Interpolation of Linear Operators on Sobolev Spaces

We are interested in the interpolation of linear operators defined on Sobolev spaces Wk = Wk(Q), 1 ? p ? oo. In particular, there is the question ([11]) whether Wk, 1 < p < o0, is an interpolation space between W1k and W!. We will answer this question affirmatively and actually do much more since we can characterize the Peetre K-functional for interpolation between W1k and Wk by means of rearrangements of f and its derivatives Dsf, Ia J ? k. Thus, the interpolation theory for the spaces Wk, 1 g p ! ??, is like that for the L. spaces (the case k = 0), which is perhaps to be expected but by no means obvious. It is a well known result [4, p. 184] that for any totally a-finite measure space (X, se), the K-functional for interpolation between L1 L1(X, Ye) and Loo = LOO(X, A) is characterized by