A note on tempered measures

Abstract

The relation between tempered distributions and measures is analysed and clarified. While this is straightforward for positive measures, it is surprisingly subtle for signed or complex measures.

Tempered distributions with translation bounded measure as Fourier transform and the generalized Eberlein decomposition

“…It is easy to see that this is equivalent to $$\mu : C_{\mathsf {c}}^\infty ({\mathbb {R}}^d) \rightarrow {\mathbb {C}}$$ being continuous with respect to the Schwartz topology induced by the embedding $$C_{\mathsf {c}}^\infty ({\mathbb {R}}^d) \hookrightarrow {\textsf {S}}({\mathbb {R}}^d)$$ (see, e.g., [1]). Remark Let us recall that a measure μ is called (1) slowly increasing if there exists a polynomial $$P \in {\mathbb {R}}[X_1,\ldots ,X_d]$$ such that $$$$\begin{equation*} \int _{{\mathbb {R}}^d} \frac{1}{1+|P(x)|}\ \mbox{d}|\mu | (x) < \infty \,, \end{equation*}$$$$ (2) strongly tempered if $$|\mu |$$ is tempered. A measure μ is strongly tempered if and only if it is slowly increasing [8, Theorem 2.6]. In this case, μ is tempered, see, for example, [1, p. 47] or [8, Lemma 2.3].…”

“…Remark Let us recall that a measure μ is called (1) slowly increasing if there exists a polynomial $$P \in {\mathbb {R}}[X_1,\ldots ,X_d]$$ such that $$$$\begin{equation*} \int _{{\mathbb {R}}^d} \frac{1}{1+|P(x)|}\ \mbox{d}|\mu | (x) < \infty \,, \end{equation*}$$$$ (2) strongly tempered if $$|\mu |$$ is tempered. A measure μ is strongly tempered if and only if it is slowly increasing [8, Theorem 2.6]. In this case, μ is tempered, see, for example, [1, p. 47] or [8, Lemma 2.3]. The converse of the last claim holds only for positive measures: Any positive tempered measure is strongly tempered [27, p. 242] [13, Theorem 2.1], and there are examples of signed tempered measures, which are not strongly tempered [1, 8].…”

“…In this case, μ is tempered, see, for example, [1, p. 47] or [8, Lemma 2.3]. The converse of the last claim holds only for positive measures: Any positive tempered measure is strongly tempered [27, p. 242] [13, Theorem 2.1], and there are examples of signed tempered measures, which are not strongly tempered [1, 8]. Also note that any translation bounded measure is slowly increasing and hence tempered [1, Theorem 7.1].…”

“…Also note that any translation bounded measure is slowly increasing and hence tempered [1, Theorem 7.1]. For a review of properties and relationships among these concepts, we refer the reader to [8].…”