SOBOLEV INEQUALITIES FOR VARIABLE EXPONENTS ATTAINING THE VALUES 1 AND n

Variable exponent spaces have been studied in many articles over the past decade; for a survey see [10, 27]. These investigations have dealt both with the spaces themselves, with related differential equations, and with applications. One typical feature is that the exponent has to be strictly bounded away from various critical values. In some recent investigations it it has been found that one needs to modify the scales of spaces at the end point to properly deal with such limiting phenomena, see [9, 17]. More concretely, consider the example of the Sobolev embedding theorem. In the constant exponent case it is well-known that the embeddings are qualitatively different according as p < n (Lebesgue space), p = n (exponential Orlicz space) or p > n (Hölder space). In the variable exponent case this has led to theorems assuming either p < n, or p− > n, where p and p− denote the greatest and least value of p, respectively. In this paper we are concerned with generalizing the former. Sobolev embeddings and embeddings of Riesz potentials have been studied, e.g., in [1, 2, 4, 6, 8, 10, 11, 12, 13, 15, 16, 20, 24, 28] in the variable exponent setting. Most proofs in the literature are based on the Riesz potential and maximal functions, and thus lead to the additional, unnatural restriction p− > 1. As far as we know, the only attempt to levy these restrictions is due to Edmunds and Rákosník [11, 12]. Their method is not based on maximal functions, and does