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

International Conference of Mathematical Sciences 2018, ICMS 2018 -- 31 July 2018 through 6 August 2018 -- 146701Let p(x) and q(y) be nondecreasing continuous functions on [0, 8) such that p(0) = q(0) = 0 and p(x), q(y) › 8 as x, y › 8. For a locally integrable function R+2 = [0,8) × [0,8), we denote its double integral by F(x,y)=?0x?0yf(t,s)dtds and its weighted mean of type (?, ß) by t?,ß(x,y)=?0x?0y(1-p(t)p(x))?(1-q(s)q(y))ßf(t,s)dtds where ? > -1 and ß > -1. We say that ?08?08f(t,s)dtds is integrable to L by the weighted mean method of type (?, ß) determined by the functions p(x) and q(x) if lim x,y›8 t ?,ß (x, y) = L exists. We prove that if lim x,y›8 t ?,ß (x, y) = L exists and t ?,ß (x, y) is bounded on R+2 for some ? > -1 and ß > -1, then lim x,y›8 t ?+h,ß+k (x, y) = L exists for all h > 0 and k > 0. Finally, we prove that if ?08?08f(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 (x)p'(x)?0yf(x,s)ds=O(1)andq(y)q'(y)?0xf(t,y)dt=O(1) hold, then lim x,y›8 F(x, y) = L exists. © 2019 Author(s)