A note on Poincaré-Sobolev type inequalities on compact manifolds

We prove a Poincaré-Sobolev type inequality on compact Riemannian manifolds where the deviation of a function from a biased average, defined using a density, is controlled by the unweighted Lebesgue norm of its gradient. Unlike classical weighted Poincaré inequalities, the density does not enter the measure or the Sobolev norms, but only the reference average. We show that the associated Poincaré constant depends quantitatively on the Lebesgue norm of the density. This framework naturally arises in the analysis of coupled elliptic systems and seems not to have been addressed in the existing literature.

💡 Research Summary

The paper establishes a new Poincaré‑Sobolev type inequality on compact Riemannian manifolds in which the deviation of a function from a density‑weighted average is controlled solely by the unweighted L^p‑norm of its gradient. The key novelty is that the density ω appears only in the definition of the reference average, not in the underlying measure or Sobolev norms, and the associated constant depends explicitly on the L^q‑norm of ω. This fills a gap in the literature where previous weighted Poincaré inequalities either required the density to define a new measure (Bakry‑Émery framework) or treated only special densities such as characteristic functions.

The authors begin by proving a Euclidean version of the inequality on bounded convex domains U ⊂ ℝ^n with smooth boundary. For ω ∈ L^q(U), ω ≥ 0, normalized so that ∫_U ω = 1, they show that for any f ∈ W^{1,p}(U) and for the exponent t = n/(n−p) (the Sobolev critical exponent) one has

‖f − E_ω f‖{L^t(U)} ≤ C ‖ω‖{L^q(U)}^{n/(p−n)} ‖∇f‖_{L^p(U)}.

The proof uses a double application of Jensen’s inequality to express |f(x)−E_ω f|^t as an ω‑weighted average of differences |f(y)−f(x)|^t, followed by a polar coordinate change and careful integration. The constant C depends only on n, p, q, and the geometry of U. By invoking the Sobolev embedding theorem and a refined Young‑ε inequality, the authors extend the estimate from the exponent t to any r ≥ 1 (subject to the usual Sobolev constraints), obtaining

‖f − E_ω f‖{L^r(U)} ≤ C₁ ‖ω‖{L^q(U)}^{n/(p−n)} ‖∇f‖_{L^p(U)}.

The dependence on ω is explicit: the constant grows like the (n/(p−n))‑th power of the L^q‑norm of ω.

The second major component is the passage from Euclidean domains to arbitrary compact manifolds (M,g). The authors construct a surjective local diffeomorphism Ψ : B → M, where B is the unit ball in ℝ^n. This is achieved by covering M with finitely many convex geodesic balls, arranging them into a spanning tree, and gluing the corresponding coordinate charts together. The resulting map Ψ has uniformly bounded Jacobian determinant, which is crucial for the subsequent change‑of‑variables arguments.

Using the coarea formula, the density ω on M is pulled back to a density \tilde ω on B via

\tilde ω(x) = ω(Ψ(x)) / |det dΨ(x)|.

Normalization is preserved (∫_B \tilde ω = 1) and the L^q‑norm satisfies

‖\tilde ω‖{L^q(B)} ≤ C₂ ‖ω‖{L^q(M)}.

Similarly, for f ∈ W^{1,p}(M) the pull‑back \tilde f = f∘Ψ satisfies

‖∇\tilde f‖{L^p(B)} ≤ C₃ ‖∇f‖{L^p(M)}.

Applying the Euclidean inequality to (\tilde f, \tilde ω) on B yields

‖\tilde f − E_{\tilde ω}\tilde f‖{L^r(B)} ≤ C₄ ‖\tilde ω‖{L^q(B)}^{n/(p−n)} ‖∇\tilde f‖_{L^p(B)}.

Because the weighted averages are compatible (E_{\tilde ω}\tilde f = E_ω f), pulling the estimate back to M gives the main result:

‖f − E_ω f‖{L^r(M)} ≤ C ‖ω‖{L^q(M)}^{n/(p−n)} ‖∇f‖_{L^p(M)}.

Here C depends only on the dimension n, the exponents p, q, r, and the Riemannian metric g (through bounds on the Jacobian of Ψ). No hidden dependence on ω remains beyond the explicit power of its L^q‑norm.

The paper’s contributions can be summarized as follows:

1. New Inequality – A Poincaré‑Sobolev type inequality where the density appears only in the reference average, not in the measure or Sobolev norms.
2. Quantitative Dependence – The constant is explicitly controlled by ‖ω‖_{L^q}^{n/(p−n)}, providing uniform estimates for families of densities.
3. Geometric Transfer – A clean method to lift Euclidean results to compact manifolds via a bounded‑Jacobian diffeomorphism and the coarea formula.
4. Elementary Proof – The argument relies on standard tools (Jensen, Young, Sobolev embedding, coarea), avoiding sophisticated curvature‑dependent techniques.
5. Potential Applications – The result is directly applicable to coupled elliptic systems, variational problems with normalization constraints, and any PDE setting where a density‑weighted average naturally arises.

Overall, the work fills a notable gap in the theory of weighted functional inequalities, offering a versatile tool for analysts working with non‑uniform averages on manifolds. Future directions could include extending the inequality to manifolds with boundary, non‑compact settings, or exploring optimality of the exponent n/(p−n) in various geometric contexts.