels in these spaces.     

Multilinear singular integral operators with generalized kernels and their multilinear commutators

In this paper, the authors study a class of multilinear singular integral operators with generalized kernels and their multilinear commutators with BMO functions. By establishing the sharp maximal estimates, the boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimate of this class of mutilinear singular integral operators is also established. These results can improve the corresponding known results of classical multilinear Calderón–Zygmund operators and multilinear Calder′on–Zygmund operators with Dini type kernels.     

Sharp inequalities of singular values of smooth kernels

New inequalities of singular values of the integral operators with smoothL ² kernels are obtained and shown by examples to be sharp if the kernels satisfy also certain boundary conditions. These results are based on an idea of Gohberg-Krein by which the singular values of the integral operators are interrelated to the eigenvalues of some two point boundary value problems.Dedicate to Professor Ky Fan on the occasion of his 85th birthday     

λ-central BMO estimates for commutators of singular integral operators with rough kernels

The authors establish λ-central BMO estimates for commutators of singular integral operators with rough kernels on central Morrey spaces. Moreover, the boundedness of a class of multisublinear operators on the product of central Morrey spaces is discussed. As its special cases, the corresponding results of multilinear Calderon-Zygmund operators and multilinear fractional integral operators can be deduced, resDectivelv.     

粗糙核抛物型奇异积分算子与外插——献给陆善镇教授75华诞

本文研究粗糙核抛物型奇异积分算子及其极大算子.借助精细的Fourier变换估计和LittlewoodPaley理论,并结合外插讨论,在积分核满足球面Hardy函数条件和相当弱的径向尺寸条件下,本文建立这些算子的L^p有界性.进一步,关于沿一般光滑曲面的奇异积分算子及其极大算子的相应结果也被建立.这些结果即使在迷向情形也是新的.     

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

We obtain estimates and convergence results with respect to -variation in spaces BV 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.     

Integral equations of the third kind with unbounded operators

We consider linear functional equations of the third kind in L ₂ with arbitrary measurable coefficients and unbounded integral operators with kernels satisfying broad conditions. We propose methods for reducing these equations by linear continuous invertible transformations either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. To the integral equations obtained after the reduction, one can apply various exact and approximate methods of solution; in particular, the two approximate methods developed in this article.     

Sharp function boundedness for vector-valued multilinear singular integral operators with non-smooth kernels

Sharp function inequalities for several vector-valued, multilinear singular integral operators with non-smooth kernels are obtained. As an application, some weighted L ^p (p > 1) norm inequalities for vector-valued multilinear operators are derived.     

Hankel integral transforms of the first kind for piecewise-homogeneous segments with application to problems of mathematical physics

We construct the operators of direct and inverse finite Hankel integral transforms of the first kind for piecewise-homogeneous segments. The kernels of these operators are derived for the most common practical cases. The application of these integral transforms to solving problems of mathematical physics in piecewise-homogeneous domains is demonstrated for the problem of determining the structure of a stationary temperature field in a nonhomogeneous orthotropic multilayer (in the radial variable) cylinder.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 24–34, 1988.     

Topological methods in solvability theory of multidimensional pair integral operators with homogeneous kernels of compact type

The problem of getting effective Fredholm conditions for operators with bihomogeneous kernels reduces to the question of invertibility for families of operators with homogeneous kernels and to the calculation of homotopy invariants for spaces of Fredholm and invertible operators of that type. The purpose of the present paper is to study integral operators with homogeneous kernels of compact type in L _p (? ⁿ ), 1 < p < +??. The classes of homotopy equivalence for the spaces of Fredholm and invertible operators in the C*-algebra of pair operators with homogeneous kernels of compact type are calculated by means of operator K-theory.