Some Tauberian theorems for weighted means of double integrals

Abstract

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 f(x,y) on R₊²=[0,∞)×[0,∞), we denote its double integral by F(x,y)=∫₀^{x}∫₀^{y}f(t,s)dtds and its weighted mean of type (α,β) by

t_{α,β}(x,y)=∫₀^{x}∫₀^{y}(1-((p(t))/(p(x))))^{α}(1-((q(s))/(q(y))))^{β}f(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 lim_{x,y→∞}t_{α,β}(x,y)=L exists. We prove that if lim_{x,y→∞}t_{α,β}(x,y)=L exists and t_{α,β}(x,y) is bounded on R₊² for some α>-1 and β>-1, then lim_{x,y→∞}t_{α+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 [displaystyle]<LaTeX>\displaystyle</LaTeX>((p(x))/(p′(x)))∫₀^{y}f(x,s)ds=O(1) and [displaystyle]<LaTeX>\displaystyle</LaTeX>((q(y))/(q′(y)))∫₀^{x}f(t,y)dt=O(1) hold, then lim_{x,y→∞}F(x,y)=L exists.

Keywords

• Divergent integrals,weighted means of double integrals,Tauberian theorems and conditions

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