Functional Analysis Seminar

In the context of Fourier analysis on the real line, a \textit{one-sided problem} involves deducing properties of a function $f$ from some information about the restriction of its Fourier transform $\widehat{f}$ to a half-line, for instance to $\mathbb{R}_- := (-\infty, 0)$. A prototypical result, which is foundational to the theory of Hardy spaces on $\mathbb{R}$, asserts that if $f \in L^2(\mathbb{R})$ is non-zero and $\widehat{f}$ vanishes on a half-line, then $f$ satisfies the \textit{Szeg\H{o} condition} $\int_{-\infty}^\infty \frac{\log |f(x)|}{1+x^2} \, dx > -\infty$. 

Various problems in operator theory involve the study of functions $f$ satisfying a weaker condition of decay of $\widehat{f}$ on a half-line. In this setting, simple examples show that the Szeg\H{o} condition need not be satisfied. However, the following local Szeg\H{o}-type conditions hold: if the decay of $\widehat{f}$ is strong enough on a half-line, then the mass of the function $f \in L^2(\mathbb{R})$ must concentrate enough for the integral $\int_E \log |f(x)| dx$ to converge on a "massive" set $E$. 

In his talk, Bartosz Malman will describe this mass condensation phenomenon and its applications to operator-theoretic problems.

The boundedness of Fourier multipliers on non-commutative $L_p$-spaces ($1 < p < \infty$) is a fundamental problem in non-commutative analysis. Building on the non-commutative Cotlar identity introduced by Mei and Ricard (2017), which yields $L_p$-boundedness ($1 < p < \infty$) of Hilbert transforms on amalgamated free products of finite von Neumann algebras, their approach relies heavily on freeness in the underlying free product structure.

In this talk, Xiaoqi Lu introduces a new strategy that overcomes this limitation. Our approach combines a generalized Cotlar identity, which holds on suitable subspaces and captures non-freeness information, with an additional condition related to the property of Rapid Decay to control the remaining components. From this framework, we establish the $L_p$-boundedness ($1 < p < \infty$) of Rademacher-type Hilbert transforms on graph products of finite von Neumann algebras. This unified framework extends earlier results for free products of finite von Neumann algebras and for graph products of groups acting on right-angled buildings. This is a joint work with Runlian Xia.

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