Sharp Sobolev Inequalities Research Articles

Second order Sobolev type inequalities in the hyperbolic spaces

In this paper, we establish several Poincaré–Sobolev type inequalities concerning to the Laplace–Beltrami operator Δg in the hyperbolic space Hn with n≥5. These inequalities could be seen as the improved second order Poincaré inequality with remainder terms involving with the sharp Rellich inequality or the sharp Sobolev inequality in Hn. The novelty of these inequalities is that it combines both the sharp Poincaré inequality and the sharp Rellich inequality or the sharp Sobolev inequality for Δg in Hn. As a consequence, we obtain the Poincaré–Sobolev inequality for the second order GJMS operator P2 in Hn. In dimension 4, we obtain an improvement of the sharp Adams inequality and an Adams inequality with exact growth for radial functions.

Sharp logarithmic Sobolev inequalities along an extended Ricci flow and applications

Sharp logarithmic Sobolev inequalities along an extended Ricci flow and applications

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

In this note, we establish a Lp-version of the Poincaré–Sobolev inequalities in the hyperbolic spaces Hn. The interest of this result is that it relates both the Poincaré (or Hardy) inequality and the Sobolev inequality with the sharp constant in Hn. Our approach is based on the comparison of the Lp-norm of gradient of the symmetric decreasing rearrangement of a function in both the hyperbolic space and the Euclidean space, and the sharp Sobolev inequalities in Euclidean spaces. This approach also gives the proof of the Poincaré–Gagliardo–Nirenberg and Poincaré–Morrey–Sobolev inequalities in the hyperbolic spaces Hn. Finally, we discuss several other Sobolev inequalities in the hyperbolic spaces Hn which generalize the inequalities due to Mugelli and Talenti in H2.

Sobolev trace inequalities of order four

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we charcaterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean 4-balls.

Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces

We extend the recent $L^{1}$ uncertainty inequalities obtained by Dall'ara-Trevisan to the metric setting. For this purpose we introduce a new class of weights, named *isoperimetric weights*, for which the growth of the measure of their level sets $\mu(\{w\leq r\})$ can be controlled by $rI(r),$ where $I$ is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new *localized Poincar\'e inequalities*, and interpolation, to prove $L^{p},1\leq p<\infty,$ uncertainty inequalities on metric measure spaces. We give an alternate characterization of the class of isoperimetric weights in terms of Marcinkiewicz spaces, which combined with the sharp Sobolev inequalities we had obtained in an earlier paper, and interpolation of weighted norm inequalities, give new uncertainty inequalities in the context of rearrangement invariant spaces.

The Sharp Sobolev Inequality on Metric Measure Spaces with Lower Ricci Curvature Bounds

We show that the sharp Sobolev inequality as known for Riemannian manifolds with a positive lower bound on the Ricci curvature holds in the same form for metric measure spaces satisfying the RCD*(K, N) condition for positive K.

Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half-spaces via mass transport and consequences

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.

Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation

We exhibit the optimal constant for Sobolev inequalities in Lorentz spaces for a mean oscillation, and its relation with a boundedness of the Hardy–Littlewood maximal operator in Sobolev spaces. Some applications to a scale invariant form of Hardyʼs inequality in a limiting case are also considered.

The sharp Sobolev inequality in quantitative form

A quantitative version of the sharp Sobolev inequality in W^{1,p} (ℝ^n) , 1 &lt; p &lt; n , is established with a remainder term involving the distance from the family of extremals.

Symmetrization and sharp Sobolev inequalities in metric 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.

Best remainder norms in Sobolev-Hardy inequalities

We exhibit the optimal norm for a remainder term in the sharp Sobolev inequality involving a Lorentz norm, and in the equivalent classical Hardy inequality. Limiting cases of the relevant inequalities are also considered.

Optimal Concentration for SU(1, 1) Coherent State Transforms and An Analogue of the Lieb-Wehrl Conjecture for SU(1, 1)

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.

From Brunn–Minkowski to sharp Sobolev inequalities

We present a simple direct proof of the classical Sobolev inequality in \(\mathbb{R}^n\) with best constant from the geometric Brunn–Minkowski–Lusternik inequality.

Sharp Sobolev inequalities in the presence of a twist

Let ( M , g ) (M,g) be a smooth compact Riemannian manifold of dimension n ≥ 3 n \ge 3 . Let also A A be a smooth symmetrical positive ( 0 , 2 ) (0,2) -tensor field in M M . By the Sobolev embedding theorem, we can write that there exist K , B &gt; 0 K,B&gt;0 such that for any u ∈ H 1 2 ( M ) u \in H_1^2(M) , \[ ( ∫ M | u | 2 ⋆ d v g ) 2 / 2 ⋆ ≤ K ∫ M A x ( ∇ u , ∇ u ) d v g + B ∫ M u 2 d v g \left (\int _M\vert u\vert ^{2^\star }dv_g\right )^{2/2^\star }\le K \int _MA_x(\nabla u, \nabla u)dv_g + B\int _Mu^2dv_g \] where H 1 2 ( M ) H_1^2(M) is the standard Sobolev space of functions in L 2 L^2 with one derivative in L 2 L^2 . We investigate in this paper the value of the sharp K K in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.

The sharp Sobolev and isoperimetric inequalities split twice

This paper shows that each of the sharp (endpoint) Sobolev inequality and the isoperimetric inequality can be split into two sharp and stronger inequalities through either the 1-variational capacity or the 1-integral affine surface area. Furthermore, some related sharp analytic and geometric inequalities are also explored.

Homogeneous sharp Sobolev inequalities on product manifolds

Let (M, g) and (N, h) be compact Riemannian manifolds of dimensions m and n, respectively. For p-homogeneous convex functions f(s, t) on [0,∞) × [0, ∞), we study the validity and non-validity of the first-order optimal Sobolev inequality on H1, p(M × N) where and Kf = Kf (m, n, p) is the best constant of the homogeneous Sobolev inequality on D1, p (Rm+n), The proof of the non-validity relies on the knowledge of extremal functions associated with the Sobolev inequality above. In order to obtain such extremals we use mass transportation and convex analysis results. Since variational arguments do not work for general functions f, we investigate the validity in a uniform sense on f and argue with suitable approximations of f which are also essential in the non-validity. Homogeneous Sobolev inequalities on product manifolds are connected to elliptic problems involving a general class of operators.

Sharp Sobolev inequalities for vector valued maps

Sharp Sobolev inequalities for vector valued maps

Sharp Sobolev inequality involving a critical nonlinearity on a boundary

We consider the solvability of the Neumann problem for the equation $$ -\Delta u+\lambda u =0, \quad \frac {\partial u}{\partial \nu}=Q(x)|u|^{q-2}u $$ on $\partial \Omega$, where $Q$ is a positive and continuous coefficient on $\partial \Omega$, $\lambda$ is a parameter and $q= {2(N-1)}/{(N-2)}$ is a critical Sobolev exponent for the trace embedding of $H^1(\Omega)$ into $L^q(\partial \Omega)$. We investigate the joint effect of the mean curvature of $\partial \Omega$ and the shape of the graph of $Q$ on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter $\lambda$ interferes with the spectrum of $-\Delta$ with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.

Sharp local embedding inequalities

In this paper we establish local versions of the Onofri and sharp Sobolev inequalities. Such local inequalities enable us to give a more direct and simpler proof of the Onofri inequality on 𝕊2, as well as an alternative proof of sharp Sobolev inequalities on 𝕊n (for n ≥ 3). © 2005 Wiley Periodicals, Inc.

Logarithmic Sobolev trace inequality

A logarithmic Sobolev trace inequality is derived. Bounds on the best constant for this inequality from above and below are investigated using the sharp Sobolev inequality and the sharp logarithmic Sobolev inequality.