Transactions of the American Mathematical Society, Series B
This paper is part of a continuing examination into the geometric and analytic properties of the ... more This paper is part of a continuing examination into the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m \mathbb {C}^n\times \mathbb {C}^m . The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We (1) investigate the geometric conditions on M M which the eigenvalue condition forces, (2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, (3) explore the L p L^p and L p L^p -Sobolev mapping properties of the associated kernels, and (4) provide examples.
This chapter produces a new class of differential operators of order k (where k is any given posi... more This chapter produces a new class of differential operators of order k (where k is any given positive integer) that satisfy an appropriate analogue of a Gagliardo–Nirenberg inequality for functions and contain the operators introduced in the works of Bourgain and Brezis and Van Schaftingen. The research in this chapter is furthermore based on div/curl-type phenomena studied by both Stein as well as one of the authors of this chapter. Thus, the chapter first introduces the notion of admissible degree increment, and describes the necessary operators and theorems. Proofs are then discussed later on in the chapter, before it concludes with further remarks on some of the problems and theorems advanced earlier on.
This paper is the fourth of a multi-part series in which we study the geometric and analytic prop... more This paper is the fourth of a multi-part series in which we study the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of ℂ^n×ℂ^m. The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We 1) investigate the geometric conditions on M which the eigenvalue condition forces, 2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, 3) explore the L^p and L^p-Sobolev mapping properties of the associated kernels, and 4) provide examples.
We discuss the positivity (or lack thereof) of the non-vanishing coefficients of the Taylor serie... more We discuss the positivity (or lack thereof) of the non-vanishing coefficients of the Taylor series expansion about z 0 = 0 for the Riemann map of a family of rectangles and rhombi with defined symmetries.
Proceedings of the American Mathematical Society, 2008
Let L γ = − 1 4 n j=1 (X 2 j + Y 2 j) + iγT where γ ∈ C, and X j , Y j and T are the left-invaria... more Let L γ = − 1 4 n j=1 (X 2 j + Y 2 j) + iγT where γ ∈ C, and X j , Y j and T are the left-invariant vector fields of the Heisenberg group structure for R n × R n × R. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the heat equation ∂ s ρ = −L γ ρ. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the b-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted ∂-operator in C n with weight exp(−τ P (z 1 ,. .. , z n)), where P (z 1 , .
Journal of Fourier Analysis and Applications, 2012
In this article, we establish Gaussian decay for the b-heat kernel on polynomial models in C 2. O... more In this article, we establish Gaussian decay for the b-heat kernel on polynomial models in C 2. Our technique attains the exponential decay via a partial Fourier transform. On the transform side, the problem becomes finding quantitative smoothness estimates on a heat kernel associated to the weighted∂-operator on L 2 (C). The bounds are established with Duhamel's formula and careful estimation.
Proceedings of the American Mathematical Society, Series B
This paper is the first of a three part series in which we explore geometric and analytic propert... more This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m \mathbb {C}^n\times \mathbb {C}^m . In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N N and the projection onto the nullspace of ◻ b \Box _b . The main application of our formulas is the critical case of codimension two quadrics in C 4 \mathbb {C}^4 where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.
Let p : C → R be a subharmonic, nonharmonic polynomial and τ ∈ R a parameter. Define Z̄τp = ∂ ∂z̄... more Let p : C → R be a subharmonic, nonharmonic polynomial and τ ∈ R a parameter. Define Z̄τp = ∂ ∂z̄ + τ ∂p ∂z̄ = e ∂ ∂z̄ e, a closed, densely defined operator on L(C). If τp = Z̄τpZ̄ ∗ τp and ̃τp = Z̄ ∗ τpZ̄τp, we solve the heat equations ∂su+ τpu = 0, u(0, z) = f(z) and ∂sũ+̃τpũ = 0, ũ(0, z) = f̃(z). We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.
In this article, we develop the theory of weighted L 2 Sobolev spaces on unbounded domains in R n... more In this article, we develop the theory of weighted L 2 Sobolev spaces on unbounded domains in R n. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the bounded, unweighted case. Contents 1. Preliminaries 2 2. Main Results 5 3. Facts for W m,p (Ω, ϕ; X), 1 < p < ∞ 9 4. Sobolev spaces on M 13 5. Weighted Besov spaces on Ω and M 18 6. Interior estimates-the proof of Theorem 2.13 27 7. Elliptic regularity at the boundary 32 8. Traces of L-harmonic functions 39 Appendix A. Background on interpolation-the real method 43 References 46
The L 2 theory of the∂ operator on domains in C n is predicated on establishing a good basic esti... more The L 2 theory of the∂ operator on domains in C n is predicated on establishing a good basic estimate. Typically, one proves not a single basic estimate but a family of basic estimates that we call a family of strong closed range estimates. Using this family of estimates on (0, q)-forms as our starting point, we establish necessary geometric and potential theoretic conditions. The paper concludes with several applications. We investigate the consequences for compactness estimates for the∂-Neumann problem, and we also establish a generalization of Kohn's weighted theory via elliptic regularization. Since our domains are not necessarily pseudoconvex, we must take extra care with the regularization.
It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, th... more It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the $\bar\partial$-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for $(0,1)$-forms) and Ben Moussa (for $(0,q)$-forms when $q\geq 1$). In this paper, we explore necessary conditions and sufficient conditions for maximal estimates in the non-pseudoconvex case. We also discuss when the necessary conditions and sufficient conditions agree and provide examples. Our results subsume the earlier known results from the pseudoconvex case.
The purpose of this paper is to prove L^p-Sobolev and Hölder estimates for the Bergman projection... more The purpose of this paper is to prove L^p-Sobolev and Hölder estimates for the Bergman projection on a class of pseudoconvex domains that admit a "good" dilation and satisfy Bell-Ligocka's Condition R. We prove that this class of domains includes the h-extendible domains, a large class of weakly pseudoconvex domains of finite type.
In this article, we establish a general sufficient condition for closed range of the Cauchy-Riema... more In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator ∂̅ in appropriately weighted L^2 and L^2-Sobolev spaces on (0,q)-forms for a fixed q on domains in C^n. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical Z(q) condition that we call weak Z(q). We provide examples that explain the necessity of working in weighted spaces both for closed range in L^2 and even more critically, in L^2-Sobolev spaces.
The goal of this paper is to introduce a class of C∞ functions whose derivatives satisfy quantita... more The goal of this paper is to introduce a class of C∞ functions whose derivatives satisfy quantitative size estimates. The estimates, called global L Gevrey estimates, first arose in the work of Boggess and Raich [BR13] when they investigated how to capture a particular type of exponential decay through estimates on the Fourier transform. In the present work, we refine the notion of global L-Gevrey functions and include a discussion of the function theory as well as the relationship to Gevrey classes and known function spaces. Additionally, we present explicit examples of global L-Gevrey functions and ways to generate new global L-Gevrey functions from old ones. We conclude with three applications: The first is solving a Carleman-type problem for constructing functions whose derivatives are a given sequence of global L-Gevrey functions. The other two applications concern extensions of a given global L-Gevrey function: the first is constructing an almost analytic extension, and the se...
The purpose of this paper is to characterize the zero sets of holomorphic functions in the Nevanl... more The purpose of this paper is to characterize the zero sets of holomorphic functions in the Nevanlinna class on a class of convex domains of infinite type in $\mathbb{C}^2$. Moreover, we also obtain $L^p$ estimates, $1 \leq p \leq \infty$, for a particular solution of the tangential Cauchy-Riemann equation on the boundaries of these domains.
Abstract. This paper provides a connection between two distinct branches of research in CR geomet... more Abstract. This paper provides a connection between two distinct branches of research in CR geometry – namely, analytic and geometric conditions that suffice to establish the closed range of the Cauchy-Riemann operator and CR invariants on CR manifolds. Specifically, we work on not necessarily pseudoconvex domains Ω ⊂ C and define third and fourth order CR invariants on bΩ and show that these invariants provide enough information to establish closed range for the ∂̄-Laplacian in L(0,q)(Ω) for a given, fixed q. The closed range estimates follow from our previously defined weak Z(q) condition. We also develop powerful linear algebra machinery to translate the information from the invariants into information about the Levi form and its eigenvalues. We conclude with several examples that demonstrate the usefulness and ease of use of the new conditions.
This paper is the second of a multi-part series in which we explore geometric and analytic proper... more This paper is the second of a multi-part series in which we explore geometric and analytic properties of the Kohn-Laplacian and its inverse on general quadric submanifolds of ℂ n × ℂ m. We have two goals in this paper. The first is to give useable sufficient conditions for a map T between quadrics to be a Lie group isomorphism that preserves □ b , and the second is to establish a framework for which appropriate derivatives of the complex Green operator are continuous in L p and L p-Sobolev spaces (and hence are hypoelliptic). We apply the general results to codimension two quadrics in ℂ 4 .
Transactions of the American Mathematical Society, Series B
This paper is part of a continuing examination into the geometric and analytic properties of the ... more This paper is part of a continuing examination into the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m \mathbb {C}^n\times \mathbb {C}^m . The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We (1) investigate the geometric conditions on M M which the eigenvalue condition forces, (2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, (3) explore the L p L^p and L p L^p -Sobolev mapping properties of the associated kernels, and (4) provide examples.
This chapter produces a new class of differential operators of order k (where k is any given posi... more This chapter produces a new class of differential operators of order k (where k is any given positive integer) that satisfy an appropriate analogue of a Gagliardo–Nirenberg inequality for functions and contain the operators introduced in the works of Bourgain and Brezis and Van Schaftingen. The research in this chapter is furthermore based on div/curl-type phenomena studied by both Stein as well as one of the authors of this chapter. Thus, the chapter first introduces the notion of admissible degree increment, and describes the necessary operators and theorems. Proofs are then discussed later on in the chapter, before it concludes with further remarks on some of the problems and theorems advanced earlier on.
This paper is the fourth of a multi-part series in which we study the geometric and analytic prop... more This paper is the fourth of a multi-part series in which we study the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of ℂ^n×ℂ^m. The goal of this article is explore the complex Green operator in the case that the eigenvalues of the directional Levi forms are nonvanishing. We 1) investigate the geometric conditions on M which the eigenvalue condition forces, 2) establish optimal pointwise upper bounds on complex Green operator and its derivatives, 3) explore the L^p and L^p-Sobolev mapping properties of the associated kernels, and 4) provide examples.
We discuss the positivity (or lack thereof) of the non-vanishing coefficients of the Taylor serie... more We discuss the positivity (or lack thereof) of the non-vanishing coefficients of the Taylor series expansion about z 0 = 0 for the Riemann map of a family of rectangles and rhombi with defined symmetries.
Proceedings of the American Mathematical Society, 2008
Let L γ = − 1 4 n j=1 (X 2 j + Y 2 j) + iγT where γ ∈ C, and X j , Y j and T are the left-invaria... more Let L γ = − 1 4 n j=1 (X 2 j + Y 2 j) + iγT where γ ∈ C, and X j , Y j and T are the left-invariant vector fields of the Heisenberg group structure for R n × R n × R. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the heat equation ∂ s ρ = −L γ ρ. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the b-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted ∂-operator in C n with weight exp(−τ P (z 1 ,. .. , z n)), where P (z 1 , .
Journal of Fourier Analysis and Applications, 2012
In this article, we establish Gaussian decay for the b-heat kernel on polynomial models in C 2. O... more In this article, we establish Gaussian decay for the b-heat kernel on polynomial models in C 2. Our technique attains the exponential decay via a partial Fourier transform. On the transform side, the problem becomes finding quantitative smoothness estimates on a heat kernel associated to the weighted∂-operator on L 2 (C). The bounds are established with Duhamel's formula and careful estimation.
Proceedings of the American Mathematical Society, Series B
This paper is the first of a three part series in which we explore geometric and analytic propert... more This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of C n × C m \mathbb {C}^n\times \mathbb {C}^m . In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N N and the projection onto the nullspace of ◻ b \Box _b . The main application of our formulas is the critical case of codimension two quadrics in C 4 \mathbb {C}^4 where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.
Let p : C → R be a subharmonic, nonharmonic polynomial and τ ∈ R a parameter. Define Z̄τp = ∂ ∂z̄... more Let p : C → R be a subharmonic, nonharmonic polynomial and τ ∈ R a parameter. Define Z̄τp = ∂ ∂z̄ + τ ∂p ∂z̄ = e ∂ ∂z̄ e, a closed, densely defined operator on L(C). If τp = Z̄τpZ̄ ∗ τp and ̃τp = Z̄ ∗ τpZ̄τp, we solve the heat equations ∂su+ τpu = 0, u(0, z) = f(z) and ∂sũ+̃τpũ = 0, ũ(0, z) = f̃(z). We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.
In this article, we develop the theory of weighted L 2 Sobolev spaces on unbounded domains in R n... more In this article, we develop the theory of weighted L 2 Sobolev spaces on unbounded domains in R n. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the bounded, unweighted case. Contents 1. Preliminaries 2 2. Main Results 5 3. Facts for W m,p (Ω, ϕ; X), 1 < p < ∞ 9 4. Sobolev spaces on M 13 5. Weighted Besov spaces on Ω and M 18 6. Interior estimates-the proof of Theorem 2.13 27 7. Elliptic regularity at the boundary 32 8. Traces of L-harmonic functions 39 Appendix A. Background on interpolation-the real method 43 References 46
The L 2 theory of the∂ operator on domains in C n is predicated on establishing a good basic esti... more The L 2 theory of the∂ operator on domains in C n is predicated on establishing a good basic estimate. Typically, one proves not a single basic estimate but a family of basic estimates that we call a family of strong closed range estimates. Using this family of estimates on (0, q)-forms as our starting point, we establish necessary geometric and potential theoretic conditions. The paper concludes with several applications. We investigate the consequences for compactness estimates for the∂-Neumann problem, and we also establish a generalization of Kohn's weighted theory via elliptic regularization. Since our domains are not necessarily pseudoconvex, we must take extra care with the regularization.
It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, th... more It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the $\bar\partial$-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for $(0,1)$-forms) and Ben Moussa (for $(0,q)$-forms when $q\geq 1$). In this paper, we explore necessary conditions and sufficient conditions for maximal estimates in the non-pseudoconvex case. We also discuss when the necessary conditions and sufficient conditions agree and provide examples. Our results subsume the earlier known results from the pseudoconvex case.
The purpose of this paper is to prove L^p-Sobolev and Hölder estimates for the Bergman projection... more The purpose of this paper is to prove L^p-Sobolev and Hölder estimates for the Bergman projection on a class of pseudoconvex domains that admit a "good" dilation and satisfy Bell-Ligocka's Condition R. We prove that this class of domains includes the h-extendible domains, a large class of weakly pseudoconvex domains of finite type.
In this article, we establish a general sufficient condition for closed range of the Cauchy-Riema... more In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator ∂̅ in appropriately weighted L^2 and L^2-Sobolev spaces on (0,q)-forms for a fixed q on domains in C^n. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical Z(q) condition that we call weak Z(q). We provide examples that explain the necessity of working in weighted spaces both for closed range in L^2 and even more critically, in L^2-Sobolev spaces.
The goal of this paper is to introduce a class of C∞ functions whose derivatives satisfy quantita... more The goal of this paper is to introduce a class of C∞ functions whose derivatives satisfy quantitative size estimates. The estimates, called global L Gevrey estimates, first arose in the work of Boggess and Raich [BR13] when they investigated how to capture a particular type of exponential decay through estimates on the Fourier transform. In the present work, we refine the notion of global L-Gevrey functions and include a discussion of the function theory as well as the relationship to Gevrey classes and known function spaces. Additionally, we present explicit examples of global L-Gevrey functions and ways to generate new global L-Gevrey functions from old ones. We conclude with three applications: The first is solving a Carleman-type problem for constructing functions whose derivatives are a given sequence of global L-Gevrey functions. The other two applications concern extensions of a given global L-Gevrey function: the first is constructing an almost analytic extension, and the se...
The purpose of this paper is to characterize the zero sets of holomorphic functions in the Nevanl... more The purpose of this paper is to characterize the zero sets of holomorphic functions in the Nevanlinna class on a class of convex domains of infinite type in $\mathbb{C}^2$. Moreover, we also obtain $L^p$ estimates, $1 \leq p \leq \infty$, for a particular solution of the tangential Cauchy-Riemann equation on the boundaries of these domains.
Abstract. This paper provides a connection between two distinct branches of research in CR geomet... more Abstract. This paper provides a connection between two distinct branches of research in CR geometry – namely, analytic and geometric conditions that suffice to establish the closed range of the Cauchy-Riemann operator and CR invariants on CR manifolds. Specifically, we work on not necessarily pseudoconvex domains Ω ⊂ C and define third and fourth order CR invariants on bΩ and show that these invariants provide enough information to establish closed range for the ∂̄-Laplacian in L(0,q)(Ω) for a given, fixed q. The closed range estimates follow from our previously defined weak Z(q) condition. We also develop powerful linear algebra machinery to translate the information from the invariants into information about the Levi form and its eigenvalues. We conclude with several examples that demonstrate the usefulness and ease of use of the new conditions.
This paper is the second of a multi-part series in which we explore geometric and analytic proper... more This paper is the second of a multi-part series in which we explore geometric and analytic properties of the Kohn-Laplacian and its inverse on general quadric submanifolds of ℂ n × ℂ m. We have two goals in this paper. The first is to give useable sufficient conditions for a map T between quadrics to be a Lie group isomorphism that preserves □ b , and the second is to establish a framework for which appropriate derivatives of the complex Green operator are continuous in L p and L p-Sobolev spaces (and hence are hypoelliptic). We apply the general results to codimension two quadrics in ℂ 4 .
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