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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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Searching for inequalities relating a convolution-type integral of functions of modulus less...

Suppose $f(x,y)$ and $g(x,y)$ are both measurable functions from $[0,1]\times[0,1]\to \mathbb{C}$ with $|f|,|g|<1$, and let $h(x,y)=\int_{0}^1 f(x,z)g(z,y) \ dz$. (So $|h(x,y)|<1$ also.) Observe tha …