Commutators for the maximal and sharp functions

The class of functions for which the commutator with the Hardy-Littlewood maximal function or the maximal sharp function are bounded on are characterized and proved to be the same.

     

Commutators of functions of operators in perturbation theory

We establish the estimate

Commutators of maximal functions on spaces of homogeneous type and their weighted,local versions

We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows that the local version of commutators also has an independent interest.     

Factorization of a class of symbols with outer functions

In this paper the Wiener–Hopf (or Riemann–Hilbert) factorization of a class of symbols important in applications is studied. The symbols in this class involve outer functions that appear in applications such as diffraction by strip gratings and infinite-dimensional integrable systems. The method proposed is based on the reduction of a vector Riemann–Hilbert to a scalar problem on an appropriate Riemann surface. Two examples are given leading to the Riemann sphere and to an elliptic curve.     

Commutators of weighted lipschitz functions and multilinear singular integrals with non-smooth kernels

This paper is devoted to studying the commutators of the multilinear singular integral operators with the non-smooth kernels and the weighted Lipschitz functions. Some mapping properties for two types of commutators on the weighted Lebesgue spaces, which extend and generalize some previous results, are obtained.     

Generating functions of even Dedekind symbols with polynomial reciprocity laws

We will obtain generating functions of even Dedekind symbols with polynomial reciprocity laws. The generating functions are expressed in terms of Kronecker’s double series. We also establish reciprocity laws satisfied by these generating functions. As an application, we demonstrate new Eisenstein series identities which involve the derivatives of the first and second order.     

On convergence determining and separating classes of functions

Herein, we generalize and extend some standard results on the separation and convergence of probability measures. We use homeomorphism-based methods and work on incomplete metric spaces, Skorokhod spaces, Lusin spaces or general topological spaces. Our contributions are twofold: we dramatically simplify the proofs of several basic results in weak convergence theory and, concurrently, extend these results to apply more immediately in a number of settings, including on Lusin spaces.     

Commutators and compressions

We present a short proof of the known theorem that every operator, not of the formλI + compact, whereλ ≢ 0, is a commutator. The second author gratefully acknowledges the support of the National Science Foundation.     

Vertical symbols,Toeplitz operators on weighted Bergman spaces over the upper half-plane and very slowly oscillating functions

Extending a recent result of Herrera Yañez, Maximenko and Vasilevski, we provide a further step in the structural analysis of algebras generated by Toeplitz operators on weighted Bergman spaces over the upper half-plane. We show that the set of “spectral” functions corresponding to Toeplitz operators generated by bounded vertical symbols is dense in the C^?$C^{?}$-algebra of very slowly oscillating functions on the positive half-line.     

Operator method for calculating Q symbols and their relation to weyl-wigner symbols and symplectic tomogram symbols

We propose a new method for calculating Husimi symbols of operators. In contrast to the standard method, it does not require using the anti-normal-ordering procedure. According to this method, the coordinate and momentum operators \(\hat q\) and \(\hat p\) are assigned other operators \(\hat X\) and \(\hat P\) satisfying the same commutation relations. We then find the result of acting with the \(\hat X\) and \(\hat P\) operators and also polynomials in these operators on the Husimi function. After the obtained expression is integrated over the phase space coordinates, the integrand becomes a Husimi function times the symbol of the operator chosen to act on that function. We explicitly evaluate the Husimi symbols for operators that are powers of \(\hat X\) or \(\hat P\) .     

Commutators and accretive operators

In this paper, singular values of commutators of Hilbert space operators are estimated. To this aim the accretivity of a transform of the operators is applied. Some recent results of Kittaneh [F. Kittaneh, Singular value inequalities for commutators of Hilbert space operators, Linear Algebra Appl. 430 (2009) 2362-2367] are extended.     

Writing Commutators of Commutators as Products of Cubes in Groups

We study the cube length of certain elements of the derived subgroup of a group G. By the cube length Cu(γ) of an element γ of a group G, we mean the least natural number k such that γ is a products of k cubes. We find an upper bound for the cube length of a commutator of commutators. If W = F?C _∞ is the wreath product of a free group F by the infinite cyclic group, we show that every element of W″ is a product of at most three cubes in W.     

Commutators on Half-Spaces

We study the boundedness and compactness of commutators on , where and are defined by and respectively. If satisfies some upper and lower estimates, then we obtain a necessary and sufficient condition for to be bounded or compact on for . The reproducing kernel of the harmonic Bergman space of can be shown to satisfy all the required estimates. Our result is the real variable analogue of the complex variable one for commutators associated with an analytic reproducing kernel.