The Intrinsic Square Function

We show that the Lusin area function and essentially all of its realvariable generalizations are pointwise dominated by an “intrinsic” square function, and that this latter function is, for all practical purposes, no larger than a “generic” square function. 0. Introduction. The Lusin area (or “square”) function is a familiar object. If f : R 7→ R is such that u(x, y) ≡ Py ∗ f(x), the Poisson integral of f , is defined for all (x, y) ∈ R + = R × (0,∞), then we define the Lusin function S(f) by S(f)(x) ≡ (∫ Γ(x) |∇u(t, y)|2 dt dy yd−1 )1/2 , where Γ(x) denotes the usual “cone of aperture one,” Γ(x) = {(t, y) : |x− t| < y}. We can similarly define a cone of aperture β for any β > 0: Γβ(x) = {(t, y) : |x− t| < βy}, and corresponding square function