The Gaussian Radon Transform in Classical Wiener Space

We study the Gaussian Radon transform in the classical Wiener space of Brownian motion. We determine explicit formulas for transforms of Brownian functionals specified by stochastic integrals. A Fock space decomposition is also established for Gaussian measure conditioned to closed affine subspaces in Hilbert spaces

A Gaussian Radon Transform for Banach Spaces

We develop a Radon transform on Banach spaces using Gaussian measure and prove that if a bounded continuous function on a separable Banach space has zero Gaussian integral over all hyperplanes outside a closed bounded convex set in the Hilbert space corresponding to the Gaussian measure then the function is zero outside this set

On a new symmetry of the solutions of the wave equation in the background of a Kerr black hole

This short paper 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. In addition, there is found a related new symmetry operator S for the solutions of the wave equation in that background. That operator is a partial differential operator with a leading order time derivative of the first order that commutes with a normal form of the wave operator. That form is obtained by multiplication of the wave operator from the left with the reciprocal of the coefficient function of its second order time derivative. It is shown that S induces an operator that commutes with the generator of time evolution in a formulation of the initial value problem for the wave equation in the setting of strongly continuous semigroups

The Gaussian Radon Transform for Banach Spaces

The classical Radon transform can be thought of as a way to obtain the density of an n-dimensional object from its (n-1)-dimensional sections in diff_x001B_erent directions. A generalization of this transform to infi_x001C_nite-dimensional spaces has the potential to allow one to obtain a function de_x001C_fined on an infi_x001C_nite-dimensional space from its conditional expectations. We work within a standard framework in in_x001C_finite-dimensional analysis, that of abstract Wiener spaces, developed by L. Gross. The main obstacle in infinite dimensions is the absence of a useful version of Lebesgue measure. To overcome this, we work with Gaussian measures. Specifically, we construct Gaussian measures concentrated on closed affine subspaces of infinite-dimensional Banach spaces, and use these measures to define the Gaussian Radon transform. We provide for this transform a disintegration theorem, an inversion procedure and explore possible applications to machine learning

Two-Weight Inequalities for Commutators

In this talk we discuss commutators with Calderon-Zygmund operators in the two-weight setting. In particular, we extend a one-dimensional result of S. Bloom for the Hilbert transform to n-dimensional Calderon-Zygmund operators, and discuss some natural extensions to iterated commutators and commutators with Riesz potentials

Commutators in the Two-Weight Setting

Let $R$ be the vector of Riesz transforms on $\mathbb{R}^n$, and let $\mu,\lambda \in A_p$ be two weights on $\mathbb{R}^n$, $1 < p < \infty$. The two-weight norm inequality for the commutator $[b, R] : L^p(\mathbb{R}^n;\mu) \to L^p(\mathbb{R}^n;\lambda)$ is shown to be equivalent to the function $b$ being in a BMO space adapted to $\mu$ and $\lambda$. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both $\lambda$ and $\mu$ being Lebesgue measure, and Bloom in the case of dimension one.Comment: v3: suggestions from two referees incorporate