A T(1) theorem for singular Radon transforms by Greenblatt M. PDF

By Greenblatt M.

We turn out an analogue for singular Radon transforms to the T (1) theoremof David and Journe.

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3, we illustrate on the left hand side the case where πB (ϕ) = x. Not so on the right hand side, where we have πB (ϕ) = 0, because there are two points in x + (B ∪ (−B)). Let us then check whether the family {πB : B ∈ G} is quasi-complete on L0 (G). It is easy to show that, for ϕ ∈ L0 (G) we have {πB (ϕ) : B ∈ G} = {x ∈ ϕ : {−x, 2x} ∩ ϕ = ∅} ∪ {0}. 3) In particular, the family of matchings {πB : B ∈ G} is complete on the large subclass of L0 (Rd ) given by D := {ϕ ∈ L0 (Rd ) : there are no distinct points x, y, z in ϕ such that x − y = y − z}.

He assumes that a given (isometry invariant) index function is injective on the points of N almost surely, which is shown to be equivalent with the fact that almost surely there is no non-trivial isometry that leaves N invariant. To stick with our setting, we will describe the definition of the graph on locally finite subsets ϕ of Rd that satisfy card(ϕ) = ∞ and ϕ = S(ϕ) for all translations or rotations S = idRd on Rd . Both properties are almost surely satisfied by the point processes described above.

Proof: The first two claims follow almost immediately from the definitions, so that we will only prove that Ψ1 ◦ Ψ2 is a local selection function. The mapping Ψ1 ◦ Ψ2 is measurable, because it is the composition of measurable mappings. Also, for ϕ ∈ L(G) and x ∈ Ψ1 ◦ Ψ2 (ϕ), we have x ∈ Ψ2 (ϕ) and then (Ψ1 ◦ Ψ2 ) ◦ θx (ϕ) = Ψ1 ◦ (Ψ2 ◦ θx )(ϕ) = Ψ1 ◦ (θx ◦ Ψ2 )(ϕ) = θx ◦ (Ψ1 ◦ Ψ2 )(ϕ), where we have used x ∈ Ψ2 (ϕ) for the second equation and x ∈ Ψ1 (Ψ2 (ϕ)) for the last equation. We have shown that the classes of (local) selection functions are stable with respect to compositions.

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A T(1) theorem for singular Radon transforms by Greenblatt M.


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