A Helmholtz-Weyl decomposition for Hölder spaces. A nonvariational approach

We prove a form of the fundamental theorem of vector calculus for Hölder continuous vector fields with no assumption on the derivatives that exploits Schauder spaces with negative exponents. Then by developing an idea of von Wahl in the context of Lebesgue spaces, we prove a form of the Helmholtz Weyl decomposition Theorem for spaces of Hölder continuous functions, with explicit formulas for the projection operators.