optimality of the time estimates in , see Section \[SectionOptimality\].

Let us also point out that, similarly as in [@je3], our approach can also be applied in the context of fully-discrete schemes: Indeed, to derive observability estimates for fully discrete approximations of that are uniform in both time and space discretization parameters, our approach shows that it is sufficient to prove observability estimates for time-continuous and space semi-discrete approximation schemes that are uniform in the space discretization parameter, see Section \[Sec-Further-Space\] for precise statements. Also note that our results can be used to recover discrete Ingham inequalities in a slightly different setting as the one in [@NegZua06], see Section \[Sec-Further-Ingham\]. We also explain how our strategy applies in the case of weak observability estimates in Section \[Sec-Further-Weak\].

Related results
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Our approach is inspired by several previous works which establish representation formula for solutions of one equation through the solution of another equation. As we have already said, this technique is called subordination (in particular in the context of functional analysis, see e.g. [@Pruss93] and references therein) or transmutation. For instance, the work [@Kannai77] provides estimates on the heat kernel thanks to an analysis of the corresponding wave equation and 