Restricted Mean Value Property on Riemannian manifolds and Carleson’s problem on Damek-Ricci spaces

Abstract

One of the most key aspects of Harmonic Analysis is the study of the boundary behaviour of solutions of Partial Differential Equations. The central theme of this thesis is to study such problems for some suitable classes of Riemannian manifolds. We first study the restricted mean value property (RMVP for short) in the context of Riemannian manifolds. For a pre-compact domain D with smooth boundary in a Riemannian manifold M, we first prove that any real-valued, bounded, continuous function u satisfying the RMVP in D, having boundary limits in a full measure subset of the boundary, is harmonic in D. We then prove another related result on D while u is any real-valued, continuous function satisfying the RMVP in D such that a suitable condition on the radius function is satisfied. We also obtain a global result for when M is a Hadamard manifold of pinched negative curvature: if u is a real-valued, bounded, continuous function which satisfies the RMVP and admits boundary limit on a full measure subset of the Gromov boundary ∂M, with respect to the harmonic measures thereon, then u is harmonic in M. We next shift our attention to the Carleson’s problem for the Schr¨odinger equation on Damek-Ricci spaces S. In this direction, we obtain the complete classification of the pairs (q, β) ∈ [1,∞] × [0,∞) such that the local (in time) Schr¨odinger maximal function satisfies the boundedness properties Hβ → Lq loc, restricted to radial functions on S. Consequently, we obtain the sharp regularity threshold β ≥ 1/4 for the Carleson’s problem for radial initial data on S. To address the well-posedness of the initial value problem, we also study some global (in space) Schr¨odinger maximal estimates on S. A corresponding sharp result is also proved for the 3-dimensional real hyperbolic space H3. Then we focus on some generalizations of the Carleson’s problem for general approach regions. We first obtain a negative result: given any compact, geodesic annulus A ⊂ H3, there exists a radial f ∈ H1/2(H3) such that its Schr¨odinger propagation is continuous on A ×(0,∞), but blows up everywhere on A , through any wider approach region. Then we introduce a class of tangential curves and prove that along these curves, the Schr¨odinger propagation converges pointwise a.e. on Damek-Ricci spaces S, for initial data satisfying the same regularity threshold β ≥ 1/4, as in the case of vertical convergence. Then using this result on S, we obtain two pointwise convergence results of the Schr¨odinger propagation along curves in the Euclidean setting.