= Simon absorption lemma
{c}
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 $F\ge0$ be bounded on $[r_0,R]$ and suppose, for every $r_0\le r<s\le R$,
$$
F(r)\le\theta F(s)+A(s-r)^{-p}+B,\qquad 0<\theta<1,\quad p>0,\quad A,B\ge0.
$$
Then
$$
F(r_0)\le C_{\theta,p}A(R-r_0)^{-p}+\frac{B}{1-\theta}.
$$
To 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 $\theta^mF(r_m)$ 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.
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