Maximal functions measuring smoothness

Maximal functions which measure the smoothness of a function are intro duced and studied from the point of view of their relationship to classical smoothness and their use in proving embedding theorems, extension theorems and various results on differentiation. New spaces of functions which generalize Sobolev spaces are introduced. AMS(MOS) subject classifications (1980). Primary 26B35, 46E35, 26Al5 Secondary 42B25