DMS Colloquium: Dr. Ji Li

Speaker: Dr. Ji Li (Macquarie University; Sydney, Australia)

Title: Schatten properties of Riesz transform commutator in the two weight setting

Abstract. Schatten class estimates of the commutator of Riesz transform in ${\mathbb{R}}^n$ link to the quantised derivative of A. Connes. A general setting for quantised calculus is a spectral triple $(\mathcal{A, H, D})$, which consists of a Hilbert space ${\mathcal{H}}$, a pre-$C^∗$-algebra ${\mathcal{A}}$, represented faithfully on ${\mathcal H}$ and a self-adjoint operator ${\mathcal D}$ acting on ${\mathcal{H}}$ such that every $a ∈ A$ maps the domain of ${\mathcal D}$ into itself and the commutator $[D, a] = Da−aD$ extends from the domain of ${\mathcal D}$ to a bounded linear endomorphism of ${\mathcal{H}}$. Here, the quantised differential \da of $a ∈ A$ is defined to be the bounded operator $i[sgn(D), a]$, $i2 = −1$.

We provide full characterisation of the Schatten properties of $[Mb, Rj ]$, the commutator of the $j$-th Riesz transform on $R^n$ with symbol $b (Mbf(x) := b(x)f(x))$, in the two weight setting. The approach is not depending on the Euclidean structure or Fourier, and hence it can be applied to other settings.

This talk is based on my recent work joint with Michael Lacey, Brett Wick, and Liangchuan Wu.