Weighted Weak-type Inequalities for the Maximal Function of Nonnegative Integral Transforms over Approach Regions

The relation between approach regions and singularities of nonnegative kernels Kt(x, y) is studied, where t ∈ (0,∞), x, y ∈ X, and X is a homogeneous space. For 1 ≤ p < q < ∞, a sufficient condition on approach regions Ωa (a ∈ X) is given so that the maximal function sup (x,t)∈Ωa ∫ X Kt(x, y)f(y) dσ(y) is weak-type (p, q) with respect to a pair of measures σ and ω. It is shown that this condition is also necessary for operators of potential type in the sense of Sawyer and Wheedon (Amer. J. Math. 114 (1992), 813–874).