Hölder norm estimate for the fractal Hilbert transform in Douglis analysis

Abstract

Douglis analysis is an alternative approach to complex methods for the investigation of linear and uniformly elliptic systems of 2n equations for 2n desired real-valued functions. The function theory associated with the Douglis operator in R2 (identifying R2 with C in the usual way) plays a very important role in problems in pure mathematics, mathematical physics, and engineering, such as plane elasticity theory and hydromechanics. The well-known Douglis system, that is, an elliptic system of first order in two inde- pendent variables, can be represented by a single "hypercomplex" equation. Solutions of such equation (null solutions of the Douglis system) are termed hyperanalytic functions. In [1] Douglis presented a complete study of the hyperanalytic function theory. For greater details the reader is directed to [2] and to [3] for a thorough treatment of this theory. In more recent times hyperanalytic function theory has been developed for solving problems of mathematical physics such as plate and shell problems. In [4, 5] the authors provided conditions for the solvability of the Riemann boundary value problem for hy- peranalytic functions on classes of fractal closed curves. Hence, this can be regarded of as a good motivation for finding conditions on the boundary, which give boundedness of certain singular integral operators, such as the Hilbert transform when the boundary is permitted to be fractal. To this end, in [6] the authors gave an estimate for the upper bound of the Hölder norm a fractal version of the Hilbert transform for domains with d-summable boundary; a geometric notion introduced in [7], which is essential for inte- gration of a form over a fractal boundary.