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- Jun 20, 2014

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Show that if $(X,\mathcal{M},\mu)$, $(Y,\mathcal{N},\nu)$ are finite measure spaces, $1 < p < \infty$, and $K$ is a measurable function on $X\times Y$, there is a bounded integral operator $I(K) : \mathscr{L}^p(\nu) \to \mathscr{L}^p(\mu)$ given by

$$I(K)(f) :x \mapsto \int_Y K(x,y)\,f(y)\, d\nu(y)\quad (f\in \mathscr{L}^p(\mu)),$$

provided that the kernel $K$ satisfies the conditions $\sup_x \int_Y \lvert K(x,y)\rvert\, d\nu(y) < \infty$ and $\sup_y \int_X \lvert K(x,y)\rvert \, d\mu(x) < \infty$.

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