On resolvents of integral and partially integral operators

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 2, pp. 88–91, March–April, 1993.     

Essential and discrete spectra of partially integral operators

Let Ω₁, Ω₂ ⊂ ℝ^ν be compact sets. In the Hilbert space L ₂(Ω₁ × Ω₂), we study the spectral properties of selfadjoint partially integral operators T ₁, T ₂, and T ₁ + T ₂, with

Stieltjes integral equations in partially ordered sets

Stieltjes integral equations are considered in partially ordered sets of real valued functions, and connections are made using integral inequalities to the existence and uniqueness problems of hereditary systems which are not Lipschitz.     

Ideals of quasi-ordered sets

We define the -ideals of a poset – or equally of a quasi-ordered set – for various collections of subsets and corresponding -ideal continuity for functions. This leads us to a choice-free -ideal continuous imbedding of a poset into a -join complete poset with an appropriate universal mapping property. Topological applications include the imbedding of Scott spaces and Alexandrov spaces into up-complete Scott spaces. Received May 26, 1998; accepted in final form June 28, 2001.     

Maximal bilinear singular integral operators associated with dilations of planar sets

We obtain square function estimates and bounds for maximal singular integral operators associated with bilinear multipliers given by characteristic functions of dyadic dilations of certain planar sets. As a consequence, we deduce pointwise almost everywhere convergence for lacunary partial sums of bilinear Fourier series with respect to methods of summation determined by the corresponding planar sets.     

Fourier integral operators, fractal sets, and the regular value theorem

We prove that if E⊂R2d, for d?2, is an Ahlfors–David regular product set of sufficiently large Hausdorff dimension, denoted by dimH(E), and ? is a sufficiently regular function, then the upper Minkowski dimension of the set does not exceed dimH(E)−m, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier integral operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are, in general, sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry.     

Homogeneous operators and homogeneous integral operators

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure—measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples—classes of homogeneous integral operators on various metric spaces—and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article—various specific cases—illustrate general statements and results given in the paper and at the same time are of interest in their own way.     

Polysingular integral operators

Summary We study a class of linear singular integral operators of the Cauchy type on the n-torus; an application is given to a boundary value problem for functions of several complex variables.     

Strong integral operators

It is shown that a strong integral (strong B-integral) operator in L₂ is a Hilbert-Schmidt (nuclear) operator.Translated from Matematicheskie Zametki, Vol. 16, No. 6, pp. 907–912, December, 1974.     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.     

Maximal operators and singular integral operators along submanifolds

In this paper we prove, for certain values of p, the L^p boundedness of the maximal operator