arXiv subjects
Publications and source records attributed to Brian Street.
We prove $L^{p_1}(\mathbb R^d)\times \dots \times L^{p_{n+2}}(\mathbb R^{d})$ polynomial growth estimates for the Christ-Journé multilinear singular integral forms and suitable generalizations.
We study a trilinear singular integral form acting on two-dimensional functions and possessing invariances under arbitrary matrix dilations and linear modulations. One part of the motivation for introducing it lies in its large symmetry groups acting on the Fourier side. Another part of the motivation is that this form stands between the bilinear Hilbert transforms and the first Calderón commutator, in the sense that it can be reduced to a superposition of the former, while it also successfully encodes the latter. As the main result we determine the exact range of exponents in which the ${L}^p$ estimates hold for the considered form.
The purpose of this paper is to study the $L^p$ boundedness of operators of the form \[ f\mapsto ψ(x) \int f(γ_t(x))K(t)\: dt, \] where $γ_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \R^N\times \R^n$, satisfying $γ_0(x)\equiv x$, $ψ$ is a $C^\infty$ cutoff function supported on a small neighborhood of $0\in \R^n$, and $K$ is a "multi-parameter singular kernel" supported on a small neighborhood of $0\in \R^N$. We also study associated maximal operators. The goal is, given an appropriate class of kernels $K$, to give conditions on $γ$ such that every operator of the above form is bounded on $L^p$ ($1<p<\infty$). The case when $K$ is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their work to the case when $K$ is (for instance) given by a "product kernel." Even when $K$ is a Calderón-Zygmund kernel, our methods yield some new results. This is the second paper in a three part series. The first paper deals with the case $p=2$, while the third paper deals with the special case when $γ$ is real analytic.
The goal of this paper is to study operators of the form, \[ Tf(x)= ψ(x)\int f(γ_t(x))K(t)\: dt, \] where $γ$ is a real analytic function defined on a neighborhood of the origin in $(t,x)\in \R^N\times \R^n$, satisfying $γ_0(x)\equiv x$, $ψ$ is a cutoff function supported near $0\in \R^n$, and $K$ is a "multi-parameter singular kernel" supported near $0\in \R^N$. A main example is when $K$ is a "product kernel." We also study maximal operators of the form, \[ \mathcal{M} f(x) = ψ(x)\sup_{0<δ_1,..., δ_N<<1} \int_{|t|<1} |f(γ_{δ_1 t_1,...,δ_N t_N}(x))|\: dt. \] We show that $\mathcal{M}$ is bounded on $L^p$ ($1<p\leq \infty$). We give conditions on $γ$ under which $T$ is bounded on $L^p$ ($1<p<\infty$); these conditions hold automatically when $K$ is a Calderón-Zygmund kernel. This is the final paper in a three part series. The first two papers consider the more general case when $γ$ is $C^\infty$.
The purpose of this paper is to study the $L^2$ boundedness of operators of the form \[ f\mapsto ψ(x) \int f(γ_t(x)) K(t) dt, \] where $γ_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \R^N\times \R^n$, satisfying $γ_0(x)\equiv x$, $ψ$ is a $C^\infty$ cutoff function supported on a small neighborhood of $0\in \R^n$, and $K$ is a "multi-parameter singular kernel" supported on a small neighborhood of $0\in \R^N$. The goal is, given an appropriate class of kernels $K$, to give conditions on $γ$ such that every operator of the above form is bounded on $L^2$. The case when $K$ is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when $K$ has a "multi-parameter" structure. For example, when $K$ is given by a "product kernel." Even when $K$ is a Calderón-Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of $L^p$ boundedness, while the third paper deals with the special case when $γ$ is real analytic.
The purpose of this announcement is to describe a development given in a series of forthcoming papers by the authors that concern operators of the form \[ f\mapsto ψ(x) \int f(γ_t(x)) K(t)\: dt, \] where $γ_t(x)=γ(t,x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \mathbb{R}^N\times \mathbb{R}^n$ satisfying $γ_0(x)\equiv x$, $K(t)$ is a "multi-parameter singular kernel" supported near $t=0$, and $ψ$ is a cutoff function supported near $x=0$. This note concerns the case when $K$ is a "product kernel". The goal is to give conditions on $γ$ such that the above operator is bounded on $L^p$ for $1<p<\infty$. Associated maximal functions are also discussed. The "single-parameter" case when $K$ is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger. The theory here extends these results to the multi-parameter context and also deals effectively with the case when $γ$ is real-analytic.
We study multi-parameter Carnot-Caratheodory balls, generalizing results due to Nagel, Stein, and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-Caratheodory balls.
We consider the problem of recovering the coefficient σ(x) of the elliptic equation \grad \cdot(σ\grad u)=0 in a body from measurements of the Cauchy data on possibly very small subsets of its surface. We give a constructive proof of a uniqueness result by Kenig, Sjöstrand, and Uhlmann. We construct a uniquely specified family of solutions such that their traces on the boundary can be calculated by solving an integral equation which involves only the given partial Cauchy data. The construction entails a new family of Green's functions for the Laplacian, and corresponding single layer potentials, which may be of independent interest.
Recently, Nagel and Stein studied the $\square_b$-heat equation, where $\square_b$ is the Kohn Laplacian on the boundary of a weakly-pseudoconvex domain of finite type in $\C^2$. They showed that the Schwartz kernel of $e^{-t\square_b}$ satisfies good "off-diagonal" estimates, while that of $e^{-t\square_b}-π$ satisfies good "on-diagonal" estimates, where $π$ is the Szegö projection. We offer a simple proof of these results, which easily generalizes to other, similar situations. Our methods involve adapting the well-known relationship between the heat equation and the finite propagation speed of the wave equation to this situation. In addition, we apply these methods to study multipliers of the form $mł(\square_b\r)$. In particular, we show that $mł(\square_b\r)$ is an NIS operator, where $m$ satisfies an appropriate Mihlin-Hörmander condition.
We present an intrinsically defined algebra of operators containing the right and left invariant Calderón-Zygmund operators on a stratified group. The operators in our algebra are pseudolocal and bounded on L^p (1<p<\infty). This algebra provides an example of an algebra of singular integrals that falls outside of the classical Calderón-Zygmund theory.