Fractional Sobolev inequalities revisited: the maximal function approach

Since the pioneering papers by E. Gagliardo and L. Nirenberg in 1959 the so called Sobolev-GagliardoNirenberg inequalities have not ceased to be one of the most useful resources for the mathematical treatment of linear and non-linear partial differential equations ([1], [3], [4], [7], [8], [10], [11], [16], [18], [19], [21], [23], [24] and [25], among an almost infinite list of references). This central role was increased with the more recent consideration of many nonlocal problems requiring non integer regularity exponents (see. e.g., the survey [20] and its many references). As usual, interpolation inequalities are obtained in a parallel way to the study of general embedding results on Sobolev spaces W (R) for different values of the exponents (for a very complete study of range of the valid exponents see [5]). The main goal of this paper is to revisit Sobolev type inequalities involving the fractional norm. It is known that the general embedding for the spaces W (R) can be obtained by interpolation theorems through the Besov space, see e.g., [1], [23], [17], [2], [21], [12], [13], and references therein. Here, we provide the proofs of the homogeneous fractional Sobolev Ẇ (R) embeddings and the trace results. Although the results below are known, our proof are self-contained and it seems to be novel by using the technique of the Hardy-Littlewood maximal functions and the sharp maximal function (see. e.g. [22] and [15]). Then, our results are as follows.