Summability on non-rectifiable Jordan curves

For a given parameterization of a Jordan curve, we define the notion of summability or classes of measurable functions on a contour where a new integral is introduced. It is shown that natural functional spaces defining summability for non-rectifiable Jordan curves are the Lebesgue spaces with the weighted norm. For non-rectifiable Jordan curves where an integral was previously defined for continuous (Hölder) functions [Y. Guseynov, Integrable boundaries and fractals for Hölder classes; the Gauss–Green theorem, Calc. Var. Partial Differential Equations 55 2016, 4, Article ID 103], a weight function is constructed which, in general, is not summable by parameter, and a weighted functional space (summability) is defined where the new integral exists.