OAI.TriangularHilbert.main_estimate

The theorem states that there is a constant C ≥ 0 such that, for all functions F, G from the plane ℝ×ℝ to ℂ that both lie in L³ with respect to Lebesgue measure, three things hold. First, for almost every point z=(x,y), the integrand t ↦ F(x+t,y)·G(x,y+t)/t is integrable on each annulus {t : 1/(n+2) < |t| < n+2}, for every natural number n (a point with this property is called good). Second, the maximal function is almost-everywhere measurable, where the maximal function at z is the supremum, over all endpoint pairs 0 < ε < R, of the absolute value of the truncated integral ∫ over ε<|t|<R of F(x+t,y)G(x,y+t)/t dt, and is defined to be 0 at points that are not good. Third, the L^{3/2} norm of this maximal function, namely (∫ maximal(z)^{3/2} dz)^{2/3} computed as a lower Lebesgue integral in the extended nonnegative reals, is at most C times the L³ norm of F times the L³ norm of G. The constant C depends on neither F nor G.