The sharp Sobolev and isoperimetric inequalities split twice

It is shown that the minimum value for c is $4/3 \sqrt 3 \pi ^2 \cong .0780$ for $\phi $ of function class $C^0 $ piecewise $C^2 $ in real Euclidean 3-space.

Let ( M , g ) (M,g) be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature K ≤ − 1 K\le -1 . If f f is a compactly supported function of bounded variation on M M , then f f satisfies the Sobolev inequality \[ 4 π ∫ M f 2 d A + ( ∫ M | f | d A ) 2 ≤ ( ∫ M ‖ ∇ f ‖ d A ) 2 . 4\pi \int _M f^2\,dA+ \left (\int _M |f|\,dA \right )^2\le \left (\int _M\|\nabla f\|\,dA \right )^2. \] Conversely, letting f f be the characteristic function of a domain D ⊂ M D\subset M recovers the sharp form 4 π A ( D ) + A ( D ) 2 ≤ L ( ∂ D ) 2 4\pi A(D)+A(D)^2\le L(\partial D)^2 of the isoperimetric inequality for simply connected surfaces with K ≤ − 1 K\le -1 . Therefore this is the Sobolev inequality “equivalent” to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces. Under the same assumptions on ( M , g ) (M,g) , if c : [ a , b ] → M c\colon [a,b]\to M is a closed curve and w c ( x ) w_c(x) is the winding number of c c about x x , then the Sobolev inequality implies \[ 4 π ∫ M w c 2 d A + ( ∫ M | w c | d A ) 2 ≤ L ( c ) 2 , 4\pi \int _M w_c^2\,dA+ \left (\int _M|w_c|\,dA \right )^2\le L(c)^2, \] which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature ≤ − 1 \le -1 .

Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concentrate especially on the conformally flat case, and obtain formulas in dimensions $2$, $4$, and $6$ for the functional determinants of operators which are well behaved under conformal change of metric. The two-dimensional formulas are due to Polyakov, and the four-dimensional formulas to Branson and Ørsted; the method is sufficiently streamlined here that we are able to present the sixdimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformal Laplacian and of the square of the Dirac operator on ${S^2}$, and in the standard conformal classes on ${S^4}$ and ${S^6}$. The ${S^2}$ results are due to Onofri, and the ${S^4}$ results to Branson, Chang, and Yang; the ${S^6}$ results are presented for the first time here. Recent results of Graham, Jenne, Mason, and Sparling on conformally covariant differential operators, and of Beckner on sharp Sobolev and Moser-Trudinger type inequalities, are used in an essential way, as are a computation of the spectra of intertwining operators for the complementary series of ${\text {S}}{{\text {O}}_0}(m + 1,1)$, and the precise dependence of all computations on the dimension. In the process of solving the extremal problem on ${S^6}$, we are forced to derive a new and delicate conformally covariant sharp inequality, essentially a covariant form of the Sobolev embedding $L_1^2({S^6})\hookrightarrow {L^3}({S^6})$ for section spaces of trace free symmetric two-tensors.

Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concentrate especially on the conformally flat case, and obtain formulas in dimensions 2 2 , 4 4 , and 6 6 for the functional determinants of operators which are well behaved under conformal change of metric. The two-dimensional formulas are due to Polyakov, and the four-dimensional formulas to Branson and Ørsted; the method is sufficiently streamlined here that we are able to present the sixdimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformal Laplacian and of the square of the Dirac operator on S 2 {S^2} , and in the standard conformal classes on S 4 {S^4} and S 6 {S^6} . The S 2 {S^2} results are due to Onofri, and the S 4 {S^4} results to Branson, Chang, and Yang; the S 6 {S^6} results are presented for the first time here. Recent results of Graham, Jenne, Mason, and Sparling on conformally covariant differential operators, and of Beckner on sharp Sobolev and Moser-Trudinger type inequalities, are used in an essential way, as are a computation of the spectra of intertwining operators for the complementary series of S O 0 ( m + 1 , 1 ) {\text {S}}{{\text {O}}_0}(m + 1,1) , and the precise dependence of all computations on the dimension. In the process of solving the extremal problem on S 6 {S^6} , we are forced to derive a new and delicate conformally covariant sharp inequality, essentially a covariant form of the Sobolev embedding L 1 2 ( S 6 ) ↪ L 3 ( S 6 ) L_1^2({S^6})\hookrightarrow {L^3}({S^6}) for section spaces of trace free symmetric two-tensors.

By adapting the mass transportation technique of Cordero-Erausquin, Nazaret and Villani, we obtain a family of sharp Sobolev and Gagliardo-Nirenberg (GN) inequalities on the half space R n−1 × R+, n ≥ 1 equipped with the weight !(x) = x a, a ≥ 0. It amounts to work with the fractional dimension na = n + a. The extremal functions in the weighted Sobolev inequalities are fully characterized. Using a dimension reduction argument and the weighted Sobolev inequalities, we can reproduce a subfamily of the sharp GN inequalities on the Euclidean space due to Del Pino and Dolbeault, and obtain some new sharp GN inequalities as well. Our weighted inequalities are also extended to the domain R n−m × R m and the weights are !(x,t) = t a1 1 ...t a m m , where n ≥ m, m ≥ 0 and a1,··· ,am ≥ 0. A weighted L p -logarithmic Sobolev inequality is derived from these inequalities.

Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy–Littlewood–Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings [Formula: see text]. We show that their argument gives a direct proof of the latter inequalities without passing through Hardy–Littlewood–Sobolev inequalities, and, moreover, a new proof of a sharp fully nonlinear Sobolev inequality involving the [Formula: see text]-curvature. Our argument relies on nice commutator identities deduced using the Fefferman–Graham ambient metric.

Brendle recently proved a sharp Sobolev inequality and logarithmic Sobolev inequality for submanifolds in Euclidean space. From the sharp Sobolev inequality, he achieved a breakthrough in the conjecture of isoperimetric inequality for minimal submanifolds. In this paper, we extend Brendle’s results to submanifolds in a smooth metric measure space. As an application, we prove some new isoperimetric-type inequalities in some smooth metric measure spaces. For example, we obtain a new isoperimetric-type inequality for self-expander.

The sharp Poincaré–Sobolev type inequalities in the hyperbolic spaces [formula omitted

Affine vs. Euclidean isoperimetric inequalities

In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182(2):307-332, 2004) proved sharp Sobolev inequalities in Euclidean spaces via Optimal Transport, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-optimal transport approach, the compact case has been successfully treated by Cavalletti and Mondino (Geom Topol 21:603-645, 2017), even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present paper we affirmatively answer the above question for noncompact Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Transport theory with quadratic distance cost, sharp Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-Sobolev and Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-logarithmic Sobolev inequalities (both for p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document} and p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}) are established, where the sharp constants contain the asymptotic volume ratio arising from precise asymptotic properties of the Talentian and Gaussian bubbles, respectively. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia (Math 140:818-826, 2004) and subsequent results, concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support Sobolev inequalities.

Second order Sobolev type inequalities in the hyperbolic spaces

We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for H¨ormander vector fields.

We derive a lower bound for the Wehrl entropy in the setting of SU(1, 1). For asymptotically high values of the quantum number k, this bound coincides with the analogue of the Lieb-Wehrl conjecture for SU(1, 1) coherent states. The bound on the entropy is proved via a sharp norm bound. The norm bound is deduced by using an interesting identity for Fisher information of SU(1, 1) coherent state transforms on the hyperbolic plane $${\mathbb{H}^{2}}$$ and a new family of sharp Sobolev inequalities on $${\mathbb{H}^{2}}$$ . To prove the sharpness of our Sobolev inequality, we need to first prove a uniqueness theorem for solutions of a semi-linear Poisson equation (which is actually the Euler-Lagrange equation for the variational problem associated with our sharp Sobolev inequality) on $${\mathbb{H}^{2}}$$ . Uniqueness theorems proved for similar semi-linear equations in the past do not apply here and the new features of our proof are of independent interest, as are some of the consequences we derive from the new family of Sobolev inequalities.

In this survey, we consider the sharp Sobolev inequality in convex cones. We also prove it by using the optimal transport technique. Then we present some results related to the Euler–Lagrange equation of the Sobolev inequality: the so-called critical p -Laplace equation. Finally, we discuss some stability result related to the Sobolev inequality.

Corrigendum to “The sharp Sobolev and isoperimetric inequalities split twice” [Adv. Math. 211 (2) (2007) 417–435