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

Source abstract

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 , a n $a_n$ a Subscript n are nonnegative integers with at least one of them positive, then f must be nonnegative a.e. and satisfy the nontrivial bound ∫ X f ≤ t Q $\int _{\mathbb {X}} f \le t_{\mathcal {Q}}\:$ integral Underscript double struck upper X Endscripts f less than or equals t Subscript script upper Q , where Q ( t ) := t − ∑ n = 2 N a n t n $\mathcal {Q}(t):=t-\sum _{n=2}^N a_n\:t^n$ script upper Q left parenthesis t right parenthesis colon equals t minus sigma summation Underscript n equals 2 Overscript upper N Endscripts a Subscript n Baseline t Superscript n and t Q $t_{\mathcal {Q}}$ t Subscript script upper Q is the unique zero of Q ′ $\mathcal {Q}'$ script upper Q prime in ( 0 , ∞ ) $(0,\infty )$ left parenthesis 0 comma infinity right parenthesis . This result is new even for R d $\mathbb {R}^d$ double struck upper R Superscript d and generalizes recent results of Carlen et al. (2021, Int. Math. Res. Not. , 24, 18604–18612) and Nakamura and Sawano (2025, J. Geom. Anal. , 35, 68) for two-fold and m -fold convolutions, respectively, to genuine polynomials. We then apply our result to obtain an a priori estimate for solutions of an integro-differential equation on symmetric spaces, which generalizes the equation on R d $\mathbb R^d$ double struck upper R Superscript d , related to the physical problem of the ground state energy of the Bose gas, studied by Carlen et al. (2020, Pure Appl. Anal. , 2, 659–684). We also obtain a surprising result on the nature of the extremizers of the convolution inequalities.