Theorems of Ingham and Chernoff on Riemannian symmetric spaces of noncompact type

Abstract

An $L^2$ version of the celebrated Denjoy-Carleman theorem regarding quasi-analytic functions was proved by Chernoff on ${\mathbb{R}}^d$ using iterates of the Laplacian. In 1934 Ingham used the classical Denjoy-Carleman theorem to relate the decay of Fourier transform and quasi-analyticity of an integrable function on $\mathbb{R}$. In this paper, we prove analogues of the theorems of Chernoff and Ingham for Riemannian symmetric spaces of noncompact type and show that the theorem of Ingham follows from that of Chernoff.

Introduction

In this paper, we will concern ourselves with the classical problem of determining the relationship between the rate of decay at infinity of the Fourier transform of an integrable function and the size of the support of the function in the context of a Riemannian symmetric space $X=G/K$ of noncompact type. Here G is a connected noncompact semisimple Lie group with finite center and K is a maximal compact subgroup of G. To understand the context better let us consider the following well-known statement: suppose $f\in L^1(X)$ and its Fourier transform satisfies the estimate

$$|\tilde{f}(\lambda ,k)|\le C\widehat{h_t}(\lambda ),\lambda \in {\mathfrak{a}}^{⁎},k\in K,$$

for some $t>0$, where $h_t$ is the heat kernel for the Laplace-Beltrami operator on X, and C is a positive number. If f vanishes on a nonempty open subset of X then f vanishes identically (we refer the reader to Section 2 for the meaning of the relevant symbols). The Euclidean version of this statement can be easily proved by showing that the very rapid decay of the Fourier transform imposes real analyticity on the function. However, the proof of this statement on symmetric spaces is not so elementary. In fact, the proof that f is still a real analytic function involves a nontrivial result such as the Kotake-Narasimhan theorem [23] and can be found in [25, p. 237]. For the real line $\mathbb{R}$ it is well-known that under much slower decay of the Fourier transform $\hat{f}$, an integrable function $f\in L^1(\mathbb{R})$ vanishes identically if it vanishes on a nonempty open set. There are classical results available in the literature which are devoted to this theme and the key idea they depend on is the notion of quasi-analyticity [20], [21], [22], [24], [28]. It is a remarkable fact that most of these results provide rather sharp conditions on the decay of the Fourier transform which impose quasi-analyticity on the function. In recent times, some of the results cited above have reappeared in the context of the study of the problem of uniqueness of solution of Schrödinger equation on Euclidean spaces [13]. This motivated us to have a fresh look at these results and explore the possibility of extending them beyond Euclidean spaces. Very recently the first and the third authors could extend the main result of [24] to Riemannian symmetric spaces X of noncompact type (see [3]) and in this paper, our objective is to do the same with the result of Ingham [21]. In its original version, Ingham's theorem concerns functions defined on the real line. A recent extension to ${\mathbb{R}}^d$ states the following.

Theorem 1.1 [5], Theorem 2.2

Let $\theta :{\mathbb{R}}^d\to [0,\infty )$ be a radially decreasing function with ${\lim }_{\Vert \xi \Vert \to \infty }\theta (\xi )=0$ and set

$$I=\underset{\Vert \xi \Vert \ge 1}{\int }\frac{\theta (\xi )}{{\Vert \xi \Vert }^d}d\xi .$$

• Let $f\in L^p({\mathbb{R}}^d)$, $p\in [1,2]$, be such that its Fourier transform $\mathcal{F}f$ satisfies the estimate

  $$|\mathcal{F}f(\xi )|\le C{e}^{-\theta (\xi )\Vert \xi \Vert },\mathit{\text{for almost every}}\xi \in {\mathbb{R}}^d.$$

  If I is infinite and f vanishes on a nonempty open set in ${\mathbb{R}}^d$ then f is the zero function on ${\mathbb{R}}^d$.

• If I is finite then given any positive number L, there exists a nontrivial radial function $f\in C_c^{\infty }({\mathbb{R}}^d)$ supported in $B(0,L)$, satisfying the estimate (1.2).

Here, the Fourier transform is given by the integral

$$\mathcal{F}f(\xi )=\underset{{\mathbb{R}}^d}{\int }f(x)e^{-2\pi ix\cdot \xi }dx,$$

with $x\cdot \xi$ being the Euclidean inner product of the vectors x and ξ.

It is important to observe that there is no assumption on the function θ which implies that the function $\xi \to \theta (\xi )\Vert \xi \Vert$ is radially increasing and hence the condition (1.2) does not automatically imply pointwise decay of the Fourier transform. However, the assumption that the integral I diverges would imply that

$$\underset{\Vert \xi \Vert \to \infty }{\lim \sup }\theta (\xi )\Vert \xi \Vert =\infty .$$

For $d=1$, using certain assumption on the function θ, Ingham [21, p. 30] showed that the condition (1.2) together with the divergence of the integral I implies that

$$\sum _{m=1}^{\infty }{\Vert \frac{d^{m}f}{d{x}^m}\Vert }_{\infty }^{-\frac{1}{m}}=\infty .$$

It then follows from the Denjoy-Carleman theorem ([30, Theorem 19.11]) that f is quasi-analytic and hence vanishes identically under the assumption that it vanishes on a nonempty open set. The proof for a general θ can then be reduced to the special case mentioned above. It is fairly natural to anticipate that an extension of Ingham's result to ${\mathbb{R}}^d$ or to a Riemannian symmetric space X will involve a suitable extension of the Denjoy-Carleman theorem to these spaces. One such result was obtained by Bochner in [6, Theorem 3] which provides an analogue of the Denjoy-Carleman theorem for ${\mathbb{R}}^d$ involving iterates of the Laplace operator. Chernoff [9, Theorem 6.1] proved the following important variant of the result of Bochner.

Theorem 1.2

Let $f:{\mathbb{R}}^d\to \mathbb{C}$ be a smooth function. Suppose that for all $m\in \mathbb{N}\cup \{0\}$, ${\mathit{\Delta }}_{{\mathbb{R}}^d}^{m}f\in L^2({\mathbb{R}}^d)$ and

$$\sum _{m\in \mathbb{N}}{\Vert {\mathit{\Delta }}_{{\mathbb{R}}^d}^{m}f\Vert }_2^{-\frac{1}{2m}}=\infty .$$

If f and all its partial derivatives vanish at the origin then f is identically zero.

In the above ${\mathrm{\Delta }}_{{\mathbb{R}}^d}$ denotes the usual Laplacian on ${\mathbb{R}}^d$. It is not hard to show that Theorem 1.2 can be used suitably to prove Ingham's theorem (Theorem 1.1). It is thus natural to look for an analogue of Chernoff's theorem on Riemannian symmetric spaces of noncompact type and then try to use it to prove an analogue of Theorem 1.1. Our first result in this paper is the following weaker version of Theorem 1.2 for a Riemannian symmetric space X of noncompact type. Here and later, Δ denotes the Laplace-Beltrami operator on X.

Theorem 1.3

Let $f\in C^{\infty }(G/K)$ be such that ${\mathit{\Delta }}^{m}f\in L^2(G/K)$, for all $m\in \mathbb{N}\cup \{0\}$ and

$$\sum _{m=1}^{\infty }{\Vert {\mathit{\Delta }}^{m}f\Vert }_2^{-\frac{1}{2m}}=\infty .$$

If f vanishes on any nonempty open set in $G/K$ then f is identically zero.

Though the assumption that f vanishes on a nonempty open set drastically changes the nature of the theorem it is still sufficient to help us in proving an analogue of Ingham's theorem on symmetric spaces which is the main goal of this paper. The main idea of the proof is to suitably use the connection between the Carleman type condition (1.3) and results regarding polynomial approximation proved in [11]. For a discussion on quasi-analyticity and polynomial approximations on Lie groups, we refer the reader to [12] and the references therein.

In the setting of a Riemannian symmetric space X, the vanishing of G-invariant differential operators applied to a function at a point of X is analogous to the vanishing of partial derivatives of a function at a point of ${\mathbb{R}}^d$. For the rank one symmetric spaces one knows that such differential operators are polynomials in the Laplace-Beltrami operator Δ. It was observed in [9] that Theorem 1.2 is false under the weaker assumption of vanishing of ${\mathrm{\Delta }}_{{\mathbb{R}}^d}^{m}f(0)$, $m\in \mathbb{N}\cup \{0\}$ instead of vanishing of all partial derivatives of f at zero. Likewise, we also construct an example (see Example 3.7) to show the impossibility of proving an exact analogue of Chernoff's theorem on symmetric spaces if we restrict ourselves only to the class of G-invariant differential operators on X.

Our next result in this paper is the following variant of the theorem of Ingham (Theorem 1.1) on a Riemannian symmetric space X of noncompact type with $\text{rank}(X)=d\ge 1$.

Theorem 1.4

Let $\theta :[0,\infty )\to [0,\infty )$ be a decreasing function with ${\lim }_{r\to \infty }\theta (r)=0$ and set

$$I=\underset{\{\lambda \in {\mathfrak{a}}_{+}^{⁎}|{\Vert \lambda \Vert }_B\ge 1\}}{\int }\frac{\theta ({\Vert \lambda \Vert }_B)}{{\Vert \lambda \Vert }_B^d}d\lambda .$$

• Suppose $f\in L^1(X)$ and its Fourier transform $\tilde{f}$ satisfies the estimate

  $$\underset{{\mathfrak{a}}^{⁎}\times K}{\int }|\tilde{f}(\lambda ,k)|e^{\theta ({\Vert \lambda \Vert }_B){\Vert \lambda \Vert }_B}|\mathbf{c}(\lambda ){|}^{-2}d\lambda dk<\infty .$$

  If f vanishes on a nonempty open set in X and I is infinite then f is the zero function.

• If I is finite then given any positive number L, there exists a nontrivial $f\in C_c^{\infty }(G//K)$ supported in $\mathcal{B}(o,L)$ satisfying the estimate (1.5).

Theorem 1.4 is then used to prove Theorem 4.2 which is an exact analogue of Ingham's theorem for Riemannian symmetric spaces of noncompact type. This completes our goal.

This paper is organized as follows: In the next section, we will recall the required preliminaries regarding harmonic analysis on Riemannian symmetric spaces of noncompact type. We will prove Theorem 1.3 and Theorem 1.4 in section 3 and section 4 respectively.