Existence of maximizers for Hardy-Littlewood-Sobolev inequalities on the Heisenberg group

In this paper, we investigate the sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group. On one hand, we apply the concentration compactness principle to prove the existence of the maximizers. While the approach here gives a different proof under the special cases discussed in a recent work of Frank and Lieb, we generalize the result to all admissible cases. On the other hand, we provide the upper bounds of sharp constants for these inequalities.Comment: To be published in Indiana University Mathematics Journa

Small scale equidistribution of random eigenbases

We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e. eigenbases) on a compact manifold M. Assume that the group of isometries acts transitively on M and the multiplicity of eigenfrequency tends to infinity at least logarithmically. We prove that, with respect to the natural probability measure on the space of eigenbases, almost surely a random eigenbasis is equidistributed at small scales; furthermore, the scales depend on the growth rate of multiplicity. In particular, this implies that almost surely random eigenbases on the n-dimensional sphere (n>=2) and the n-dimensional tori (n>=5) are equidistributed at polynomial scales.Comment: 13 page

Small scale quantum ergodicity in negatively curved manifolds

The main theorem has been slightly generalized to include a larger class of symbols.Comment: 22 page

Nodal geometry of eigenfunctions on smooth manifolds and hardy-littlewood-sobolev inequalities on the heisenberg group

Part I: Let (M,g) be a n dimensional smooth, compact, and connected Riemannian manifold without boundary, consider the partial differential equation on M: -Δu=Λu, in which Δ is the Laplace-Beltrami operator. That is, u is an eigenfunction with eigenvalue Λ. We analyze the asymptotic behavior of eigenfunctions as Λ go to ∞ (i.e., limit of high energy states) in terms of the following aspects. (1) Local and global properties of eigenfunctions, including several crucial estimates for further investigation. (2) Write the nodal set of u as N={u=0}, estimate the size of N using Hausdorff measure. Particularly, surrounding the conjecture that the n-1 dimensional Hausdorff measure is comparable to square root of Λ, we discuss separately on lower bounds and upper bounds. (3) BMO (bounded mean oscillation) estimates of eigenfunctions, and local geometric estimates of nodal domains (connected components of nonzero region). (4) A covering lemma which is used in the above estimates, it is of independent interest, and we also propose a conjecture concerning its sharp version. Part II: O the Heisenberg group with homogeneous dimension Q=2n+2, we study the Hardy-Littlewood-Sobolev (HLS) inequality, and particularly its sharp version. Weighted Hardy-Littlewood-Sobolev inequalities with different weights shall also be investigated, and we solve the following problems. (1) Establish the existence results of maximizers. (2) Provide a upper bound of sharp constants

Uniformly bounded spherical harmonics and quantum ergodicity on S2S^2

On the two-dimensional unit sphere, we construct uniformly bounded spherical harmonics of arbitrary degree, under a condition of point distribution on the sphere. It extends the results on odd-dimensional spheres by Bourgain, Shiffman, and Marzo-Ortega-Cerda. Moreover, we show that the spherical harmonics constructed in this paper are equidistributed in the phase space, i.e., they are quantum ergodic. It provides the first example of Laplacian eigenfunctions which are both uniformly bounded and quantum ergodic.Comment: 16 page