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

TL;DR

The paper generalizes real, integrable solutions of convolution inequalities from monomial to genuine polynomial iterates on $\mathbb{R}^d$ and on non-compact type symmetric spaces. It analyzes real $f\in L^1$ satisfying $f \ge \sum_{n=2}^N a_n\,*^n f$, introduces the polynomial $\mathcal{Q}(t)=t-\sum_{n=2}^N a_n t^n$ and its unique positive critical point $t_{\mathcal{Q}}$, and proves $f$ is nonnegative with $\int f \le t_{\mathcal{Q}}$. The proofs blend Fourier analysis, group/Helgason Fourier transform techniques, and a Vivanti-Pringsheim-type complex-analytic argument to handle the polynomial nonlinearity, with an analogous result on symmetric spaces. An application yields a priori estimates for an integro-differential equation linked to the ground-state energy of the Bose gas in the classical setting. Together, these results deepen the understanding of nonlinear convolution inequalities and provide tools for energy estimates on homogeneous spaces.

Abstract

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.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof : Proof of Theorem \ref{['thm']}
• proof : Proof of Theorem \ref{['Rn-thm']}
• Remark 3.1
• proof : Proof of Theorem \ref{['thm1']}
• proof : Proof of Theorem \ref{['thm2']}