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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.

We consider the initial-boundary value problem for intermediate long-wave equation on a halfline. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.