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The main goal of this paper is to estimate the Hölder norm of a fractal version of the Hilbert transform in the Douglis analysis context acting from Hölder spaces of Douglis algebra valued functions defined on h-summable curves.

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We investigate an electromagnetic Dirichlet type problem for the 2D quaternionic time-harmonic Maxwell system over a great generality of fractal closed type curves, which bound Jordan domains in R2. The study deals with a novel approach of h-summability condition for the curves, which would be extremely irregular and deserve to be considered fractals. Our technique of proofs is based on the intimate relations between solutions of time-harmonic Maxwell system and those of the Dirac equation through some nonlinear equations, when both cases are reformulated in quaternionic forms.

desplazado y su estrecha relación con los principales operadores del cálculo vectorial, es posible establecer una conexión directa entre las soluciones del sistema de Maxwell tiempo-armónico y dos ecuaciones cuaterniónicas. Además, se expone la aplicación de la condición de Lorentz para transformar el sistema de Maxwell tiempo-armónico en una simple ecuación cuaterniónica en función de los potenciales escalar y vectorial.