Some Tauberian Theorems for Weighted Means of Double Integrals on R-+(2)

Let p(x) and q(y) be nondecreasing continuous functions on [0, infinity) such that p(0) = q(0) = 0 and p(x), q(y) -> infinity as x, y -> infinity. For a locally integrable function f(x,y) on R-+(2) = [0, infinity) x [0, infinity), we denote its double integral by F(x,y) = integral(x)(0) integral(y)(0) f(t, s)dtds and its weighted mean of type (alpha, beta) by t(alpha,beta)(x,y) = integral(x)(0) integral(y)(0) (1- p(t)/p(x))(alpha)(1-q(s)/q(y))(beta) f(t, s)dtds where alpha > -1 and beta > -1. We say that integral(infinity)(0)integral(infinity)(0) f(t, s)dtds is integrable to L by the weighted mean method of type (alpha, beta) determined by the functions p(x) and q(x) if lim(x,y -> infinity) t(alpha,beta)(x, y) = L exists. We prove that if lim(x,y -> infinity )t(alpha,beta)(x, y) = L exists and t a p(x, y) is bounded on R-+(2) for some alpha > -1 and beta > -1, then lim(x,y -> infinity )t(alpha+h,beta+k)(x, y)= L exists for all h > 0 and k > 0. Finally, we prove that if integral(infinity)(0) integral(infinity)(0) f(t, s)dtds is integrable to L by the weighted mean method of type (1, 1) determined by the functions p(x) and q(x) and conditions p(x)/p'(x) integral(y)(0) f(x, s)ds = O(1) and q(y)/q'(y) integral(x)(0) f(t, y)dt = O(1) hold, then lim(x,y ->infinity) F(x, y) = L exists.