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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\u{\i} 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., Comment: 45 pages

Let $\alpha>0$, $\beta>\alpha$, and let $X_1,\ldots, X_q$ be $\mathscr{C}^{\alpha}_{\mathrm{loc}}$ vector fields on a $\mathscr{C}^{\alpha+1}$ manifold which span the tangent space at every point, where $\mathscr{C}^{s}$ denotes the Zygmund-H\"older space of order $s$. We give necessary and sufficient conditions for when there is a $\mathscr{C}^{\beta+1}$ structure on the manifold, compatible with its $\mathscr{C}^{\alpha+1}$ structure, with respect to which $X_1,\ldots, X_q$ are $\mathscr{C}^{\beta}_{\mathrm{loc}}$. This strengthens previous results of the first author which dealt with the setting $\alpha>1$, $\beta>\max\{ \alpha, 2\}$., Comment: 53 pages. Final version, to be appeared in Journal of Functional Analysis

We present and validate a novel method for noise injection of arbitrary spectra in quantum circuits that can be applied to any system capable of executing arbitrary single qubit rotations, including cloud-based quantum processors. As the consequences of temporally-correlated noise on the performance of quantum algorithms are not well understood, the capability to engineer and inject such noise in quantum systems is paramount. To date, noise injection capabilities have been limited and highly platform specific, requiring low-level access to control hardware. We experimentally validate our universal method by comparing to a direct hardware-based noise-injection scheme, using a combination of quantum noise spectroscopy and classical signal analysis to show that the two approaches agree. These results showcase a highly versatile method for noise injection that can be utilized by theoretical and experimental researchers to verify, evaluate, and improve quantum characterization protocols and quantum algorithms for sensing and computing., Comment: 8 pages, 6 figures. Updated References

Silicon Carbide is a promising host material for spin defect based quantum sensors owing to its commercial availability and established techniques for electrical and optical microfabricated device integration. The negatively charged silicon vacancy is one of the leading spin defects studied in silicon carbide owing to its near telecom photoemission, high spin number, and nearly temperature independent ground state zero field splitting. We report the realization of nanoTesla shot-noise limited ensemble magnetometry based on optically detected magnetic resonance with the silicon vacancy in 4H silicon carbide. By coarsely optimizing the anneal parameters and minimizing power broadening, we achieved a sensitivity of 3.5 nT/$\sqrt{Hz}$. This was accomplished without utilizing complex photonic engineering, control protocols, or applying excitation powers greater than a Watt. This work demonstrates that the silicon vacancy in silicon carbide provides a low-cost and simple approach to quantum sensing of magnetic fields.

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., Comment: v5: 62 pages, final version, to appear in Adv. Math

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., Comment: v3: 39 pages, final version, to appear in J. Funct. Anal

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., Comment: 44 pages; final version; to appear in Asian Jour. Math.; third part in a three-part series; part 1: arXiv:1709.04528; part 2: arXiv:1808.04159

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., Comment: v6: fixed minor typos, 36 pages. Final version, to appear in Amer. Jour. Math., Part 2 in a 3 parts series. Part 1: arXiv:1709.04528 Part 3: arXiv:1808.04635

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., Comment: 56 pages; this version was significantly revised from the previous version

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-Carath\'eodory) 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., Comment: Part 1 in a 3 part series, 64 pages. final version, to appear in GAFA; part 2: arXiv:1808.04159, part 3: arXiv:1808.04635

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., Comment: To appear in Annales de l'Institut Henri Poincar\'e (C) Analyse Non Lin\'eaire

We prove $L^{p_1}(\mathbb R^d)\times \dots \times L^{p_{n+2}}(\mathbb R^{d})$ polynomial growth estimates for the Christ-Journ\'e multilinear singular integral forms and suitable generalizations., Comment: To appear in Memoirs of the AMS

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 Calder\'on 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., Comment: 12 pages

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., Comment: 36 pages

The purpose of this paper is to study the smoothing properties (in $L^p$ Sobolev spaces) of operators of the form $f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt$, where $\gamma_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in\mathbb{R}^N\times \mathbb{R}^n$, satisfying $\gamma_0(x)\equiv x$, $\psi$ 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 Calder\'on-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 $\gamma$ were given under which the above operator is bounded on $L^p$ ($1

The goal of this paper is to study operators of the form, \[ Tf(x)= \psi(x)\int f(\gamma_t(x))K(t)\: dt, \] where $\gamma$ is a real analytic function defined on a neighborhood of the origin in $(t,x)\in \R^N\times \R^n$, satisfying $\gamma_0(x)\equiv x$, $\psi$ 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, \[ \mathcal{M} f(x) = \psi(x)\sup_{0<\delta_1,..., \delta_N<<1} \int_{|t|<1} |f(\gamma_{\delta_1 t_1,...,\delta_N t_N}(x))|\: dt. \] We show that $\mathcal{M}$ is bounded on $L^p$ ($1

The purpose of this paper is to study the $L^p$ boundedness of operators of the form \[ f\mapsto \psi(x) \int f(\gamma_t(x))K(t)\: dt, \] where $\gamma_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \R^N\times \R^n$, satisfying $\gamma_0(x)\equiv x$, $\psi$ 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 $\gamma$ such that every operator of the above form is bounded on $L^p$ ($1

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 \[ f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt, \] where $\gamma_t(x)=\gamma(t,x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \mathbb{R}^N\times \mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $K(t)$ is a "multi-parameter singular kernel" supported near $t=0$, and $\psi$ 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 $\gamma$ such that the above operator is bounded on $L^p$ for $1

The purpose of this paper is to study the $L^2$ boundedness of operators of the form \[ f\mapsto \psi(x) \int f(\gamma_t(x)) K(t) dt, \] where $\gamma_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \R^N\times \R^n$, satisfying $\gamma_0(x)\equiv x$, $\psi$ 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 $\gamma$ such that every operator of the above form is bounded on $L^2$. The case when $K$ is a Calder\'on-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 Calder\'on-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 $\gamma$ is real analytic., Comment: 60 pages; part 1 of a 3 part series; to appear in Journal d'Analyse Mathematique

We consider the problem of recovering the coefficient \sigma(x) of the elliptic equation \grad \cdot(\sigma \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, Sj\"ostrand, 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., Comment: Final version, 17 pages, to appear in Comm in PDE

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., Comment: Final version, to appear in Rev. Mat. Ibero

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 $e^{-t\square_b}-\pi$ satisfies good "on-diagonal" estimates, where $\pi$ is the Szeg\"o 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 $m\l(\square_b\r)$. In particular, we show that $m\l(\square_b\r)$ is an NIS operator, where $m$ satisfies an appropriate Mihlin-H\"ormander condition., Comment: 29 pages; minor corrections

We present an intrinsically defined algebra of operators containing the right and left invariant Calder\'on-Zygmund operators on a stratified group. The operators in our algebra are pseudolocal and bounded on L^p (1

The development of adult literacy provision in the United Kingdom from the 1960s onwards can be divided into three parts that correspond to significant shifts in approaches to adult literacy. First, the discovery of adult "illiteracy" during the 1960s led to government grants, a national Right to Read Campaign, and the development of local practice and experience. Second, there was a period of consolidation during the 1970s and early 1980s around the principle of learner-centered approaches, with minimal assessment procedures and central direction, and a growing body of expertise among practitioners who also began to undertake their own action research. The government-funded agency, Adult Literacy and Basic Skills Unit, consolidated and became expert in the production of materials, guidelines for good practice, and small research projects. A membership organization emerged for bridging academic/research and practitioner interests: Research and Practice in Adult Literacy. The third phase, which began in the late 1980s and continues currently, has involved a considerable shift of policy and focus, under pressure from a government concerned with ensuring that education generally responds to national and economic needs. The major finding is that literacy programs, curricula, and assessment should be addressed to the specificity of experience in different places and times. (Contains 82 references.) (YLB)