Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If I have the three bounds $$\Bigg{|}\iiint_{\mathbb{R}^3} K(x,y,z)f(y,z)g(x,z)h(x,y)dxdydz\Bigg{|}\le \begin{cases} & C_1 \|f\|_2 \|g\|_4 \|h\|_4 \\ & C_2 \|f\|_4 \|g\|_2 \|h\|_4 \\ & C_3 \|f\|_4 \|g\|_4 \|h\|_2, \end{cases}$$ is there some way to interpolate between them? For example, is the integral bounded by $C\|f\|_3 \|g\|_3\|h\|_3$, where $C$ is a convex combination of $C_1, C_2, C_3$?

The only interpolation theorems I can find are for operators, not functionals, and I am not sure how to think of my integral as an operator in order to apply them. Note that I'm assuming the constants above are much better than 1, so I don't care about applying H\"older's inequality to get $some$ $C$.

If this is false, can you provide a counterexample?