Sobolev inequalities with jointly concave weights on convex cones

Balogh, Zoltán M.; Gutiérrez, Cristian E.; Kristály, Alexandru (2021). Sobolev inequalities with jointly concave weights on convex cones. Proceedings of the London Mathematical Society, 122(4), pp. 537-568. Oxford University Press 10.1112/plms.12384

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Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form
(\int_E|u(x)|q ω(x) dx)^1/q\leq K_0(\int_E|∇u(x)|p σ(x) dx)^1/p, u∈ C_0^∞ (R^n), (WSI)
where p \geq 1 and q > 0 is the corresponding Sobolev critical exponent. Here E ⊆ Rn is an open convex cone, and ω, σ : E → (0,∞) are two homogeneous weights verifying a general concavity-type structural condition. The constant K_0 = K_0(n, p, q, ω, σ) > 0 is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that K_0 is optimal in (WSI) if and only if ω and σ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to partial differential equations are also provided.

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