Regularity of the Beurling Transform in Smooth Domains

Restricted Beurling transforms on Campanato spaces

“…As shown in [17], Proposition 2 is, in a sense, sharp in the scale of Lipschitz spaces. Moreover, general Calderón-Zygmund operators map Λ ω into Λ ω even in the case, where the smoothness of ∂Ω does not affect the computations (see, for example, [9] for various results of this type).…”

“…[17, Theorem 2]). Let ω be a regular modulus of continuity and let Ω ⊂ C be a bounded domain with a C 1,ω -smooth boundary.…”

A T1 theorem and Calderón–Zygmund operators in Campanato spaces on domains

“…It is worth mentioning that Proposition 2 is also sharp. It is proved in [17] in a particular, but important case of the Beurling transform in plane domains.…”

“…To check the condition , we use a modification of Theorem 1 in . To do this, we fix a constant .…”

“…It is worth mentioning that Proposition is also sharp. It is proved in in a particular, but important case of the Beurling transform on plane domains and Dini regular moduli of continuity. The same argument may be modified for certain homogeneous Calderón–Zygmund operators with a smooth even kernel.…”