Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
Sharp inequalities in unit ball with constraints on moments.
problem Establishing Sobolev trace inequalities with constraints.
method Constructing smooth test functions for higher order moments.
result Almost optimal Sobolev trace inequalities for 2nd and 4th orders.
Derives sharp Sobolev trace inequalities for higher order derivatives on balls and half-spaces.
problem Sharp Sobolev trace inequalities for higher order derivatives.
method Scattering theory on hyperbolic spaces, generalized Poisson kernel, explicit formulas of extremal functions.
result Explicit formulas of extremal functions and sharp trace Sobolev inequalities.
The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.
problem Proving higher order trace inequalities on Poincaré-Einstein manifolds.
method Introducing conformally covariant boundary operators and using them to set up higher order Dirichlet problems.
result Sharp higher order Sobolev trace inequalities on the ball are obtained.
The paper classifies minimizers for a specific inequality in Euclidean balls.
problem Classifying minimizers for a specific inequality in Euclidean balls.
method Adaptation of Frank-Lieb proof for the sharp Sobolev inequality.
result Implication of a fully nonlinear sharp Sobolev trace inequality.
Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.
problem Establishing a Sobolev trace inequality on a specific domain.
method Using weighted norms and fractional powers of sub-Laplacian on Heisenberg group.
result Sharp Sobolev trace inequality on Siegel domain involving weighted norms.
The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
problem Establishing higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
method Introducing conformally covariant boundary operators, proving extension theorems, and establishing trace inequalities.
result Generalized CR Sobolev trace inequalities for all γ ∈ (0, n+1) \mathbb{N}.
Study bounds for Brownian motion on manifolds with sticky boundary conditions.
problem Proving geometric bounds for Brownian motion on manifolds with sticky boundary conditions.
method Interpolation involving energy interactions between boundary and interior of the manifold.
result Explicit geometric bounds on Steklov eigenvalues, boundary trace operators, and boundary trace logarithmic Sobolev constants.
Generalizes Escobar-Riemann mapping problem for smooth metric measure spaces.
problem Finding a function that attains the Escobar weighted constant.
method Introducing Escobar quotient, infimum, and resolving the problem when the weighted constant is negative.
result Obtained an Aubin type inequality connecting weighted Escobar constant and optimal constant for trace inequality.
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 ch…
New boundary operators for sixth-order GJMS operator on manifolds.
problem Developing boundary operators for sixth-order GJMS operator.
method Conformally covariant boundary operators and fractional GJMS operators.
result New realization of fractional GJMS operators and Sobolev trace inequalities.
The paper examines rigidity of Einstein metrics using curvature functionals.
problem Characterizing rigidity of Einstein metrics.
method Critical points of quadratic curvature functionals and integral inequalities involving Weyl curvature, trace-less Ricci curvature, and Sobolev constant.
result Rigidity results for Einstein metrics on complete manifolds.
The paper explores how Finsler manifolds differ from Riemannian ones in functional inequalities.
problem Analytic phenomena on Finsler manifolds differ from Riemannian ones.
method Comparative analysis of Finsler and Riemannian manifolds, focusing on Sobolev spaces, Hardy inequalities, and uncertainty principles.
result Functional inequalities (Hardy, uncertainty) break down on Finsler Cartan-Hadamard manifolds, while Caffarelli-Kohn-Nirenberg inequality exhibits a sharp threshold.
The paper explores how Finsler manifolds differ from Riemannian ones in functional inequalities.
problem Analytic phenomena on Finsler manifolds differ from Riemannian ones.
method Comparative analysis of Finsler and Riemannian manifolds.
result Functional inequalities (Hardy, uncertainty, CKN) behave differently on Finsler manifolds.
The paper proves rigidity and vanishing theorems for translating solitons.
problem Understanding the properties of translating solitons in geometry.
method Using Sobolev inequalities and Lq-norms, the paper proves rigidity and vanishing theorems. result Translating solitons are shown to be hypersurfaces under certain conditions.
We study the horizontal Laplacian ΔH associated to the Hopf fibration S3→S2 with arbitrary Chern number k. We use representation theory to calculate the spectrum, describe the heat kernel and obtain the complete heat trace asymptotics of ΔH. We express the Green functions for associated Poisson semigroup…
Under a spectral assumption on the Laplacian of a Poincaré--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order 2γ∈(0,2) or 2γ∈(2,4) and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption…
The study proves Liouville theorems on curved manifolds with convex boundaries.
problem Proving Liouville theorems on manifolds with nonnegative curvature and strictly convex boundary.
method Analyzing smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary.
result Derives Liouville theorems and verifies a conjecture about eigenvalues and inequalities.
Research characterizes traces of Sobolev mappings into manifolds, identifying analytical obstructions and proving surjectivity under certain conditions.
problem Characterizing traces of Sobolev mappings into compact manifolds.
method Analytical and topological methods, including homotopy and fundamental groups.
result Identification of an analytical obstruction when p<m is an integer and πp(N) is non-trivial, and proof of surjectivity under specific conditions. New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.
problem Proving Sobolev inequalities on Kähler manifolds.
method Using a classical Bidaut-Véron and Véron approach.
result Proved new Sobolev inequalities on compact Kähler manifolds with positive Ricci curvature.
Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.
problem Developing inequalities for submanifolds in curved spaces.
method Connecting Fenchel-Willmore and logarithmic Sobolev inequalities for mean-convex submanifolds.
result Established extensions of Fenchel-Willmore inequality and derived new Sobolev-type inequalities.
Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.
problem Optimal functional inequalities and their stability.
method Compactness theorems, characterization of optimizers, and quantitative stability analysis.
result Characterization and stability of optimizers for Sobolev inequalities and their fractional generalizations.
The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.
problem Proving (p,q)-Sobolev and Nash inequalities on Finsler metric measure manifolds. method Global p-Poincaré inequality, (p,q)-Sobolev inequality, Nash inequality derivation. result Established global optimal (p,q)-Sobolev inequality with a sharp constant. Frank and Lieb proved sharp Sobolev inequalities without rearrangements.
problem Proving sharp Sobolev inequalities for function spaces.
method Using conformal covariance and commutator identities from the Fefferman-Graham ambient metric.
result Direct proof of sharp Sobolev inequalities and new nonlinear inequality.
Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.
problem Establishing inequalities for tensor fields on curved manifolds.
method Applying the ABP method to symmetric tensor fields on manifolds with nonnegative sectional curvature.
result Log Sobolev and Michael Simon inequalities for tensor fields.
Sharp inequality for submanifolds in Euclidean space.
problem Logarithmic Sobolev inequality for submanifolds.
method Proved a sharp inequality with mean curvature term.
result Submanifolds in Euclidean space have a logarithmic Sobolev inequality.
Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.
problem Proving the Michael-Simon-Sobolev inequality for submanifolds of codimension 2.
method Optimal transport techniques.
result Sharpness of the inequality for submanifolds of codimension 2.
Sharp Sobolev and Michael-Simon inequalities on curved manifolds.
problem Proving inequalities on manifolds with nonnegative curvature.
method Analyzing asymptotic volume ratio and curvature properties.
result Sharp Sobolev and Michael-Simon inequalities established.
Study noncommutative Sobolev inequalities using quantum state metrics.
problem Establishing Sobolev inequalities in noncommutative settings.
method Generalizing monotone metrics in quantum states.
result Developed new matrix-valued Beckner inequalities.
Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type m…
In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the W1,2 Sobolev inequality along the Ricci flow …
We define the "sum of squares of the wavelengths" of a Riemannian surface (M,g) to be the regularized trace of the inverse of the Laplacian. We normalize by scaling and adding a constant, to obtain a "mass", which is scale invariant and vanishes at the round sphere. This is an anlaog for closed surfaces of the ADM mass…
Proves Sobolev inequality on manifolds with specific curvature properties.
problem Proving Sobolev inequality on manifolds with asymptotically nonnegative Bakry-Émery Ricci curvature.
method Density and Bakry-Émery Ricci curvature.
result Proves Sobolev inequality on manifolds with asymptotically nonnegative Bakry-Émery Ricci curvature.
Proves new Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.
problem Proving Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.
method Using the Alexandrov-Bakelman-Pucci method to prove Michael-Simon type inequalities.
result Extends existing inequalities to the k-Ricci curvature setting and provides isoperimetric inequalities. New Sobolev inequalities found for curved spaces.
problem Sobolev inequalities in curved spaces with specific decay conditions.
method Used ABP method developed by Cabré and Brendle.
result Established Sobolev inequalities for compact domains and submanifolds.
Sharp inequalities derived from fractional Sobolev inequalities on spheres.
problem Deriving sharp inequalities from fractional Sobolev inequalities.
method Alternative elementary argument using fractional Sobolev inequalities.
result Sharp Moser-Trudinger-Onofri inequalities derived from fractional Sobolev inequalities.
The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Generalized Lions' concentration compactness and rigidity results of Sobolev inequalities on singular spaces.
result Almost extremal functions are close to extremal functions on the round sphere and Euclidean Sobolev inequality.
Improved CR Sobolev inequalities on CR sphere established.
problem Establishing CR Sobolev inequalities on CR sphere.
method Nice commutator identities involving CR intertwining operators.
result Simpler proof of existence and classification of minimizers.
Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.
problem Proving logarithmic Sobolev inequalities on manifolds with nonnegative curvature.
method Employing the ABP method developed by Brendle.
result Sharp L2 and Lp logarithmic Sobolev inequalities established. The paper improves CR Sobolev inequalities and classifies minimizers.
problem Higher-order CR Sobolev inequalities on the CR sphere.
method Improvement through vanishing higher order moments of the volume element.
result New direct proof of minimizers' classification and existence of minimizers in C2k(N). The isoperimetric inequality and related inequalities are explored.
problem Proving the isoperimetric inequality and related inequalities.
method Discussing classical and recent proofs.
result Various proofs of the isoperimetric inequality and Sobolev inequality.
Sharp inequalities on curved spaces with bounded curvature.
problem Establishing inequalities on curved spaces with curvature constraints.
method Using Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below.
result Best constants for inequalities on curved spaces with curvature constraints.
Study shows how close functions are to optimal in Riemannian manifolds.
problem Understanding how close functions are to optimal in Riemannian manifolds.
method Analyzes quantitative stability of Sobolev inequalities on compact Riemannian manifolds.
result Functions that nearly saturate a critical Sobolev inequality are quantitatively close to extremal functions.
Poincaré and Sobolev inequalities for differential forms on Heisenberg balls are derived.
problem Establishing inequalities for differential forms on Heisenberg balls.
method Using Rumin's differentials and a global homotopy of Rumin's complex.
result Global homotopy improves differentiability of Rumin forms on bounded geometry contact manifolds.
Sharp Sobolev inequality derived for Riemannian manifolds with bounded Ricci curvature.
problem Deriving a sharp Sobolev inequality for Riemannian manifolds with bounded Ricci curvature.
method Reduction to functions with small volume support, first order uniform asymptotic expansion of isoperimetric profile, local uniform Sobolev inequality.
result Sharp Sobolev inequality for W1,p(M) into Ln−pnp(M) is derived. New Sobolev inequality found for mean convex spacelike submanifolds in Minkowski space.
problem Finding a Sobolev inequality for mean convex spacelike submanifolds in Minkowski space.
method Applying the ABP estimate method to spacelike submanifolds in Rn,1. result Obtained a Sobolev inequality without a mean curvature term for mean convex hypersurfaces.
Upper bounds on constants for Brownian motion with sticky boundary.
problem Bounding constants for Brownian motion with sticky boundary.
method Interpolation approach based on energy interactions and Reilly formula.
result Upper bounds on Poincaré and Logarithmic Sobolev constants.
Sharp fractional Sobolev inequalities on closed manifolds identified.
problem Critical fractional Sobolev embedding on closed Riemannian manifolds.
method Intrinsic heat-kernel based framework, determining optimal coefficients, proving sharp inequalities.
result Sharp p-power inequality and almost sharp inequality established.