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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 that if $h(x,y)$ has modulus close to one most of the time (for example, if $\int \int |h(x,y)|\ dx\ dy\approx 1$ or if $h$ has modulus close to 1 in measure), then each of $f$ and $g$ must also have modulus close to one most of the time.

I'd like to know if there's some measurement of "how close to one in modulus" each of these functions is to 1 which will yield some form of inequality relating $f$ and $g$ to $h$. Ideally, I'm hoping for some useful function $D$ which measures how close a function is to one in modulus in some sense, so that if $D(h)$ is this big or small, then $D(f)D(g)$ or $D(f)+D(g)$ is also big or small, implying that $f$ and $g$ are also close to one in modulus.

Ideally, this inequality would extend to a similar convolution of many functions. In my particular situation, I have that $|\int \int h(x,y)\ dx\ dy|$ is close to one, which is quite strong and implies that $h$ is close to constant, but this doesn't pass to $f$ or $g$ and so I assume I should be looking at something weaker.

It sure seems close enough to a convolution that I'd expect some version of a Fourier transform to convert to multiplication, but if there is one, I haven't worked it out.

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Actually, it is the kernel of the product of integral operators with kernels $f$ and $g$. So, just use the operator norm in some space or any estimate thereof (like the Hilbert-Schmidt norm). – fedja Mar 23 '12 at 9:42

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