Measure Theory/Integral operators

Proposition (Schur's lemma):

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

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

Then the operator

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

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