Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn

The main results of this paper concern sharp constants for the Moser‐Trudinger inequalities on spheres in complex space ?ⁿ. We derive Moser‐Trudinger inequalities for smooth functions and holomorphic functions with different sharp constants (see Theorem 1.1). The sharp Moser‐Trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the CR setting. Though there is a close connection in spirit between inequalities proven here on complex spheres and those on the Heisenberg group for functions with compact support in any finite domain proven earlier by the same authors [17], derivation of the sharp constants for Moser‐Trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the Heisenberg group. Variants of Moser‐Onofri‐type inequalities are also given on complex spheres as applications of our sharp inequalities (see Theorems 1.2 and 1.3). One of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients (see Theorem 1.4). © 2004 Wiley Periodicals, Inc.     

Balls have the worst best Sobolev inequalities. Part II: variants and extensions

We continue our previous study of sharp Sobolev-type inequalities by means of optimal transport, started in (Maggi and Villani J. Geom. Anal. 15(1), 83–121 (2005)). In the present paper, we extend our results in various directions, including Gagliardo–Nirenberg, Faber–Krahn, logarithmic-Sobolev or Moser–Trudinger inequalities with trace terms. We also identify a class of domains for which there is no need for a trace term to cast the Sobolev inequality.     

Sharp Weighted Multidimensional Integral Inequalities for Monotone Functions

We prove sharp weighted inequalities for general integral operators acting on monotone functions of several variables. We extend previous results in one dimension, and also those in higher dimension for particular choices of the weights (power weights, etc.). We introduce a new kind of conditions, which take into account the more complicated structure of monotone functions in dimension n > 1, and give an example that shows how intervals are not enough to characterize the boundedness of the operators (contrary to what happens for n = 1). We also give several applications of our techniques.     

Direct approach to the problem of strong local minima in calculus of variations

Let (M,g) be a two-dimensional compact Riemannian manifold. In this paper, we use the method of blowing up analysis to prove several Moser–Trudinger type inequalities for vector bundle over (M,g). We also derive an upper bound of such inequalities under the assumption that blowing up occurs. The research of the second author was partially supported by NSFC grant and the Foundation of Shanghai for Priority Academic Discipline.     

Fractional Sobolev,Moser–Trudinger,Morrey–Sobolev inequalities under Lorentz norms

We consider the Sobolev type inequalities under Lorentz norms on bounded open domains for fractional derivatives (−∆) ^s/2 in the following three cases: n > ps, n = ps, and n < ps, whence establishing the weak type Sobolev inequalities, Moser–Trudinger and Morrey–Sobolev inequalities for fractional derivatives in Lorentz norms. Applying these inequalities, we obtain the trace forms of six related functional inequalities. Bibliography: 44 titles.     

On Hardy-Littlewood-Pólya and Taikov type inequalities for multiple operators in Hilbert spaces

We present a unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Pólya and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemannian manifolds and derive the well-known Taikov and Hardy-Littlewood-Pólya inequalities for functions defined on the d-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of another class. In addition, we establish sharp Solyar type inequalities for unbounded closed operators with closed range.

     

Isocapacity estimates for Hessian operators

Through a new powerful potential-theoretic analysis, this paper is devoted to discovering the geometrically equivalent isocapacity forms of Chou–Wang's Sobolev type inequality and Tian–Wang's Moser–Trudinger type inequality for the fully nonlinear 1≤k≤n/2$1\le k\le n/2$ Hessian operators.     

Affine Moser–Trudinger and Morrey–Sobolev inequalities

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L ⁿ energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L ⁿ Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L ^p energy, with p > n, is replaced by the affine energy.     

A thermodynamical formalism for Monge–Ampère equations,Moser–Trudinger inequalities and Kähler–Einstein metrics

We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kähler manifold X. This functional can be seen as a generalization of Mabuchi?s K-energy functional and its twisted versions to more singular situations. Applications to Monge–Ampère equations of mean field type, twisted Kähler–Einstein metrics and Moser–Trudinger type inequalities on Kähler manifolds are given. Tian?s α-invariant is generalized to singular measures, allowing in particular a proof of the existence of Kähler–Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning Kähler metrics with conical singularities confirms a recent conjecture of Donaldson). As another application we show that if the Calabi flow in the (anti-)canonical class exists for all times then it converges to a Kähler–Einstein metric, when a unique one exists, which is in line with a well-known conjecture.     

Multilinear commutators of BMO functions and multilinear singular integrals with non-smooth kernels

This paper is concerned with certain multilinear commutators of BMO functions and multilinear singular integral operators with non-smooth kernels. By the sharp maximal functions estimates, the weighted norm inequalities for this kind of commutators are established.     

The heat flow with a critical exponential nonlinearity

We analyze the possible concentration behavior of heat flows related to the Moser–Trudinger energy and derive quantization results completely analogous to the quantization results for solutions of the corresponding elliptic equation. As an application of our results we obtain the existence of critical points of the Moser–Trudinger energy in a supercritical regime.     

Mixed Norm Inequalities for Directional Operators Associated to Potentials

We study mixed norm inequalities for directional operators which appear applying the method of rotations to homogeneous operators with variable kernel and with the homogeneity of Riesz potentials. The results are sharp for a range of values of the parameter and for all its values when the inequalities are restricted to radial functions.     

Logarithmic estimates for the Hilbert transform and the Riesz projection

We study sharp Llog L inequalities for the Hilbert transform and the Riesz projection acting on vector-valued functions defined on the unit circle.     

A sharp form of trace Moser–Trudinger inequality on compact Riemannian surface with boundary

In this paper, a sharp form of trace Moser–Trudinger inequality is established on the boundary of compact Riemannian surface, and the existence of extremal function is proved via the method of blowing up analysis.     

Sharp inequalities of singular values of smooth kernels

New inequalities of singular values of the integral operators with smoothL ² kernels are obtained and shown by examples to be sharp if the kernels satisfy also certain boundary conditions. These results are based on an idea of Gohberg-Krein by which the singular values of the integral operators are interrelated to the eigenvalues of some two point boundary value problems.Dedicate to Professor Ky Fan on the occasion of his 85th birthday     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators

This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For L in some class of elliptic operators, we study weighted norm Lp inequalities for singular “non-integral” operators arising from L; those are the operators φ(L) for bounded holomorphic functions φ, the Riesz transforms ∇L−1/2 (or (−Δ)^1/2L−1/2) and its inverse L1/2(−Δ)^−1/2, some quadratic functionals gL and GL of Littlewood-Paley-Stein type and also some vector-valued inequalities such as the ones involved for maximal Lp-regularity. For each, we obtain sharp or nearly sharp ranges of p using the general theory for boundedness of Part I and the off-diagonal estimates of Part II. We also obtain commutator results with BMO functions.     

Differentiated shift-invariant integral operators,univariate case

In a recent paper [1] the author, along with H. Gonska, introduced some wavelet type integral operators over the whole real line and studied their properties such as shift-invariance, global smoothness preservation, convergence to the unit, and preservation of probability distribution functions. These operators are very general and they are introduced through a convolution-like iteration of another general operator with a scaling type function. In this paper the author provides sufficient conditions, so that the derivatives of the above operators enjoy the same nice properties as their originals. A sufficient condition is also given so that the “global smoothness preservation” related inequality becomes sharp. At the end several applications are given, where the derivatives of the very general specialized operators are shown to fulfill all the above properties. In particular it is shown that they preserve continuous probability density functions.     

New Improved Moser�CTrudinger Inequalities and Singular Liouville Equations on Compact Surfaces

We consider a singular Liouville equation on a compact surface, arising from the study of Chern–Simons vortices in a self-dual regime. Using new improved versions of the Moser–Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.     

Weighted integral and integro-differential inequalities

A general weighted integral inequality for two continuous functions on an interval [a,b] is presented. The equality conditions are given. This result implies the new inequalities for the incomplete beta and gamma functions as well as the related estimates for the confluent hypergeometric function, error function, and Dawson's integral. Also it implies various weighted integro-differential inequalities, those of the Opial type included, and some inequalities which involve the Erdélyi–Kober and Riemann–Liouville fractional integrals.     

Extremal functions for sharp Moser–Trudinger type inequalities in the whole space RN

In this paper, we prove the existence of maximizers for the sharp Moser–Trudinger type inequalities in whole space ${\mathbb{R}}^N$, $N\ge 2$ with more general nonlinearity. The main key in our proof is a precise estimate of the concentrating level of the Moser–Trudinger functional associated with our inequalities on the normalized concentrating sequences. This estimate solves a heavily non-trivial and open problem related to the sharp Moser–Trudinger inequality. Our method gives an alternative proof of the existence of maximizers for the Moser–Trudinger inequality and singular Moser–Trudinger inequality in whole space ${\mathbb{R}}^N$ due to Li and Ruf [30] and Li and Yang [31] without using blow-up analysis argument.