Notes for "Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic o...

H. Kekkonen, M. Lassas, S. Siltanen, Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators, Inverse Problems, 32, 2016, 085005

Page 10, line 6, Section 3 Generalised random variables

The connection between and is

Notes:
Consider , then we have

where . Since and , we obtain

Hence, we find that

which implies

Page 11, line (from end) 7, Section 3.2

Condition guarantees that .

Notes:
Taking be eigensystem of on , we have

which is just the required estimation.
Here, in this part, we may see () and , which coincides with formula (3.3) and (3.4) on page 10. Taking in the first formula on page 12, we have

Question: The operator is a self-adjoint elliptic operator with smooth coefficients (defined on closed manifold), the eigenvalues are irrelevant to the definition function space of the operator ?

Proof of Lemma 3, Page 15

and we can write

Notes:
Here, we assume that . By my understanding, the equality means that for we have

Since , we obtain and

holds for every . Because for any , we can deduce that the corresponding is continuous. Hence, we find that

holds on for any . That is to say,

holds on space , which implies .

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2020/02