TBA by Joris Roos
- Series
- Analysis Seminar
- Time
- Wednesday, April 16, 2025 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Joris Roos – University of Massachusetts Lowell – joris_roos@uml.edu
We develop a theory of Hilbert-space valued stochastic integration with respect to cylindrical martingale-valued measures. As part of our construction, we expand the concept of quadratic variation, to the case of cylindrical martingale-valued measures that are allowed to have discontinuous paths; this is carried out within the context of separable Banach spaces. Our theory of stochastic integration is applied to address the existence and uniqueness of solutions to stochastic partial differential equations in Hilbert spaces.
For \( c\in(1,2) \) we consider the following operators
\[
\mathcal{C}_{c}f(x) \colon = \sup_{\lambda \in [-1/2,1/2)}
\bigg| \sum_{n \neq 0} f(x-n) \frac{e^{2\pi i\lambda \lfloor |n|^{c} \rfloor}}{n} \bigg|\text{,}
\]
\[
\mathcal{C}^{\mathsf{sgn}}_{c}f(x) \colon = \sup_{\lambda \in [-1/2,1/2)}
\bigg| \sum_{n \neq 0} f(x-n) \frac{e^{2\pi i\lambda \mathsf{sign}(n) \lfloor |n|^{c} \rfloor}}{n} \bigg| \text{,}
\]
and prove that both extend boundedly on \( \ell^p(\mathbb{Z}) \), \( p\in(1,\infty) \).
The second main result is establishing almost everywhere pointwise convergence for the following ergodic averages
\[
A_Nf(x)\colon =\frac{1}{N}\sum_{n=1}^N f(T^n S^{\lfloor n^c\rfloor} x) \text{,}
\]
where $T,S\colon X\to X$ are commuting measure-preserving transformations on a $\sigma$-finite measure space $(X,\mu)$, and $f\in L_{\mu}^p(X), p\in(1,\infty)$.
The point of departure for both proofs is the study of exponential sums with phases $\xi_2 \lfloor |n^c|\rfloor+ \xi_1n$ through the use of a simple variant of the circle method.
This talk is based on joint work with Leonidas Daskalakis.
It is well known that $H^2(\mathbb{D}^2)$ is a RKHS with the reproducing kernel $K( \lambda, z) = \frac{1}{(1-\overline{\lambda_1}z_1)(1 - \overline{\lambda_2}z_2)}$ and that for any submodule $M \subseteq H^2(\mathbb{D}^2)$ its reproducing kernel is $K^M( \lambda, z) = P_M K( \lambda, z)$ where $P_M$ is the orthogonal projection onto $M$. Associated with any submodule $M$ are the core function $G^M( \lambda, z) = \frac{K^M( \lambda, z)}{K( \lambda, z)}$ and the core operator $C_M$, an integral transform on $H^2(\mathbb{D}^2)$ with kernel function $G^M$. The utility of these constructions for better understanding the structure of a given submodule is evident from the various works in the past 20 years. In this talk, we will discuss the relationship between the rank, codimension, etc. of a given submodule and the properties of its core function and core operator. In particular, we will discuss the longstanding open question regarding whether we can characterize all submodules whose core function is bounded. This is a joint project with Rongwei Yang.
We establish pointwise convergence for nonconventional ergodic averages taken along $\lfloor p^c\rfloor$, where $p$ is a prime number and $c\in(1,4/3)$ on $L^r$, $r\in(1,\infty)$. In fact, we consider averages along more general sequences $\lfloor h(p)\rfloor$, where $h$ belongs in a wide class of functions, the so-called $c$-regularly varying functions. A key ingredient of our approach are certain exponential sum estimates, which we also use for establishing a Waring-type result. Assuming that the Riemann zeta function has any zero-free strip upgrades our exponential sum estimates to polynomially saving ones and this makes a conditional result regarding the behavior of our ergodic averages on $L^1$ to not seem entirely out of reach. The talk is based on joint work with Erik Bahnson, Abbas Dohadwala and Ish Shah.
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.
Our technique bypasses the use of Fourier or the dependence of group structure. Direct applications include the the (global) weak type endpoint estimate for multi-parameter Calderon–Zygmund operators and maximal function characterisation of multi-parameter Hardy spaces.
This talk is based on the recent progress: Ji Li, ``Fefferman–Stein type inequality'', Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2024.
I will review some recent results in the theory of differentiation of integrals.