Some Tauberian theorems for weighted means of double integrals on ℝ+2

Let p(x) and q(y) be nondecreasing continuous functions on [0, ∞) such that p(0) = q(0) = 0 and p(x), q(y) → ∞ as x, y → ∞. For a locally integrable function ℝ+2 = [0,∞) × [0,∞), we denote its double integral by F(x,y)=∫0x∫0yf(t,s)dtds and its weighted mean of type (α, β) by tα,β(x,y)=∫0x∫0y(1−p(t)p(x))α(1−q(s)q(y))βf(t,s)dtds where α > −1 and β > −1. We say that ∫0∞∫0∞f(t,s)dtds is integrable to L by the weighted mean method of type (α, β) determined by the functions p(x) and q(x) if limx,y→∞ tα,β(x, y) = L exists. We prove that if limx,y→∞ tα,β(x, y) = L exists and tα,β(x, y) is bounded on ℝ+2 for some α > −1 and β > −1, then limx,y→∞ tα+h,β+k(x, y) = L exists for all h > 0 and k > 0. Finally, we prove that if ∫0∞∫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)∫0yf(x,s)ds=O(1)andq(y)q′(y)∫0xf(t,y)dt=O(1) hold, then limx,y→∞ F(x, y) = L exists.