Commutators, Generalized Eigenfunction Expansions and Singular Integral Operators

Bilinear decompositions and commutators of singular integral operators

Let b be aBMO-function. It is well-known that the linear commutator [b, T ] of a Calderón-Zygmund operator T does not, in general, map continuously H(R) into L(R). However, Pérez showed that if H(R) is replaced by a suitable atomic subspace H b(R ) then the commutator is continuous from H b(R ) into L(R). In this paper, we find the largest subspace H b (R ) such that all commutators of Calderón...

Convergence of Generalized Eigenfunction Expansions

We present a simplified theory of generalized eigenfunction expansions for a commuting family of bounded operators and with finitely many unbounded operators. We also study the convergence of these expansions, giving an abstract type of uniform convergence result, and illustrate the theory by giving two examples: The Fourier transform on Hecke operators, and the Laplacian operators in hyperboli...

Hardy Spaces, Commutators of Singular Integral Operators Related to Schrödinger Operators and Applications

Let L = −∆+ V be a Schrödinger operator on R, d ≥ 3, where V is a nonnegative function, V 6= 0, and belongs to the reverse Hölder class RHd/2. The purpose of this paper is three-fold. First, we prove a version of the classical theorem of Jones and Journé on weak∗-convergence in H L(R ). Secondly, we give a bilinear decomposition for the product space H L(R )×BMOL(R). Finally, we study the commu...

Generalized Eigenfunction Expansions for Operator Algebras^)

A generalized eigenfunction expansion may be regarded as a representation for the spectral theorem by a transform technique. These representations have been presented in many forms, an early version of which was the von Neumann "direct integral" decomposition for a class of operator algebras [l9]. In 1953 [17], Mautner applied the von Neumann technique to the class of operators acting in an L2-...

Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators