Revisiting Microlocal Analysis: A Bridge Between Fractal Cohomology and Classical Function Spaces

We hereafter go further than in our previous works ( David & Lapidus, 2024b , 2024e ), in the characterization of the cohomology groups via Taylor-like expansions and identify the corresponding coefficients in terms of discrete (and complex) fractional derivatives. We also prove that the generators of the cohomology groups constitute an orthogonal eigenvector basis of the fractal Hodge Laplacian (acting on the complexified total fractal cohomology space). Towards the end, we introduce suitable function spaces, called the local and global Hölder–Zygmund spaces, as well as of local Sobolev spaces—and their connections with the fractal cohomology spaces from David and Lapidus (2024e)—paves the way for the development of a fractal analog of the classic microlocal analysis.