×

Behaviour of the Brascamp-Lieb constant. (English) Zbl 1377.26030

In this paper, the authors prove the continuity of the Brascamp-Lieb constant with respect to the underlying linear transformations.
Let \(H\) and \(H_{j}\) for \(j=1,\dotsc,m\) denote Euclidean spaces of finite dimensions \(n\) and \(n_{j}\), where \(n_{j}\leq n\), equipped with Lebesgue measure for each \(1\leq j\leq m\). Let \(L_{j}: H\to H_{j}\) be surjective linear transformations and \(p_{j}\) real numbers with \(0\leq p_{j}\leq 1\). Then, for every set of \(m\) nonnegative functions \(f_{j}: H_{j}\to\mathbb{R}\) the Brascamp-Lieb inequatlity states that \[ \int_{H}\prod^{m}_{j=1}(f_{j}\circ L_{j})^{p_{j}}\leq C\prod^{m}_{j=1}\left( \int_{H_{j}}f_{j}\right)^{p_{j}} \] for some constant \(C\).
The Brascamp-Lieb constant denoted by \(\mathrm{BL}(\mathbf{L},\mathbf{p})\) is the smallest constant \(C\) for which the Brascamp-Lieb inequality holds for all nonnegative functions \(f_{j}\in L^{1}(H_{j})\), \(j=1,\dotsc,m\). The authors write \(\mathbf{p}\) for \((p_{1},p_{2},\dotsc,p_{m})\) and \(\mathbf{L}\) for \((L_{1},L_{2},\dotsc,L_{m})\). The theorem proved in the paper is as follows:
Theorem. For each \(\mathbf{p}\), the Brascamp-Lieb constant \(\mathrm{BL}(\cdot,\mathbf{p})\) is a continuous function.
In contrast, they provide a counterexample to the claim that the Brascamp-Lieb constant is everywhere differentiable.
For the proof of the theorem, they first deal with the case that each \(L_{j}\) is of rank one. The argument is based on the availability of a simple formula for \(\mathrm{BL}(\mathbf{L},\mathbf{p})\) due to Barthe. In the general-rank case, they proceed first to establish a suitable Barthe-type formula which holds in full generality.
The authors conjecture that in general the dependence of \(\mathrm{BL}(\mathbf{L},\mathbf{p})\) with respect to \(\mathbf{L}\) is at least locally Hölder continuous.
Also, they formulate a conjecture for the case that the linear surjections \(L_{j}: \mathbb{R}^{n}\to \mathbb{R}^{n_{j}}\) are replaced by local submersions \(B_{j}: U\to \mathbb{R}^{n_{j}}\), defined on a neighbourhood \(U\) of a point \(x_{0}\in\mathbb{R}^{n}\). The suspected result is:
Given \(\varepsilon >0\), there exists \(\delta>0\) (depending on the maps \(B_{j}\)) such that \[ \int_{B(x_{0},\delta)}\prod^{m}_{j=1} (f_{j}\circ B_{j})^{p_{j}}\leq (1+\varepsilon) \mathrm{BL}(\mathbf{L},\mathbf{p})\prod^{m}_{j=1}\left(\int_{\mathbb{R}^{n_{j}}}f_{j}\right)^{p_{j}}, \] a conjecture that is intimately related to the continuity of the general Brascamp-Lieb constant.

MSC:

26D20 Other analytical inequalities
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
PDFBibTeX XMLCite
Full Text: DOI arXiv