Some Tauberian theorems for Ces\`{a}ro summability of double integrals over $\mathbb{R}_{+}^{2}$

Abstract

In this paper, we obtain one-sided and two-sided Tauberian conditions of Landau and Hardy types for $(C,1,0)$ and $(C,0,1)$ summability methods for improper double integrals under which convergence of improper double integrals follows from $(C,1,0)$ and $(C,0,1)$ summability of improper double integrals. We give similar results for $(C, 1, 1)$ summability method of improper double integrals. In general, we obtain Tauberian conditions in terms of the concepts of slowly decreasing (resp. oscillating) and strong slowly decreasing (resp. oscillating) functions in different senses for Ces\`aro summability methods of real or complex-valued locally integrable functions on $[0,\infty)\times[0,\infty)$ in different senses.