A Weak–type Orthogonality Principle

We are interested in the relationships between three different concepts, first and foremost is that of the phase space, by which we generally mean the Euclidean space formed from the cross product of the spatial variable with the dual frequency variable. Next, we want to associate subsets of that space with functions, the subset describing the location of the function in natural ways. And finally, we want to understand the extent to which orthogonality of the functions can be quantified by geometric conditions on the corresponding sets in the phase plane. These concerns are not currently very much in the forefront of harmonic analysis, but rather the means towards an end. We treat them as a subject in their own right because the inequalities that we obtain are of an optimal nature and they refine the basic orthogonality issues in the proof of the bilinear Hilbert transform inequalities [4], and complement investigations into “best basis” signal or image processing [7], including the directional issues that arise in the context of brushlets [5]. We state the principal results, and then turn to complementary issues and discussion. For a set W ⊂ R of finite volume we define λW := {c(W ) + λ(x− c(W )) | x ∈ W} where c(W ) is the center of mass of W . We say that W is symmetric if −(W − c(W )) = W − c(W ). We call the product s = Ws × Ωs of a symmetric convex set Ws and a second set a tile. Here, we need not assume that the second set lies in R, it could lie in some other set altogether. We use tiles to study the connection between geometry of the phase plane and orthogonality which is the intention of the following definition.