A NOTE ON THE SUBLINEAR SOBOLEV INEQUALITY

Abstract

In this note we show that the classical Sobolev inequality cannot in unrestricted form hold for exponents p ∈ (0, 1). 1 Main Result Let d ≥ 2. A classical inequality due to Sobolev [10] states that for each p ∈ (1, d) there exists a universal constant C = C(p, d) > 0 such that ‖u‖Lq(Rd) ≤ C‖∇u‖Lp(Rd;Rd) (1.1) for all u ∈ Lloc(R) with sufficient decay at infinity and whose distributional derivative∇u ∈ L(R;R), where the exponents p, q satisfy the relation