On the regular convergence of multiple integrals of locally Lebesgue integrable functions over R- ~+ m

摘要

Let the function. f:R- + m→C be such that. f∈Lloc1(R- + m), where. m≥. 2 is a fixed integer. We investigate the convergence behavior of the. m-multiple integral. ∫ _0 v1∫ _0 v2...∫ _0 vmf(t _1,t _2,...,t m)dt _1dt _2...dtmas min{v _1,v _2,...,v _m}→∞, while using two notions of convergence: the one in Pringsheim's sense and the one in the regular sense. For the sake of brevity, we present our main result in the case. m=. 2 as follows: If. f∈L _(loc) ~1(R- _+ ~2) and the double integral (*) converges regularly, then the finite limits. lim _(v2)→∞∫ _0 v1(∫ _0 ~(v2f)(t _1,t _2)dt _2)dt _1=:J _1(v _1) and. limv1→∞∫ _0 ~(v2)(∫ _0 ~(v1f)(t _1,t _2)dt _1)dt _2=:J _2(v 2) exist uniformly in. 0