Anthony Carbery

We consider some integral-geometric quantities that have recently arisen in harmonic analysis and elsewhere, derive some sharp geometric inequalities relating them, and place them in a wider context.

Let \Sigma be a strictly convex, compact patch of a C^{2} hypersurface in \mathbb{R}^{n} , with non-vanishing Gaussian curvature and surface measure d\sigma induced by the Lebesgue measure in \mathbb{R}^{n} . The Mizohata–Takeuchi conjecture states that \int |\widehat{g d\sigma}|^{2} w \leq C \|Xw\|_{\infty} \int |g|^{2} for all g\in L^{2}(\Sigma) and all weights w \colon \mathbb{R}^{n}\rightarrow [0,+\infty) , where X denotes the X -ray transform. As partial progress towards the conjecture, we show, as a straightforward consequence of recently-established decoupling inequalities, that for every \varepsilon>0 , there exists a positive constant C_{\varepsilon} , which depends only on \Sigma and \varepsilon , such that for all R \geq 1 and all weights w \colon \mathbb{R}^{n}\rightarrow [0,+\infty) , we have \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\varepsilon} \sup_{T} \Big(\int_{T} w^{(n+1)/2}\Big)^{2/(n+1)}\int |g|^{2}, where T ranges over the family of tubes in \mathbb{R}^{n} of dimensions R^{1/2}\times \cdots \times R^{1/2}\times R . From this we deduce the Mizohata–Takeuchi conjecture with an R^{(n-1)/(n+1)} -loss; i.e., that \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\frac{n-1}{n+1}+ \varepsilon}\|Xw\|_{\infty} \int |g|^{2} for any ball B_{R} of radius R and any \varepsilon>0 . The power (n-1)/(n+1) here cannot be replaced by anything smaller unless properties of \widehat{g d\sigma} beyond ‘decoupling axioms’ are exploited. We also provide estimates which improve this inequality under various conditions on the weight, and discuss some new cases where the conjecture holds.

The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately. On the one hand, the theorem gives a uniform approach to classical results including Maurey's factorisation theorem and Lozanovski\u{\i}'s factorisation theorem, and, on the other hand, it underpins the duality theory for multilinear norm inequalities developed in our previous two papers. In this paper we give a simple proof of this basic disentanglement theorem. Whereas the approach of our previous paper was rather involved - it relied on the use of minimax theory together with weak*-compactness arguments in the space of finitely additive measures, and an application of the Yosida-Hewitt theory of such measures - the alternate approach of this paper is rather straightforward: it instead depends upon elementary perturbation and compactness arguments.

We develop a framework for a duality theory for general multilinear operators which extends that for transversal multilinear operators which has been established in arXiv:1809.02449. We apply it to the setting of joints and multijoints, and obtain a "factorisation" theorem which provides an analogue in the discrete setting of results of Bourgain and Guth (arXiv:0811.2251 and arXiv:1012.3760) from the Euclidean setting.

In previous work we established a multilinear duality and factorisation theory for norm inequalities for pointwise weighted geometric means of positive linear operators defined on normed lattices. In this paper we extend the reach of the theory for the first time to the setting of general linear operators defined on normed spaces. The scope of this theory includes multilinear Fourier restriction-type inequalities. We also sharpen our previous theory of positive operators. Our results all share a common theme: estimates on a weighted geometric mean of linear operators can be disentangled into quantitative estimates on each operator separately. The concept of disentanglement recurs throughout the paper. The methods we used in the previous work - principally convex optimisation - relied strongly on positivity. In contrast, in this paper we use a vector-valued reformulation of disentanglement, geometric properties (Rademacher-type) of the underlying normed spaces, and probabilistic considerations related to p-stable random variables.