Symmetric Functionals and Singular Traces

We study the construction and properties of positive linear functionals on symmetric spaces of measurable functions which are monotone with respect to submajorization. The construction of such functionals may be lifted to yield the existence of singular traces on certain non-commutative Marcinkiewicz spaces which generalize the notion of Dixmier trace.     

Singular symmetric functionals and stabilizing subspaces of Marcinkiewicz spaces

We consider Marcinkiewicz spaces of functions measurable on a semiaxis that admit a wide set of singular symmetric Dixmier functionals. For elements of these spaces we study the measurability property introduced by A. Connes. We establish that this property is closely connected with the Tauberian property (which is more strong) but is not reduced to it. We specify the maximal subspace of the Marcinkiewicz space such that for its elements both properties are equivalent. We prove that this subspace is not reducible to other known subspaces of the Marcinkiewicz space and that it plays an important role in the theory of Dixmier functionals.     

Sublinear operators on block-type spaces

This study establishes the boundedness of sublinear operators on block spaces built on Banach function spaces. These results are used to study the boundedness of the Marcinkiewicz integrals, singular integral operators and fractional integral operators with homogeneous kernels on block-type spaces.     

Disjointly strictly singular inclusions of symmetric spaces

In this paper, the disjoint strict singularity of inclusions of symmetric spaces of functions on an interval is considered. A condition for the presence of a “gap” between spaces sufficient for the inclusion of one of these spaces into the other to be disjointly strictly singular is found. The condition is stated in terms of fundamental functions of spaces and is exact in a certain sense. In parallel, necessary and sufficient conditions for an inclusion of Lorentz spaces to be disjointly strictly singular (and similar conditions for Marcinkiewicz spaces) are obtained and certain other assertions are proved. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 3–14, January, 1999.     

Strictly singular and strictly co-singular inclusions between symmetric sequence spaces

Strict singularity and strict co-singularity of inclusions between symmetric sequence spaces are studied. Suitable conditions are provided involving the associated fundamental functions. The special case of Lorentz and Marcinkiewicz spaces is characterized. It is also proved that if E?F are symmetric sequence spaces with E≠?₁ and F≠c₀ and ?_∞ then there exist a intermediate symmetric sequence space G such that E?G?F and both inclusions are not strictly singular. As a consequence new characterizations of the spaces c₀ and ?₁ inside the class of all symmetric sequence spaces are given.     

Correction theorem for Sobolev spaces constructed by a symmetric space

A sharp correction theorem is established for Sobolev spaces in which the norm (quasinorm) of generalized derivatives is calculated in an arbitrary symmetric space. The exact relation between the norm of a corrected function in the Lipschitz space and the measure of the set on which the corrected and original functions are different makes it possible to characterize the Sobolev spaces constructed on the basis of the Marcinkiewicz space in terms of correctability. This opens a way to constructing Sobolev-Marcinkiewicz spaces for functions with an arbitrary domain of definition.     

Some Rough Operators on Product Spaces

In this survey report, we shall mainly summarize some recent progress, interesting problems and typical methods used in the theory related to rough Marcinkiewicz integrals and rough singular integrals on product spaces. In addition, we give new proofs for some known results.     

Continuous linear functionals on certain function spaces satisfying a system of two singular second order differential equations

In this paper, we study a system of two singular second order differential equations, which arises from the theory of harmonic analysis on complex symmetric spaces. First of all, the distributional solutions on an neighborhood of zero in R² are determined. Next, some new function spaces are introduced and the system is solved in the duals of these new spaces.     

Some Rough Operators on Product Spaces

In this survey report, we shall mainly summarize some recent progress, interesting problems and typical methods used in the theory related to rough Marcinkiewicz integrals and rough singular integrals on product spaces. In addition, we give new proofs for some known results.     

Dixmier traces of operators on Banach and Hilbert spaces

We present a simple approach to the theory of traces on the Marcinkiewicz operator ideal Particular attention is paid to the Dixmier traces, which may be obtained from both dilation‐invariant functionals and shift‐invariant functionals on $\mathfrak l_\infty$. Since the concept of positivity does not make sense for traces defined on operator ideals over Banach spaces, traces are not assumed to be positive in the Hilbert space setting, as well. This standpoint raises some new problems. The present studies will be continued in 35 , 36 .     

Fully symmetric functionals on a Marcinkiewicz space are Dixmier traces

As a consequence of the exposition of Dixmier type traces in the book of A. Connes (1994) [2], we were led to ask how general is this class of functionals within the space of all unitarily invariant functionals on the corresponding Marcinkiewicz ideal Mψ. In this paper we prove the surprising result that the set of all Dixmier traces on Mψ coincides with the set of all fully symmetric functionals on this space.     

Norm Inequalities in Generalized Morrey Spaces

We prove that Calderón-Zygmund operators, Marcinkiewicz operators, maximal operators associated to Bochner-Riesz operators, operators with rough kernel as well as commutators associated to these operators which are known to be bounded on weighted Morrey spaces under appropriate conditions, are bounded on a wide family of function spaces.     

几类具有粗糙核的算子在加权Morrey空间上的有界性

设Ω∈L~q(S~(n-1))且1相似文献   

О ДИФФЕРЕНцИРОВАНИИ ИНтЕгРАлОВ От ФУНкцИ И Иж сИММЕтРИЧНых пРОс тРАНстВ ДИФФЕРЕНцИАльНыМИ Б АжИсАМИ

We consider new methods of constructing differential bases. Symmetric spaces with essentially different fundamental functions at zero can be defined by means of differential bases. Even the Lorentz and Marcinkiewicz or the Lebesgue and Marcinkiewicz spaces can be defined by means of differential bases.     

Multilinear integral operators and mean oscillation

In this paper, the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces are obtained. The operators include Calderón—Zygmund singular integral operator, fractional integral operator, Littlewood—Paley operator and Marcinkiewicz operator.     

Marcinkiewicz积分交换子在加权Morrey空间上的有界性

Marcinkiewicz积分是分析中的一类被广泛研究的重要算子.利用Marcinkiewicz积分算子μΩ与Lipschitz函数b生成的交换子μΩ,b在加权L~p空间上的有界性,研究了它在加权Morrey空间上的有界性.     

Tractability of Tensor Product Linear Operators

This paper deals with the worst case setting for approximating multivariate tensor product linear operators defined over Hilbert spaces. Approximations are obtained by using a number of linear functionals from a given class of information. We consider the three classes of information: the class of all linear functionals, the Fourier class of inner products with respect to given orthonormal elements, and the standard class of function values. We wish to determine which problems are tractable and which are strongly tractable. The complete analysis is provided for approximating operators of rank two or more. The problem of approximating linear functionals is fully analyzed in the first two classes of information. For the third class of standard information we show that the possibilities are very rich. We prove that tractability of linear functionals depends on the given space of functions. For some spaces all nontrivial normed linear functionals are intractable, whereas for other spaces all linear functionals are tractable. In “typical” function spaces, some linear functionals are tractable and some others are not.     

On the Charshiladze-Lozinski theorem for compact topological groups and homogeneous spaces

This paper is concerned with an extension of the Charshiladze-Lozinski theorem to compact (not necessarily abelian) topological groups G and symmetric compact homogeneous spaces G/H. The proof is based on a generalized Marcinkiewicz — Berman formula. As an application, some divergence theorems for expansions of continuous resp. integrable complex — valued functions on Euclidean spheres and projective spaces in series of polynomial functions on these spaces are established.