Iterated convolution inequalities on $\mathbb{R}^d$ and Riemannian Symmetric Spaces of non-compact type

In a recent work (Int Math Res Not 24:18604-18612, 2021), Carlen-Jauslin-Lieb-Loss studied the convolution inequality $f \ge f*f$ on $\mathbb{R}^d$ and proved that the real integrable solutions of the above inequality must be non-negative and satisfy the non-trivial bound $\int_{\mathbb{R}^d} f \le \frac{1}{2}$. Nakamura-Sawano then generalized their result to $m$-fold convolution (J Geom Anal 35:68, 2025). In this article, we replace the monomials by genuine polynomials and study the real-valued solutions $f \in L^1(\mathbb{R}^d)$ of the iterated convolution inequality \begin{equation*} f \ge \displaystyle\sum_{n=2}^N a_n \left(*^n f\right) \:, \end{equation*} where $N \ge 2$ is an integer and for $2 \le n \le N$, $a_n$ are non-negative integers with at least one of them positive. We prove that $f$ must be non-negative and satisfy the non-trivial bound $\int_{\mathbb{R}^d} f \le t_{\mathcal{Q}}\:$ where $\mathcal{Q}(t):=t-\displaystyle\sum_{n=2}^N a_n\:t^n$ and $t_{\mathcal{Q}}$ is the unique zero of $\mathcal{Q}'$ in $(0,\infty)$. We also have an analogue of our result for Riemannian Symmetric Spaces of non-compact type. Our arguments involve Fourier Analysis and Complex analysis. We then apply our result to obtain an a priori estimate for solutions of an integro-differential equation which is related to the physical problem of the ground state energy of the Bose gas in the classical Euclidean setting.