CHAPTER 1 Overview of principal results We are interested in singular integral operators on functions of two variables, which act by performing a one dimensional transform along a particular line in the plane. The choice of lines is to be variable. Thus, for a measurable map, v from R2 to the unit circle in the plane, that is a vector field, and a Schwartz function f on R2, define Hv, f(x) := p.v. f(x yv(x)) dy y . This is a truncated Hilbert transform performed on the line segment {x + tv(x) : |t| 1}. We stress the limit of the truncation in the definition above as it is important to different scale invariant formulations of our questions of interest. This is an example of a Radon transform, one that is degenerate in the sense that we seek results independent of geometric assumptions on the vector field. We are primarily interested in assumptions of smoothness on the vector field. Also relevant is the corresponding maximal function (1.1) Mv, f := sup 0t≤ (2t)−1 t −t |f(x sv(x))| ds The principal conjectures here concern Lipschitz vector fields. Zygmund Conjecture 1.2. Suppose that v is Lipschitz. Then, for all f L2(R2) we have the pointwise convergence (1.3) lim(2t)−1 t→0 t −t f(x sv(x)) ds = f(x) a. e. More particularly, there is an absolute constant K 0 so that if −1 = K v Lip , we have the weak type estimate (1.4) sup λ0 λ|{Mv, f λ}|1/2 f 2 . The origins of this question go back to the discovery of the Besicovitch set in the 1920’s, and in particular, constructions of this set show that the Conjecture is false under the assumption that v is older continuous for any index strictly less than 1. These constructions, known since the 1920’s, were the inspiration for A. Zygmund to ask if integrals of, say, L2(R2) functions could be differentiated in a Lipschitz choice of directions. That is, for Lipschitz v, and f L2, is it the case that lim(2 →0 )−1 f(x yv(x)) dy = f(x) a.e.(x) These and other matters are reviewed in the next chapter. Much later, E. M. Stein [25] raised the singular integral variant of this conjec- ture. 1
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