Iterated convolution inequalities on R d $\mathbb {R}^d$ double struck upper R Superscript d and Riemannian symmetric spaces of non-compact type

Abstract Let X $\\mathbb {X}$ double struck upper X be a Riemannian symmetric space of non-compact type. For a real-valued f ∈ L 1 ( X ) $f \\in L^1(\\mathbb {X})$ f element of upper L Superscript 1 Baseline left parenthesis double struck upper X right parenthesis with ∫ X f ≥ 0 $\\int _{\\mathbb {X}} f \\ge 0$ integral Underscript double struck upper X Endscripts f greater than or equals 0 , we prove that if f satisfies the iterated convolution inequality: f ≥ ∑ n = 2 N a n ( ∗ n f ) , a.e. on X , $$ \\begin{align*} f \\ge \\sum_{n=2}^N a_n \\left(*^n f\\right),\\:\\:\\text{ a.e. on } \\mathbb{X}, \\end{align*} $$ where N ≥ 2 $N \\ge 2$ upper N greater than or equals 2 is an integer and for 2 ≤ n ≤ N $2 \\le n \\le N$ 2 less than or equals n less than or equals upper N </