Hölder type estimates for Gaussian multiplicative chaos

We investigate the right tail behavior of a certain class of GMC ratios, reminiscent of Hölder’s inequality. We start with a heuristic argument to justify the optimal exponent in the tail estimate. Since Kahane’s convexity inequality does not apply to GMC ratios, implementing the heuristic in the continuous setting is nontrivial from the viewpoint of GMC theory. We address the problem by enlarging the class of GMC ratios considered, and deduce the upper and lower bounds for the right tail of GMC ratios.