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 K-t(x, y) is studied, where t is an element of (0,infinity), x, y is an element of X, and X is a homogeneous space. For 1 less than or equal to p < q < infinity, a sufficient condition on approach regions Omega(a), (a is an element of X) is given so that the maximal function ((x,t)is an element of Omega alpha)(sup)integral(X)K(t)(x,y)f(y)d sigma(Y) is weak-type (p, q) with respect to a pair of measures sigma and omega. 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).