The Hardy–Morrey & Hardy–John–Nirenberg inequalities involving distance to the boundary

We strengthen the classical inequality of C.B. Morrey concerning the optimal Hölder continuity of functions in W1,pW1,p when p>np>n, by replacing the LpLp-modulus of the gradient with the sharp Hardy difference involving distance to the boundary. When p=np=n we do the same strengthening in the integral form of a well known inequality due to F. John and L. Nirenberg.