The KKL theorem shows that every Boolean function can be biased by a coalition of $o(n)$ players. Russel, Saks and Zuckerman extended this result to multiround protocols having $o(\log^*n)$ rounds.
We extend both results to arbitrary product distributions on the Boolean hypercube.
The KKL theorem fails for highly biased coordinates. Indeed, such distributions exhibit qualitatively different behavior. Whether unbiased functions can be biased both ways with respect to the uniform distribution, the same doesn’t hold for highly biased distributions (for example, consider the OR function with respect to $\mu_p$ for $p=1/n$). This especially complicated the inductive step in the multiround setting. Our proof uses a novel boosting argument to overcome this difficulty.