Elements of functional calculus andL 2 regularity for some classes of fourier integral operators

We employ the method of slices to develop a rudimentary calculus describing the nature of operators T*T (respectively, TT*), where T are Fourier integral operators with one-sided right (respectively, left) singularities; this idea has its roots in the work of Greenleaf and Seeger. Such a result allows us to reduce the L² regularity problem for operators in n dimensions with one-sided singularities to the L² regularity problem for operators with two-sided singularities in n − 1 dimensions. As a consequence we deduce almost sharp L²-Sobolev estimates for operators in three-dimensions; an interesting special case is provided by certain restricted X-ray transforms associated to line complexes which are not well curved. We also provide a proof of almost-sharpness by looking at a restricted X-ray transform associated to the line complex generated by the curve t → (t, t^k). Appropriate notions of singularity, strong singularity, and type are also developed.     

Multiple fourier integral and multiple fourier series under square summation

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 1, pp. 39–52, January–February, 1990.     

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.     

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

     

On resolvents of integral and partially integral operators

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

Strongly omnipresent integral operators

An operatorT on the spaceH(G) of holomorphic functions on a domainG is strongly omnipresent whenever there is a residual set of functionsfH(G) such thatT f exhibits an extremely wild behaviour near the boundary. The concept of strong omnipresence was recently introduced by the first two authors. In this paper it is proved that a large class of integral operators including Volterra operators with or without a perturbation by differential operators has this property, completing earlier work about differential and antidifferential operators.The work of the first two authors has been partially supported by DGES grant PB96-1348 and the Junta de Andalucía.