Hölder norm estimate for a Hilbert transform in Hermitian Clifford analysis

A Hilbert transform for Holder continuous circulant (2 x 2) matrix functions, on the d-summable (or fractal) boundary I" of a Jordan domain Omega in a"e(2n) , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the Holder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the Holder exponents, the diameter of I" and a specific d-sum (d > d) of the Whitney decomposition of Omega. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary