A classical Fefferman-Stein inequality relates the distributional estimate for a square function for a harmonic function u to a non-tangential maximal function of u. We extend this ineuality to certain multiparameter settings, including the Shilov boundaries of tensor product domains, and the Heisenberg groups with flag structure.
I will review some recent results in the theory of differentiation of integrals.
We prove that the planar Bochner Riesz mean converges almost everywhere for any L^p function in the optimal range, for 5/3
Let $S^{2d-1}$ be the unit sphere in $\mathbb{R}^{2d}$, and $\sigma_{2d-1}$ the normalized spherical measure in $S^{2d-1}$. The (scale t) bilinear spherical average is given by
$$\mathcal{A}_{t}(f,g)(x):=\int_{S^{2d-1}}f(x-ty)g(x-tz)\,d\sigma_{2d-1}(y,z).$$
There are geometric motivations to study bounds for such bilinear spherical averages, in connection to the study of some Falconer distance problem variants. Sobolev smoothing bounds for the operator
$$\mathcal{M}_{[1,2]}(f,g)(x)=\sup_{t\in [1,2]}|\mathcal{A}_{t}(f,g)(x)|$$
We go over some relevant history and related problems to motivate the study of the Carleson-Radon operator and the difficulty exhibiting in the planar case. Our main result confirms that the planar Carleson-Radon operator along homogenous curve with general monomial \(t^\alpha\) term modulation admits full range \(L^p\) bound assuming the natural non-resonant condition. In the talk, I'll provide a brief overview of the three key ingredients of the LGC based proof:
This presentation is dedicated to extending both defocusing and focusing Calogero–Moser–Sutherland derivative nonlinear Schrödinger equations (CMSdNLS), which are introduced in Abanov–Bettelheim–Wiegmann [arXiv:0810.5327], Gérard-Lenzmann [arXiv:2208.04105] and R.
We prove a wavelet T(1) theorem for compactness of multilinear Calderón -Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator. This talk is based on joint work with Walton Green and Brett Wick.
The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. We show that it is bounded in the Banach range.
We shall discuss the quantum dynamics associated with ergodic Schroedinger operators. Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) has been obtained for a variety of ergodic operator families, but it is well known that Anderson localization is highly unstable and can also be destroyed by generic rank one perturbations.
This will be an expository talk which aims to introduce some problems in harmonic analysis and geometric measure theory concerning the geometry of a measure for which an associated integral operator is well behaved. As an example, we shall prove a result of Mattila and Preiss concerning the relationship between the rectifiability of a measure and the existence of the Riesz transform in the sense of principle value.
