 Simon absorption lemma

The Simon absorption lemma absorbs a sufficiently small multiple of a stronger local quantity by iterating over nested regions. A useful abstract form is the following. Let $F\ge 0$ be bounded on $[r_0,R]$ and suppose, for every $r_0\le r<s\le R$,ThenTo prove this, choose $q$ with ${\theta }^{1/p}<q<1$ and let $r_j=r_0+(R-r_0)(1-q^j)$. Iterating gives the geometric series with ratio $\theta q^{-p}<1$, since $r_{j+1}-r_j=(R-r_0)(1-q)q^j$. The residual ${\theta }^{m}F(r_m)$ tends to zero by boundedness; the $B$ terms sum to $B/(1-\theta )$.

In a Schauder estimate, $F(r)$ can be a scaled Hölder seminorm on a ball and $A$ a lower-order data bound. Covering arguments produce the nested-radius inequality before this absorption step. Boundary versions use regions cut by a half-space and need a compatible covering; the mere appearance of a small coefficient in one inequality does not by itself justify absorption at the same radius.