A fully anisotropic Sobolev inequality

Let n ≥ 2 and let A : Rn → [0,∞] be any convex function satisfying the following properties: A(0) = 0 and A(ξ) = A(−ξ) for ξ ∈ R; (1.1) for every t > 0, {ξ ∈ R : A(ξ) ≤ t} (1.2) is a compact set whose interior contains 0. Observe that A need not depend on the length |ξ| of ξ nor be the sum of functions of its components ξi, i = 1, . . . , n. The purpose of this note is to exhibit an inequality of Sobolev type, for real-valued weakly differentiable functions u on Rn decaying to 0 at infinity, which involves the gradient ∇u through the integral ∫ Rn A(∇u) dx. In the relevant inequality, a role will be played by the function B : [0,∞) → [0,∞] associated with A as follows. Denote by A : [0,∞] → [0,∞] the left-continuous increasing function satisfying |{ξ ∈ R : A(ξ) ≤ 1}| = |{ξ ∈ R : A (|ξ|) ≤ t}| for every t ≥ 0, where | · | stands for Lebesgue measure. Assume that ∫