Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type

Abstract

Abstract. We prove Beurling's theorem for rank 1 Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space.

Beurling's theorem for Riemannian symmetric spaces II

“…Some of the latter theorems (which follow from Beurling's) were proved independently on symmetric spaces in recent years by many authors (see [15,3,12,14,16,6], etc.). The statements of those theorems and proofs of the above implications in the case of rank 1 symmetric spaces can be found in [13]. The proofs in the case of higher rank are similar.…”

“…In Section 5 we have indicated why Beurling's theorem should be regarded as the master theorem in the quantitative uncertainty principle. This paper is a sequel to [13], where we established Beurling's theorem for rank 1 symmetric spaces.…”

“…We can eliminate the Plancherel density µ(λ) in the above inequality basically because µ(λ) −→ ∞ as |λ| −→ ∞ (see [13] for details). Now the Euclidean Beurling theorem implies that F (λ) = f δ (λ) = 0 for all λ ∈ a * .…”

Beurling’s Theorem and $L^p$−$L^q$ Morgan’s Theorem for Step Two Nilpotent Lie Groups

“…Recently an analogue of Beurling's theorem is proved for Riemannian symmetric spaces in [18]. Due to the structural difference, the statement as well as the method of proving the theorem is different and it involves decomposing the hypothesis in K-types and treating each component separately.…”

An Analogue of Beurling–Hörmander’s Theorem Associated with Partial Differential Operators

“…More recently, R.P. Sarkar and J. Sengupta have proved an analogue of BeurlingHörmander's Theorem for rank one Riemannian symmetric spaces of the noncompact type (see [21]). In this paper, for α > 0 we consider the following system of partial differential operators…”