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Brian Street

Publications and source records attributed to Brian Street.

At least 19 recordsLinked to original sources
arXiv2026

Higher-order Gaussian bounds for maximally subelliptic boundary value problems

We establish higher-order Gaussian upper bounds for the heat semigroups associated with a broad class of sectorial maximally subelliptic quadratic forms on manifolds with boundary. Near non-characteristic boundary points and in the interior, we obtain pointwise bounds for all mixed derivatives of the heat kernel in time and along the H\"ormander vector fields, expressed in the associated Carnot--Carath\'eodory geometry. The results apply to operators of arbitrary even order, systems, nonsymmetric forms, and boundary conditions beyond the Dirichlet case.

math.AP↗
arXiv2026

Non-uniqueness for a differential equation and a proof by ChatGPT

Let $f(t,x),M(t,x)\in C([0,1]^2)$ with $M(t,x)>0$. We consider differential equations of the form \[ \frac{\partial f}{\partial t}(t,x)=\frac{M(t,x)f(t,x)-M(t,0)f(t,0)}{x},\quad x>0. \] For a fixed positive weight $M$, we ask whether the condition $f(0,x)=0$ forces $f\equiv 0$. We show the answer is negative for smooth functions: there exist $f(t,x),M(t,x)\in C^{\infty}([0,1]^2)$ with $f(0,x)=0$, $f(t,0)\not\equiv 0$, and $M(t,x)>0$ satisfying the above equation. However, we show that for a large class of $M(t,x)$, the equation does have uniqueness. We relate this to uniqueness/non-uniqueness theorems for weighted Laplace transforms. A key example originated in an output by ChatGPT-5.5-Pro, and we include a discussion of its output as well as a complete proof.

math.AP↗
arXiv2026

A Priori Estimates for Maximally Subelliptic Quadratic Forms

We prove a priori subelliptic estimates, near a non-characteristic boundary point, for the heat operators associated to a wide class of maximally subelliptic quadratic forms. This is the third paper in a series devoted to studying general maximally subelliptic boundary value problems.

math.AP↗
arXiv2025

Function spaces and trace theorems for maximally subelliptic boundary value problems

We introduce Besov and Triebel--Lizorkin spaces on a manifold with boundary adapted to H\"ormander vector fields, near a so-called non-characteristic point of the boundary. We prove sharp results in these spaces for the corresponding restriction and trace operators, show these operators are retractions, and other related results. This is the second paper in a forthcoming series devoted to a general theory of maximally subelliptic boundary value problems, and lays the function space foundation for this general theory.

math.AP↗
arXiv2025

Carnot-Carath\'eodory Balls on Manifolds with Boundary

Nagel, Stein, and Wainger introduced a detailed quantitative study of Carnot--Carath\'eodory balls on a smooth manifold without boundary. Most importantly, they introduced scaling maps adapted to Carnot--Carath\'eodory balls and H\"ormander vector fields. Their work was extended by many authors and has since become a key tool in the study of the interior theory of subelliptic PDEs; in particular, the study of maximally subelliptic PDEs. We introduce a generalization of this quantitative theory to manifolds with boundary, where we have scaling maps both on the interior and on the part of the boundary which is non-characteristic with respect to the vector fields. This is the first paper in a forthcoming series devoted to studying maximally subelliptic boundary value problems.

math.CA↗
arXiv2024

Hypoellipticity and Higher Order Gaussian Bounds

Let $(\mathfrak{M},\rho,\mu)$ be a metric measure space satisfying a doubling condition, $p_0\in (1,\infty)$, and $T(t):L^{p_0}(\mathfrak{M},\mu)\rightarrow L^{p_0}(\mathfrak{M},\mu)$, $t\geq 0$, a strongly continuous semi-group. We provide sufficient conditions under which $T(t)$ is given by integration against an integral kernel satisfying higher-order Gaussian bounds of the form \[ \left| K_t(x,y) \right| \leq C \exp\left( -c \left( \frac{\rho(x,y)^{2\kappa}}{t} \right)^{\frac{1}{2\kappa-1}} \right) \mu\left( B_\rho\left(x,\rho(x,y)+t^{1/2\kappa}\right) \right)^{-1}, \] where $B_\rho$ denotes the metric ball. We also provide conditions for similar bounds on ``derivatives'' of $K_t(x,y)$ and our results are localizable. If $A$ is the generator of $T(t)$ the main hypothesis is that $\partial_t -A$ and $\partial_t-A^{*}$ satisfy a hypoelliptic estimate at every scale, uniformly in the scale. We present applications to subelliptic PDEs.

math.AP↗
arXiv2022

Spaces of Besov-Sobolev type and a problem on nonlinear approximation

We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki\uı spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best $n$-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov.

math.FA↗
arXiv2022

Improving the Regularity of Vector Fields

Let $α>0$, $β>α$, and let $X_1,\ldots, X_q$ be $\mathscr{C}^α_{\mathrm{loc}}$ vector fields on a $\mathscr{C}^{α+1}$ manifold which span the tangent space at every point, where $\mathscr{C}^{s}$ denotes the Zygmund-Hölder space of order $s$. We give necessary and sufficient conditions for when there is a $\mathscr{C}^{β+1}$ structure on the manifold, compatible with its $\mathscr{C}^{α+1}$ structure, with respect to which $X_1,\ldots, X_q$ are $\mathscr{C}^β_{\mathrm{loc}}$. This strengthens previous results of the first author which dealt with the setting $α>1$, $β>\max\{ α, 2\}$.

math.CA↗
arXiv2021

Coordinates Adapted to Vector Fields III: Real Analyticity

Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are real analytic. We give necessary and sufficient, coordinate-free conditions for the existence of such a coordinate system. Moreover, we present a quantitative study of these coordinate charts. This is the third part in a three-part series of papers. The first part, joint with Stovall, lay the groundwork for the coordinate system we use in this paper and showed how such coordinate charts can be viewed as scaling maps for sub-Riemannian geometry. The second part dealt with the analogous questions with real analytic replaced by $C^\infty$ and Zygmund spaces.

math.DG↗
arXiv2021

Coordinates Adapted to Vector Fields II: Sharp Results

Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are $\mathscr{C}^{s+1}$ for $s\in (1,\infty]$, where $\mathscr{C}^s$ denotes the Zygmund space of order $s$. We give necessary and sufficient, coordinate-free conditions for the existence of such a coordinate system. Moreover, we present a quantitative study of these coordinate charts. This is the second part in a three-part series of papers. The first part, joint with Stovall, addressed the same question, though the results were not sharp, and showed how such coordinate charts can be viewed as scaling maps in sub-Riemannian geometry. When viewed in this light, these results can be seen as strengthening and generalizing previous works on the quantitative theory of sub-Riemannian geometry, initiated by Nagel, Stein, and Wainger, and furthered by Tao and Wright, the author, and others. In the third part, we prove similar results concerning real analyticity.

math.DG↗
arXiv2020

Sub-Hermitian Geometry and the Quantitative Newlander-Nirenberg Theorem

Given a finite collection of $C^1$ complex vector fields on a $C^2$ manifold $M$ such that they and their complex conjugates span the complexified tangent space at every point, the classical Newlander-Nirenberg theorem gives conditions on the vector fields so that there is a complex structure on $M$ with respect to which the vector fields are $T^{0,1}$. In this paper, we give intrinsic, diffeomorphic invariant, necessary and sufficient conditions on the vector fields so that they have a desired level of regularity with respect to this complex structure (i.e., smooth, real analytic, or have Zygmund regularity of some finite order). By addressing this in a quantitative way we obtain a holomorphic analog of the quantitative theory of sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. We call this sub-Hermitian geometry. Moreover, we proceed more generally and obtain similar results for manifolds which have an associated formally integrable elliptic structure. This allows us to introduce a setting which generalizes both the real and complex theories.

math.CV↗
arXiv2019

Endpoint Lebesgue estimates for weighted averages on polynomial curves

We establish optimal Lebesgue estimates for a class of generalized Radon transforms defined by averaging functions along polynomial-like curves. The presence of an essentially optimal weight allows us to prove uniform estimates, wherein the Lebesgue exponents are completely independent of the curves and the operator norms depend only on the polynomial degree. Moreover, our weighted estimates possess rather strong diffeomorphism invariance properties, allowing us to obtain uniform bounds for averages on curves satisfying a natural nilpotency hypothesis.

math.CA↗
arXiv2019

Sharp Regularity for the Integrability of Elliptic Structures

As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of $\mathbb{R}^r\times \mathbb{C}^n$ (for some $r$ and $n$) in such a way that the structure is locally the span of $\frac{\partial}{\partial t_1},\ldots, \frac{\partial}{\partial t_r},\frac{\partial}{\partial \overline{z}_1},\ldots, \frac{\partial}{\partial \overline{z}_n}$; where $\mathbb{R}^r\times \mathbb{C}^n$ has coordinates $(t_1,\ldots, t_r, z_1,\ldots, z_n)$. In this paper, we give optimal regularity for the coordinate charts which achieve this realization. Namely, if the manifold has Zygmund regularity of order $s+2$ and the structure has Zygmund regularity of order $s+1$ (for some $s>0$), then the coordinate chart may be taken to have Zygmund regularity of order $s+2$. We do this by generalizing Malgrange's proof of the Newlander-Nirenberg Theorem to this setting.

math.CV↗
arXiv2018

Coordinates Adapted to Vector Fields: Canonical Coordinates

Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields have a higher level of smoothness. For example, when is there a coordinate system in which the vector fields are smooth, or real analytic, or have Zygmund regularity of some finite order? We address this question in a quantitative way, which strengthens and generalizes previous works on the quantitative theory of sub-Riemannian (aka Carnot-Carathéodory) geometry due to Nagel, Stein, and Wainger, Tao and Wright, the second author, and others. Furthermore, we provide a diffeomorphism invariant version of these theories. This is the first part in a three part series of papers. In this paper, we study a particular coordinate system adapted to a collection of vector fields (sometimes called canonical coordinates) and present results related to the above questions which are not quite sharp; these results from the backbone of the series. The methods of this paper are based on techniques from ODEs. In the second paper, we use additional methods from PDEs to obtain the sharp results. In the third paper, we prove results concerning real analyticity and use methods from ODEs.

math.DG↗
arXiv2017

Singular integrals and a problem on mixing flows

We prove a result related to Bressan's mixing problem. We establish an inequality for the change of Bianchini semi-norms of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which we prove bounds on Hardy spaces. We include additional observations about the approach and a discrete toy version of Bressan's problem.

math.CA↗
arXiv2017

Differential Equations with a Difference Quotient

The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of such a differential equation is, for a polynomial $P$ and continuous functions $f(t,x):[0,1]\times [0,1]\rightarrow \mathbb{R}$, \begin{equation*} \frac{\partial}{\partial t} f(t,x) = \frac{ P(f(t,x))-P(f(t,0))}{x}, \quad x>0. \end{equation*} These differential equations are related to inverse problems.

math.CA↗
arXiv2016

Sobolev spaces associated to singular and fractional Radon transforms

The purpose of this paper is to study the smoothing properties (in $L^p$ Sobolev spaces) 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 $(t,x)\in\mathbb{R}^N\times \mathbb{R}^n$, satisfying $γ_0(x)\equiv x$, $ψ$ is a $C^\infty$ cut-off function supported on a small neighborhood of $0\in \mathbb{R}^n$, and $K$ is a "multi-parameter fractional kernel" supported on a small neighborhood of $0\in \mathbb{R}^N$. When $K$ is a Calderón-Zygmund kernel these operators were studied by Christ, Nagel, Stein, and Wainger, and when $K$ is a multi-parameter singular kernel they were studied by the author and Stein. In both of these situations, conditions on $γ$ were given under which the above operator is bounded on $L^p$ ($1<p<\infty$). Under these same conditions, we introduce non-isotropic $L^p$ Sobolev spaces associated to $γ$. Furthermore, when $K$ is a fractional kernel which is smoothing of an order which is close to $0$ (i.e., very close to a singular kernel) we prove mapping properties of the above operators on these non-isotropic Sobolev spaces. As a corollary, under the conditions introduced on $γ$ by Christ, Nagel, Stein, and Wainger, we prove optimal smoothing properties in isotropic $L^p$ Sobolev spaces for the above operator when $K$ is a fractional kernel which is smoothing of very low order.

math.CA↗