#### Access Type

Open Access Dissertation

#### Date of Award

January 2012

#### Degree Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics

#### First Advisor

Guozhen Lu

#### Abstract

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.

#### Recommended Citation

Han, Xiaolong, "Nodal geometry of eigenfunctions on smooth manifolds and hardy-littlewood-sobolev inequalities on the heisenberg group" (2012). *Wayne State University Dissertations*. 507.

http://digitalcommons.wayne.edu/oa_dissertations/507