A Remark on Fine Differentiability

Generalized Lipschitz numbers, fine differentiability, and quasiconformal mappings

“…In this section we study the fine differentiability of Sobolev and BV functions, as well as of the Hardy-Littlewood maximal function. For some previous studies of fine differentiability and finely harmonic functions, see Gardiner [19,20] and Lávička [42,43,44]. These references study the case p = 2, whereas we will keep working with the 1-fine topology.…”

Finely continuously differentiable functions

“…However, it is natural to ask if a much stronger result might be true; namely, might fine−C 1 (U ) actually coincide with C 1 f−loc (U )? When n = 2, Lávička [9] showed that this is the case by adapting work of Fuglede on finely holomorphic functions. The proof is based on a special property of thin sets in the plane, and in higher dimensions much less is known [10].…”

“…This paper concerns the notion of fine differentiability of functions of the form f : U → R, where U is a finely open subset of R n . Previous papers dealing with this topic include Mastrangelo [12], Mastrangelo and Dehen [13], Raynaud-Pimenta [14], and Lávička [9,10].…”

“…The proof is based on a special property of thin sets in the plane, and in higher dimensions much less is known [10]. In fact, Lávička [9] has pointed out that, in this case, it is not even known whether a function that has zero fine differential at every point of a fine domain must be constant. (By a fine domain we mean a non-empty finely connected finely open set.…”