Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by $$\Phi_Kf(y)=\sum_{x\in X}K(x,y)f(x).$$
We also have the adjoint which is going the other way: $$\Phi_K^*g(x)=\sum_{y\in Y}K(x,y)g(y).$$
The duality principle, which is often useful in number theory, says that these two transforms have the same norm when viewed as maps between $L^2(X)$ and $L^2(Y)$. Specifically, we have $$\sum_{y\in Y}(\Phi_Kf(y))^2\leq C\sum_{x\in X}f(x)^2$$ for any $f$ if and only if $$\sum_{x\in X}(\Phi_K^*g(x))^2\leq C\sum_{y\in Y}g(y)^2$$ for any $g$. Now suppose that for some $f:X\to\mathbb R$ we have a lower bound for $\sum_{y\in Y}(\Phi_Kf(y))^2$. The duality principle then tells us that we can deduce a corresponding lower bound for $\sum_{x\in X}(\Phi_K^*g(x))^2$ for some function $g:Y\to \mathbb R$. My question is then this: can we have any more information about the function $g$, perhaps with further assumptions on $K$? For instance, we might want to avoid $g$ being highly oscillatory, or too concentrated, etc.
