Haar Shifts, Commutators, and Hankel Operators

Hankel operators lie at the junction of analytic and real-variables. We will explore this junction, from the point of view of Haar shifts and commutators. 1. Haar Functions We consider operators which satisfy invariance properties with respect to two well-known groups. The first group we take to the translation operators (1.1) Try f (x) := f (x − y) , y ∈ R . Note that formally, the adjoint operator is (Try) = Tr−y. The collection of operators {Try : y ∈ R} is a representation of the additive group (R,+). It is an important, and very general principle that a linear operator L acting on some vector space of functions, which is assumed to commute with all translation operators, is in fact given as convolution, in general with respect to a measure or distribution, thus, L f (x) = ∫ f (x − y) μ(dy) . For instance, with the identity operator, μ would be the Dirac pointmass at the origin. The second group is the set of dilations on Lp, given by (1.2) Dil λ f (x) := λ −1/p f (x/λ) , 0 < λ, p < ∞ . Here, we make the definition so that ‖ f ‖p = ‖Dil λ f ‖p. The scale of the dilation Dil (p) λ is said to be λ, and these operators are a representation of the multiplicative group (R+, ∗). The Haar measure of of this group is dy/y. Underlying this subject are the delicate interplay between local averages and differences. Some of this interplay can be encoded into the combinatorics of grids, especially the dyadic grid, defined to be D := {2( j, j + 1) : j, k ∈ Z}. The Haar functions are a remarkable class of functions indexed by the dyadic grid D. Set h(x) = −1(−1/2,0) + 1(0,1/2) , 2000 Mathematics Subject Classification. Primary: 47B35, 42B20 Secondary: 47B47,