[Picture] [Picture] [Picture] [Picture] [Picture] [Picture] [Picture] The remaining integral gives exponent n-alpha. So alpha < n-1 is the condition for the function to have a weak derivative in L1. If alpha < n/p -1 it is also in Lp. [Picture] The formula below is one of the fundamental theorems of calculus and tells how to recover u from its derivative. We generalise this to weak derivatives: [Picture] So we get the fundamental theorem in a formulation with test functions: [Picture] The upper bounds in the integrals should be 1 and not t! For the last statement we need approximation with smooth functions. Once we have this, the basic Sobolev inequalities (GNS and Morrey) can be proved directly without regularising first. For step 2 in the proof of Theorem 1 in 5.6.1 use Exercise 16.