Proposition (Schur's lemma):

Let ${\displaystyle k:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ be an integration kernel, and suppose that ${\displaystyle p,q}$ are measurable functions such that

${\displaystyle \int _{\mathbb {R} ^{n}}|k(x,y)p(x)|dx\leq C_{1}q(y)}$ and ${\displaystyle \int _{\mathbb {R} ^{n}}|k(x,y)q(y)|dy\leq C_{2}q(x)}$.

Then the operator

${\displaystyle Kf(x):=\int _{\mathbb {R} ^{n}}k(x,y)f(y)dy}$

is bounded, and explicitly ${\displaystyle \|K\|\leq {\sqrt {C_{1}C_{2}}}}$.