A simple application, sometimes called Schur's lemma: Let X, Y be measurable spaces, K(x,y) a measurable function on X × Y. Suppose Kx(y) = K(x, y) is uniformly interable for a.e. x and Ky(x) = K(x, y) is uniformly interable for a.e. y, with uniform bound C. Then the integral operator
is bounded with it operator norm
This is sometimes called the "strong-type" estimate (as opposed to a weak-type estimate derived from, for example, the Hardy-Littlewood maximal function).
Schur's lemma can be applied to show that an L_p function can be recovered in L_p norm from its Fourier data, via a suitable summability kernel. Also, Young's inequality for convolutions.