Derivative loss in an estimate

An estimate for $T$ has loss of $r$ derivatives if controlling the output through order $k$ requires input control through order $k+r$ , for example

p_{K,k}(Tf)\le C_{K,k}\,p_{K',k+r}(f),

where $p_{K,k}$ is a derivative seminorm. The source and target spaces, sets $K,K'$ , and permitted dependence of $C_{K,k}$ are part of the estimate.

Example and a different use of loss

Differentiation $D$ obeys $\|Df\|_{C^k}\le\|f\|_{C^{k+1}}$ . An inverse PDE operator may also lose derivatives if its estimate requires more regularity of the forcing than it returns for the solution.

A factor $\varepsilon^{-r}$ in a parameter estimate is a loss of powers of a small scale, not necessarily a loss of differentiability. Both losses can occur together and must be tracked separately.

Finite compositions

Derivative requirements on a finite acyclic computation graph remain finite at each fixed output order, while possibly growing with the number of stages.