Proposition (Schur's lemma):
Let k : R n → R n {\displaystyle k:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} be an integration kernel, and suppose that p , q {\displaystyle p,q} are measurable functions such that
Then the operator
is bounded, and explicitly ‖ K ‖ ≤ C 1 C 2 {\displaystyle \|K\|\leq {\sqrt {C_{1}C_{2}}}} .