The (L ^p, L ^q) bilinear Hardy-Littlewood function for the tail

Abstract

Let (X, B, µ, T) be a measure preserving dynamical system on a finite measure space. Consider the maximal function

$${R^*}:(f,g) \in {L^P} \times {L^q} \to {R^*}(f,g)(x) = \mathop {\sup }\limits_n {{f({T^n}x)g({T^{2n}}x)} \over n}$$

. We prove that if p and q are greater than or equal to one and ${1 \over p} + {1 \over q} < 2$ < 2 then R* maps L ^p × L ^q into any L ^r as long as 0 < r < 1/2. This implies that R*(f, g) is finite almost everywhere and ${{f({T^n}x)g({T^{2n}}x)} \over n} \to 0$ → 0 for a.e. x as n → ∞.