This paper studies observability inequalities for heat equations on both bounded domains and the whole space \({\mathbb {R}}^d\) . The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge functions closely related to the heat kernel. On a bounded domain, we derive the observability inequality for observation sets of positive log-type Hausdorff content. Notably, the aforementioned inequality holds not only for all sets with Hausdorff dimension s for any \(s\in (d-1,d]\) , but also for certain sets of Hausdorff dimension exactly \(d-1\) . On the whole space \({\mathbb {R}}^d\) , we establish the observability inequality for observation sets that are thick at the scale of the log-type Hausdorff content. Furthermore, we prove that for the 1-dimensional heat equation on an interval, the Hausdorff content we have chosen is an optimal scale for the observability inequality. To obtain these observability inequalities, we use the adapted Lebeau-Robbiano strategy of Duyckaerts and Miller (J. Funct. Anal. 2012). For this purpose, we prove the following results at scale of the log-type Hausdorff content, the former being derived from the latter: we establish a fractal version of spectral inequality/Logvinenko-Sereda uncertainty principle; we develop a quantitative propagation of smallness of analytic functions; we build up a Remez-type inequality; and more fundamentally, we provide an upper bound for the log-type Hausdorff content of a set where a monic polynomial is small, based on an estimate by Lubinsky (J. Inequal. Appl. 1997), which is ultimately traced back to the classical Cartan Lemma. In addition, we set up a capacity-based slicing lemma (related to the log-type gauge functions) and establish a quantitative relationship between Hausdorff contents and capacities. These tools are crucial in the studies of the aforementioned propagation of smallness in high-dimensional situations.
