Distributions singulières, dimension du support et symétrie de la transformation de Fourier (Q462789)

A theorem of Frostman characterises the Hausdorff dimension, dim \(K\), of a compact subset \(K\) of the circle \(\mathbb{T}\) in terms of measures supported in \(K\). In Fourier analytic terms, the result says that \(\text{dim}K\) is the supremum of all \(\alpha\) such that there is a measure \(\mu\) supported in \(K\) with \(\Sigma_{n\in \mathbb{Z}} \frac{|\hat{\mu}(n)|^2}{(1+|n|^{1-\alpha})}<\infty\). Beurling proved the following theorem showing that the measures in Frostman's result may be replaced by (Schwartz) distributions: For \(0<\alpha \leq 1\), if there is a distribution \(S\) with support in \(K\) such that \(\Sigma_{n\in \mathbb{Z}} \frac{|\hat{S}(n)|^2}{(1+|n|^{1-\alpha})}<\infty\), then dim \(K \geq \alpha\). One of the main results of the current paper is the strengthening of this theorem, showing that it is sufficient to assume a one-sided estimate for \(\hat{S}\): If there is a distribution \(S\) with support in \(K\) such that \(\Sigma_{n<0} \frac{|\hat{S}(n)|^2}{(1+|n|^{1-\alpha})}<\infty\), then \(\text{dim}K \geq \alpha\). In an earlier work, the authors have proved the following remarkable result: there is a nontrivial singular pseudo-function \(S\) such that \(\hat S\in \ell^2(\mathbb{Z}_{-})\). (Recall that \(S\) is singular means \(\hat S(n) = 0\) for \(n\geq 0\) and that \(S\) is a pseudo-function means \(\lim_{|n|\rightarrow \infty} \hat S(n) = 0\).) The second important result of the paper is on the size of the support of \(S\) in this result: there is a nontrivial singular pseudo-function \(S\) such that \(\hat S\in \ell^2(\mathbb{Z}_{-})\) supported on a compact set of finite \(t \log 1/t\)-Hausdorff measure (and this is the critical value). The final result shows the possibility of certain non-symmetry. It involves the Fourier dimension: \(\text{dim}_F(K):= \sup \alpha\), the supremum being taken over all \(\alpha\) for which there is a positive measure \(\mu\) supported in \(K\) with \(|\hat{\mu}(n)|^2 = O(1/n^{\alpha})\). Here is the result: let \(\text{dim}_F(K)=d>0\). Then for every \(p>2/d\), there is a complex measure \(\nu \in M(K)\) such that \(\hat{\nu}\in \ell^p(\mathbb{Z}_{-})\) and \(\hat{\nu}\notin \ell^p(\mathbb{Z}_{+})\). Many open questions are raised at appropriate places in the paper.