Two Elementary Proofs of the L 2 Boundedness of Cauchy Integrals on Lipschitz Curves

Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^p$ boundedness

“…For a vector field v, if one truncates (1.3) as 11) then it is reasonable to ask for pure regularity assumption on v in order to bound H v, 0 . Indeed, a counterexample in [22] based on the Perron tree construction of the Besicovitch-Kakeya set (see [23] and [12]) shows that no bounds are possible for v being Hölder continuous of an exponent less than one, and it is a long standing open problem in harmonic analysis that whether Lipschitz regularity suffices.…”

Boundedness and applications of singular integrals and square functions: a survey

“…The main outline of the proof then follows that of the T 1 Theorem. We refer the reader to [34] or to [32], pp 64-67, for the details of an argument that is somewhat simpler than the original one in [45].…”

b-weighted dyadic BMO from dyadic BMO and associatedT(b) theorems

“…In order to define this norm, we must look at Haar wavelets which are adapted to the function b as in [2,5]. Definition 2.6 Let I ⊆ ∆ be a collection of intervals and b a function which has nonzero mean on all intervals in I.…”

“…This is a natural extension of the previous work, in particular in [2] where the dyadic T (b) theorems are proven and in [13], where sharp growth bounds for b-input theorems are proven. Our approach utilizes b-adapted Haar wavelets as in [5]. Associated b-output T (b) theorems in both local and global cases follow as corollaries from theorems which compare b-weighted dyadic BMO with the dyadic BMO developed in [9].…”