A theory of multilinear duality and factorisation
A talk in the Mathematisches Kolloquium (SFB 1283) series by
Tony Carbery from Edinburgh
| Abstract: | Classical linear duality theory in functional analysis concerns itself (inter alia) with statements "dual" to those such as boundedness of a linear operator $T$ between Banach spaces $Y$ and $X$. A typical such dual statement is that the adjoint $T^\ast$ maps $X^\ast$ to $Y^\ast$ boundedly. Classical Maurey--Nikishin--Stein factorisation theory asks when a bounded linear operator $T: Y \to X$ must automatically factorise through a third space $Z$. Motivated by a range of problems arising in harmonic analysis, we study analogues of these questions where the linear operator $T$ is replaced by a $\textbf{pointwise geometric mean}$ of a collection of positive linear operators. The theory finds its expression in terms of certain pointwise factorisation properties of function spaces which are naturally associated to the problem under consideration. The development of the theory involves tools such as convex optimisation and minimax theory, as well as functional-analytic considerations concerning the dual of $L^\infty$ and the Yosida--Hewitt theory of finitely additive measures. If time permits, we shall discuss ramifications of the theory in the context of concrete families of geometric inequalities, including Loomis--Whitney inequalities, Brascamp--Lieb inequalities and multilinear Kakeya inequalities.
This is joint work with Timo Hänninen and Stefán Valdimarsson. |