Tangential Cauchy-Riemann Operator Research Articles

The quaternionic Monge–Ampère operator and plurisubharmonic functions on the Heisenberg group

Many fundamental results of pluripotential theory on the quaternionic space $${\mathbb {H}}^n$$ are extended to the Heisenberg group. We introduce notions of a plurisubharmonic function, the quaternionic Monge–Ampere operator, differential operators $$d_0$$ and $$d_1$$ and a closed positive current on the Heisenberg group. The quaternionic Monge–Ampere operator is the coefficient of $$ (d_0d_1u)^n$$ . We establish the Chern–Levine–Nirenberg type estimate, the existence of quaternionic Monge–Ampere measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternionic Monge–Ampere operator. Unlike the tangential Cauchy–Riemann operator $$ {\overline{\partial }}_b $$ on the Heisenberg group which behaves badly as $$ \partial _b{\overline{\partial }}_b\ne -{\overline{\partial }}_b\partial _b $$ , the quaternionic counterpart $$d_0$$ and $$d_1$$ satisfy $$ d_0d_1=-d_1d_0 $$ . This is the main reason that we have a good theory for the quaternionic Monge–Ampere operator than $$ (\partial _b{\overline{\partial }}_b)^n$$ .

A note on the closed range of $${\bar{\partial }}_b$$ on q-convex manifolds

We prove that the tangential Cauchy–Riemann operator $${\bar{\partial }}_b$$ has closed range on Levi-pseudoconvex CR manifolds that are embedded in a q-convex complex manifold X. Our result generalizes the known case when X is a Stein manifold (in particular, when $$X={\mathbb C}^n$$ ).

The Tangential $$\varvec{k}$$-Cauchy–Fueter Operator and $$\varvec{k}$$-CF Functions Over the Heisenberg Group

In this paper, we investigate quaternionic analysis on the $$(4n+1)$$-dimensional Heisenberg group. The tangential k-Cauchy–Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables, respectively. We give the Penrose integral formula for k-CF functions and establish the Bochner–Martinelli formula for the tangential k-Cauchy–Fueter operator. We also construct the tangential k-Cauchy–Fueter complex.

On Transversally Elliptic Operators and the Quantization of Manifolds with f-Structure

An $f$-structure on a manifold $M$ is an endomorphism field $\phi\in\Gamma(M,\End(TM))$ such that $\phi^3+\phi=0$. Any $f$-structure $\phi$ determines an almost CR structure $E_{1,0}\subset T_\C M$ given by the $+i$-eigenbundle of $\phi$. Using a compatible metric $g$ and connection $\nabla$ on $M$, we construct an odd first-order differential operator $D$, acting on sections of $\S=\Lambda E_{0,1}^*$, whose principal symbol is of the type considered in arXiv:0810.0338. In the special case of a CR-integrable almost $\S$-structure, we show that when $\nabla$ is the generalized Tanaka-Webster connection of Lotta and Pastore, the operator $D$ is given by $D = \sqrt{2}(\dbbar+\dbbar^*)$, where $\dbbar$ is the tangential Cauchy-Riemann operator. We then describe two "quantizations" of manifolds with $f$-structure that reduce to familiar methods in symplectic geometry in the case that $\phi$ is a compatible almost complex structure, and to the contact quantization defined in \cite{F4} when $\phi$ comes from a contact metric structure. The first is an index-theoretic approach involving the operator $D$; for certain group actions $D$ will be transversally elliptic, and using the results in arXiv:0810.0338, we can give a Riemann-Roch type formula for its index. The second approach uses an analogue of the polarized sections of a prequantum line bundle, with a CR structure playing the role of a complex polarization.

Global Homotopy Formulas on q-Concave CR Manifolds for Large Degrees

Using functional analysis and a Friedrichs approximation lemma for first order differential operators, we derive a global homotopy formula in large degrees for the tangential Cauchy-Riemann operator from local homotopy formulas without loss of regularity.

The [formula omitted] equation on weakly pseudo-convex CR manifolds of dimension 3

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in C N ( N ⩾ 2 ) , of codimension one or more, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator ∂ ¯ b has closed range in L 2 ( M ) in order to rule out the Rossi example, we push regularity up to show ∂ ¯ b has closed range in H s ( M ) for all s > 0 . We then use the Szegö projection to show there is a smooth solution for the ∂ ¯ b problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.

Global regularity for [formula omitted] on weakly pseudoconvex CR manifolds

Let M 2 n - 1 be a compact, orientable, weakly pseudoconvex manifold of dimension at least five, embedded in C N ( n ⩽ N ) , of codimension one or more, and endowed with the induced CR structure. We show the tangential Cauchy–Riemann operator ∂ ¯ b has closed range on such a manifold M , hence we get global existence and regularity results for the ∂ ¯ b problem. We also show the middle (i.e. 1 ⩽ q ⩽ n - 2 ) ∂ ¯ b cohomology groups of M , H 0 p , q ( M , ∂ ¯ b ) , H s p , q ( M , ∂ ¯ b ) , and H ∞ p , q ( M , ∂ ¯ b ) with respect to L 2 , Sobolev s norm, and C ∞ coefficients respectively are finite and isomorphic to each other. The results are obtained by microlocalization using a new type of weight function called strongly CR plurisubharmonic.

DERIVATIONS ON CR MANIFOLDS

We studied the relation between the tangential Cauchy-Riemann operator <TEX>${\={\partial}}_b$</TEX> CR-manifolds and the derivation <TEX>$d^{{\pi}^{0,\;1}}$</TEX> associated to the natural projection map <TEX>${\pi}^{0.1}\;:\;TM\;{\bigotimes}\;{\mathbb{C}}\;=\;T^{1,0}\;{\bigoplus}\;T^{0,\;1}\;{\rightarrow}\;T^{0,\;1}$</TEX>. We found that these two differential operators agree only on the space of functions <TEX>${\Omega}^0(M),\;unless\;T^{1,\;0}$</TEX> is involutive as well. We showed that the difference is a derivation, which vanishes on <TEX>${\Omega}^0(M)$</TEX>, and it is induced by the Nijenhuis tensor associated to <TEX>${\pi}^{0.1}$</TEX>.

A differential criterium for regularity of quaternionic functions

Let Ω⊆ C 2 . We prove that there exist differential operators T and N, with complex coefficients, such that a function f : Ω→ H of class C 1 is regular if and only if ( N− jT) f=0 on ∂Ω ( j a basic quaternion) and f is harmonic on Ω. At the same time we generalize a result of Kytmanov and Aizenberg. We show that a complex harmonic function h on Ω ( ∂Ω connected) is holomorphic if and only if ∂ ̄ nh= aL(h) on ∂Ω, where ∂ ̄ n is the normal part of ∂ ̄ , L is a tangential Cauchy–Riemann operator and a∈ C . To cite this article: A. Perotti, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Local exactness of top-degree forms in abstract Cauchy-Riemann complexes

We show that the Poincare lemma for the tangential Cauchy-Riemann operator in the top degree is valid in abstract 1-concave CR manifolds.

On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian

Let M be the boundary of a (smoothly bounded) pseudoconvex domain in C n ( n ≥ 3), or more generally any compact pseudoconvex CR-manifold of dimension 2 n - 1 for which the range of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] is closed in L 2 . In this article, we study the L p -Sobolev and Hölder regularity properties of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] and □ b near a point of finite type under a comparable eigenvalues condition on the Levi form. We show that if all possible sums of q 0 eigenvalues of the Levi matrix are comparable to its trace near a point of finite commutator type ("Condition D ( q 0 )"), then the inverse K q of □ b on (0, q )-forms for q 0 ≤ q ≤ n - 1 - q 0 satisfies sharp kernel estimates in terms of the quasi-distance associated to the Hörmander sum of squares operator. In particular, we obtain the "maximal L p estimates" for □ b which were conjectured in the 1980s. We also prove sharp estimates for certain parts of the kernels of K q 0 -1 and K n-q 0 and give some applications concerning domains with at most one degenerate eigenvalue. Finally, we establish the composition and mapping properties of a class of singular integral (nonisotropic smoothing) operators that arises naturally in complex analysis. These results yield optimal regularity of K q (and related operators) in the Sobolev and Lipschitz norms, both isotropic and nonisotropic.

Boundary regularity for the local $$\bar \partial _b - complex$$ on certain weakly pseudoconvex surfaceson certain weakly pseudoconvex surfaces

Let U be a neighborhood of 0in ℂn, 1 ≤s ≤ n − 1,m ∈ ℕ,M = {(z1,..., zn) ∈ U:Imzn = ¦z1¦2 + ...¦zs¦2 + (¦zs+1¦2 +...+ ¦zn−1¦2)m +some error} term be a hypersurface in U. In this paper, the author constructs homotopy operators for the local tangential Cauchy-Riemann operator on the neighborhoods Mδ:= M ∩ {¦zn¦ < δ} of 0in M. C0 -estimates up to the boundary are proved for the operators. The constants in the estimates do not depend on δ.

Regularity of the tangential Cauchy-Riemann complex and applications

In a series of papers Webster ([26], [27], [28]) has shown how C-estimates for the tangential Cauchy-Riemann complex can be applied to several non-linear problems in Complex Analysis. For example, he gave a simplification of the proof of Kuranishi’s embedding theorem and an application to the integrability problem for almost CR vector bundles. Connected with Webster’s approach are some regularity assertions, whose parameters follow from the C-estimates. Therefore, an improvement of these estimates would lead to an improvement in the applications. Now the author of this article, together with Lan Ma, has given in [12] such an improvement. In two subsequent papers this has been applied to Webster’s approach. The results are formulated in two theorems at the end of this introduction. Here we want to briefly describe the estimates for the tangential CR equations. Let G be a strictly pseudoconvex domain in C with C-smooth boundary and 0 ∈ bG. By a local homotopy formula for the tangential Cauchy-Riemann operator ∂b we mean the following. There exists a neighborhood base {M} of relatively open sets M , with 0 ∈ M ⊂ bG, and on each M there are given operators Rq (q = 1, 2, . . . , n− 2), which fullfill the equation

On the removal of singular sets for the tangential Cauchy—Riemann operator

On the removal of singular sets for the tangential Cauchy—Riemann operator

The range of the tangential Cauchy-Riemann operator over a small ball

In this paPer we study the closed range property of tangential Cauchy-Riemann equations of an abstract CR structure over a small ball. Let D be the set {x E R*“’ 1 1x1 4, Kuranishi [ 141 and Akahori [ 11 have studied the L* theory

The range of the tangential Cauchy-Riemann operator

The range of the tangential Cauchy-Riemann operator

Toeplitz operators on certain weakly pseudoconvex domains

AbstractLet Ω be the weakly pseudoconvex domainand let ∂Ω be its boundary. If ϕ ∈ L∞ (∂Ω), we denote by Tϕ, the Toephtz operator with symbol ϕ acting on the Hardy space H2(∂Ω), and by J(∂Ω) the C*-subalgebra of B(H2(∂Ω)) generated by the Toeplitz operators with continuous symbol. Our main theorem asserts that J(∂Ω) contains the ideal K of all compact operators on H2(∂Ω), and that the symbol map ϕ→Tϕ induces an isomorphism of C(∂Ω) onto the quotient C*-algebra ℑ(∂Ω)/K. Similar results have been established before for other domains, and in particular when Ω is strongly pseudoconvex. The main interest of our results lies in their proofs: ours are elementary, whereas those used in the strongly pseudoconvex case depend heavily on the theory of the tangential Cauchy-Riemann operator.

Higher order analogues to the tangential Cauchy-Riemann equations for real submanifolds of 𝐶ⁿ with C.R. singularity

An infinite succession of higher order operators are developed for real C ∞ {\mathcal {C}^\infty } submanifolds of C n {C^n} , with possible C.R. singularity, which reduce to the tangential Cauchy-Riemann operator in the case of a C.R. submanifold. Certain known holomorphic approximation and extension results for C.R. submanifolds are then “extended” to the non-C.R. case.

The complex Frobenius Theorem and uniqueness of solutions to the tangential Cauchy-Riemann equations

Suppose M is a C ∞ real k-dimensional CR-submanifold of C n , n > 1, and suppose that ∂ ̄ t6 M is the tangential Cauchy-Riemann operator on M. Let S be a C 1 real ( k − 1)-dimensional submanifold of M which is noncharacteristic for ∂̄t6 M at p ϵ S. Conditions are found so that a C ∞ solution f of ∂ ̄ t6 Mf = 0 which vanishes on one side of S in M must vanish in a neighborhood of p in M. If M is a real hypersurface, it is known that such unique continuation always exists. If the codimension of M in C n is greater than 1, and if the excess dimension of the Levi algebra on M is constant, then it is proved that CR-functions on M which vanish on one side of S must vanish in a full neighborhood of p. The assumption on the dimension of the Levi algebra allows us to use the Complex Frobenius Theorem. Other methods to prove such unique continuation results are also developed.

A generalization of a theorem of Hans Lewy

Let p be a real-valued Coo function defined in a neighborhood of the origin 0 in Cn, such that p(O) =O, dp(O) #0. Then, near zero, M= {z; p(z)=O} is a real submanifold of Cn of dimension 2n-1. If 30p(O) 5O, then M has a holomorphic hull which contains an open set. We shall prove an L2 version of this fact. Let 0b represent the tangential Cauchy-Riemann operator on M introduced by Kohn [1]. By L2 on M, we mean the space of functions which are square integrable with respect to surface area.