Irina Holmes: Research NSF Postdoctoral Fellow Department of Mathematics Washington University in St. Louis irina.holmes@email.wustl.edu
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### Overview

Postdoctoral Research: My current research interests are in dyadic harmonic analysis, weighted inequalities, multiparameter harmonic analysis, and operator theory. For more details, see the most recent publications below.

Graduate Research: My graduate research interests were infinite-dimensional analysis, probability, functional analysis and machine learning. Together with my adviser, I worked on developing the Gaussian Radon transform for Banach spaces, an infinite-dimensional generalization of the classical Radon transform, and applications of this transform to machine learning. For more details, see my Graduate Research Statement, or my thesis.

Undergraduate Research: As an undergraduate at LSU, I worked as a research student at CCT. In particular, I worked with Horst Beyer on the Kerr metric of rotating black holes.

### Publications

• Commutators with Fractional Integral Operators - joint with Robert Rahm and Scott Spencer (Studia Mathematica 233, no.3, 2016).
• In this paper we investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $\mu, \lambda \in A_{p,q}$ and $\alpha/n + 1/q = 1/p$, the norm $\| [b, I_{\alpha}] : L^p(\mu^p) \to L^q(\lambda^q) \|$ is equivalent to the norm of $b$ in the weighted BMO space $BMO(\nu)$, where $\nu = \mu\lambda^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.
• Two Weight Inequalities for Iterated Commutators with Calderón-Zygmund Operators - joint with Brett D. Wick (Submitted).
• Given a Calderón-Zygmund operator $T$, a classic result of Coifman-Rochberg-Weiss relates the norm of the commutator $[b, T]$ with the BMO norm of $b$. We focus on a weighted version of this result, obtained by Bloom and later generalized by Lacey and the authors, which relates $\| [b, T]: L^p(\mathbb{R}^n; \mu) \rightarrow L^p(\mathbb{R}^n; \lambda) \|$ to the norm of $b$ in a certain weighted BMO space determined by $A_p$ weights $\mu$ and $\lambda$. We extend this result to higher iterates of the commutator and recover a one-weight result of Chung-Pereyra-Perez in the process.
• Commutators in the Two-Weight Setting - joint with Michael T. Lacey and Brett D. Wick (Mathematische Annalen, 2016).
• Generalizes the previous paper, by proving the boundedness of the commutator $[b, T] : L^p(\mu) \rightarrow L^p(\lambda)$ for any Calderón-Zygmund operator $T$, where $\mu, \lambda$ are Muckenhoupt $A_p$ weights on $\mathbb{R}^n$, provided that $b$ is in a special weighted $BMO$ class. Conversely, specializing to Riesz transforms, we characterize this $BMO$ class in terms of the boundedness of the commutators.
• Bloom's Inequality: Commutators in a Two-Weight Setting - joint with Michael T. Lacey and Brett D. Wick (Archiv der Mathematik 106, no. 1, 2016).
• Gives a modern proof of a result by Bloom which characterizes the boundedness of the commutator $[b, H] : L^2(\mu) \rightarrow L^2(\lambda)$, where $H$ is the Hilbert transform and $\mu, \lambda$ are Muckenhoupt $A_2$ weights on $\mathbb{R}$, in terms of a novel weighted $BMO$ condition.
• The Gaussian Radon Transform in Classical Wiener Space - joint with Ambar Sengupta (Communications on Stochastic Analysis 8, no. 2, 2014).
• Provides concrete examples and computations of the Gaussian Radon transform in the setting of the classical Wiener space of Brownian motion, and establishes a Fock space decomposition for Gaussian measure conditioned to closed affine subspaces in Hilbert spaces.
• The Gaussian Radon Transform and Machine Learning - joint with Ambar Sengupta (Infinite Dimensional Analysis, Quantum Probability and Related Topics, 18, no. 3, 2015).
• This work is motivated by an increasing number of recent publications exploring probabilistic interpretations of kernel-based prediction methods. While there is a clear Bayesian interpretation to some of these methods when working in finite dimensions, the absence of Lebesgue measure in infinite-dimensional Hilbert spaces poses an obstacle. We show that the Gaussian Radon transform can be used to provide a stochastic interpretation to the ridge regression problem in infinite dimensions.
• An Inversion Formula for the Gaussian Radon Transform for Banach Spaces (Infinite Dimensional Analysis, Quantum Probability and Related Topics, 16, no. 4, 2013).
• Proves a disintegration formula leading to the expression of the Gaussian Radon transform as a conditional expectation, and an inversion procedure involving the Segal-Bargmann transform.
• A Gaussian Radon Transform for Banach Spaces - joint with Ambar Sengupta (Journal of Functional Analysis 263, no. 11, 2012).
• Establishes the existence and uniqueness of the Gaussian Radon transform and proves an analogue of the Helgason support theorem.
• On a new symmetry of the solutions of the wave equation in the background of a Kerr black hole - joint with Horst R. Beyer (Classical and Quantum Gravity, 25, no. 13, 2008)
• Derives the constant of motion of a scalar field in the gravitational field of a Kerr black hole which is associated to a Killing tensor of that space-time.
• On a Problem in the Stability Discussion of Rotating Black Holes (Proceedings of The National Conference On Undergraduate Research (NCUR) 2006)
• This is an undergraduate paper which I presented at the 2006 National Conference for Undergraduate Research in Asheville, North Carolina.