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 $f\ge f\ast f$ 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 ${\mathcal{Q}}^{\mathrm{\prime }}$ in $(0,\mathrm{\infty })$. 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.