d-dimensional orthogonal polynomials: Commutator theorem and other results

We give new and simplified proofs of three basic theorems in the theory of orthogonal polynomials associated to a classical, [Formula: see text]-valued random variable [Formula: see text] with all moments, namely: (1) The characterization of [Formula: see text] in terms of commutators among the creation–annihilation–preservation (CAP) operators in its quantum decomposition. (2) The characterization, in terms of the same objects, of the fact that the distribution of [Formula: see text] is a product measure. (3) The equivalence of the symmetry of [Formula: see text] with the vanishing of the associated preservation operator. Our new formulation of these results allows one to obtain a stronger form of the above statements.

Essential Self-Adjointness of Anticommutative Operators

The self-adjoint extensions of symmetric operators satisfying anticommutation relations are considered. It is proven that an anticommutative type of the Glimm-Jaffe-Nelson commutator theorem follows. Its application to an abstract Dirac operator is also considered.

Commutators in real interpolation with quasi-power parameters

The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi-power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the real interpolation methods.

A commutator theorem for fractional integrals in spaces of homogeneous type

We give a new proof of a commutator theorem for fractional integrals in spaces of homogeneous type.