The Brascamp-Lieb inequalities: recent developments. (Q2846671)

Let \(B_j\:\mathbb{R}^n\to \mathbb{R}^{n_j}\) be linear maps and \(p_j\), \(1\leq j\leq m\), be non-negative exponents. Author considers inequalities of the form NEWLINE\[NEWLINE \int_{\mathbb{R}^n}\prod \limits ^m_{j=1} f_j^{p_j}(B_jx)\,\text{d}x\leq C\prod \limits ^m_{j=1}\left(\int_{\mathbb{R}^{n_j}} f_j(y^j)\,\text{d}y^{j}\right)^{p_j}, NEWLINE\]NEWLINE \(x=(y^1,\dots,y^m)\), \(y^j\in\mathbb{R}^{n_j}\). These inequalities contain famous Hölder and Young convolution inequalities. This survey discusses several theorems from joint papers [Geom. Funct. Anal. 17(2007), No. 5, 1343--1415 (2008; Zbl 1132.26006)] and [Math. Res. Lett. 17, No. 4, 647--666 (2010; Zbl 1247.26029)] of \textit{J. M. Bennett} et al. For example, the following theorem is proved.NEWLINENEWLINELet \(1\leq i\leq m\) and \(A_i\) be a positive semidefinite real matrix, \(\mu_i\) be a positive finite measure on \(\mathbb{R}^n\), \(p_i>0\), \(p=(p_1,\dots,p_n)\). For \(t\geq 0\) let us define NEWLINE\[NEWLINEf_i(x,t)=\int_{\mathbb{R}^n}\exp(-\pi\left\langle A_i(x-tv),(x-tv)\right\rangle)\,\text{d}\mu_i(v); \quad F^p(x,t)=\prod^m_{i=1}f_i^{p_i}(x,t). NEWLINE\]NEWLINE If \(\sum_{i=1}^m p_iA_i\geq A_l\) for all \(l\) and \(\sum_{i=1}^m p_iA_i\) is invertible, then \(\int_{\mathbb{R}^n}F^p(x,t)\,\text{d}x\) is a decreasing function of \(t\).NEWLINENEWLINEFor the entire collection see [Zbl 1180.46003].