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.
","abstract_html":"<p>Part I: Let (M,g) be a n dimensional smooth, compact, and connected Riemannian manifold without boundary, consider the partial differential equation on M:</p> <p>-Δu=Λu,</p> <p>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.</p> <p>(1) Local and global properties of eigenfunctions, including several crucial estimates for further investigation.</p> <p>(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.</p> <p>(3) BMO (bounded mean oscillation) estimates of eigenfunctions, and local geometric estimates of nodal domains (connected components of nonzero region).</p> <p>(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.</p> <p>Part II: O the Heisenberg group with homogeneous dimension Q=2n+2, we study the Hardy-Littlewood-Sobolev (HLS) inequality,</p> <p>and particularly its sharp version. Weighted Hardy-Littlewood-Sobolev inequalities with different weights shall also be investigated, and we solve the following problems.</p> <p>(1) Establish the existence results of maximizers.</p> <p>(2) Provide a upper bound of sharp constants.</p>","abstract_has_math":false,"creators":["Han, Xiaolong"],"institution":null,"degree_name":"Ph.D.","degree_level":"Open Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Guozhen Lu"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-02T08:00:00Z","date_published":"2012-01-02T08:00:00Z","updated_at":"2026-07-24T05:59:04Z","subjects":["eigenfunctions of Laplacian on smooth manifolds, existence of maximizers, geometric estimates of nodal sets and nodal domains, Hardy-Littlewood-Sobolev inequalities, Heisenberg group","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wayne.edu/oa_dissertations/507","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Guozhen Lu"]},{"key":"dc:creator","label":"Author","values":["Han, Xiaolong"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2012-01-01T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["eigenfunctions of Laplacian on smooth manifolds, existence of maximizers, geometric estimates of nodal sets and nodal domains, Hardy-Littlewood-Sobolev inequalities, Heisenberg group","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wayne.edu/oa_dissertations/507"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Nodal geometry of eigenfunctions on smooth manifolds and hardy-littlewood-sobolev inequalities on the heisenberg group"]}]}],"canonical_facts":{"dc:contributor":["Guozhen Lu"],"dc:creator":["Han, Xiaolong"],"dc:date.available":["2012-01-01T08:00:00Z"],"dc:description.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.
"],"dc:identifier":["https://digitalcommons.wayne.edu/oa_dissertations/507"],"dc:subject":["eigenfunctions of Laplacian on smooth manifolds, existence of maximizers, geometric estimates of nodal sets and nodal domains, Hardy-Littlewood-Sobolev inequalities, Heisenberg group","Mathematics"],"dc:title":["Nodal geometry of eigenfunctions on smooth manifolds and hardy-littlewood-sobolev inequalities on the heisenberg group"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Open Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T05:59:04Z"}