On integral representations and a priori Lipschitz estimates for the canonical solution of the $$\bar \partial $$ -equation

Lieb, Ingo; Range, R. Michael

doi:10.1007/bf01460799

Cited by 22 publications

(45 citation statements)

References 16 publications

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“…We will transfer the kernel EkH¡ from the model to the domain Q via a special Heisenberg coordinate system which was introduced by Phong and Stein [26,27]. We just summarize Phong and Stein's results and compare them to the results obtained by Lieb and Range [18,19]. We have mentioned standard Heisenberg coordinate system in §1 already.…”

Section: Transferredmentioning

confidence: 98%

“…Let U be a boundary neighborhood of 0 e ÖD c Cn+X ,fe C(~1)i0(C/) n dom(¿0 with df=0. In this section we apply the Lieb-Range's theorem (see [18,19]) to find the integral representation for the Kohn solution of the Cauchy-Riemann equation on the Siegel upper-half space. Using the same notations as in [18,19], we define operators T and E as follows:…”

Section: Amentioning

confidence: 99%

“…The purpose of this paper is to discuss the integral representation for the Kohn solution of the Cauchy-Riemann equation, using more direct methods than those described above. We look at the problem in the following way (as [1,5,8,18,19]): When n > 1 , the equation dp = f is over-determined. It can be solved only when / satisfies the consistency condition df -0.…”

Section: Introductionmentioning

confidence: 99%

“…In those papers, they also open a new direction by discussing singular integrals whose singularity is not a point! On the other hand, Lieb and Range [18,19,20] studied this problem by different methods via looking at the integral representations. They found an integral representation for a function u which defined on Q.…”

Section: Introductionmentioning

confidence: 99%

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Optimal 𝐿^{𝑝} and Hölder estimates for the Kohn solution of the \overline∂-equation on strongly pseudoconvex domains

Chang

1989

Trans. Amer. Math. Soc.

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Abstract.Let Í2 be an open, relatively compact subset in C+1 , and assume the boundary of ß , dCl, is smooth and strongly pseudoconvex. Let Op(£) be an integral operator with mixed type homogeneities defined on ii : i.e., K has the form as follows:where Ek is a homogeneous kernel of degree -A: in the Euclidean sense and H/ is homogeneous of degree -/ in the Heisenberg sense. In this paper, we study the optimal U and Holder estimates for the kernel K . We also use LiebRange's method to construct the integral kernel for the Kohn solution 3'N of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to d*N. On the other hand, we prove Lieb-Range's kernel gains 1 in "good" directions (hence gains 1/2 in all directions) via Phong-Stein's theory. We also discuss the transferred kernel from the Siegel upper-half space to fi .

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Section: Transferredmentioning

confidence: 98%

Section: Amentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal 𝐿^{𝑝} and Hölder estimates for the Kohn solution of the \overline∂-equation on strongly pseudoconvex domains

Chang

1989

Trans. Amer. Math. Soc.

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“…In recent years integral formulas and their applications have attracted a lot of attention in several complex variables; see for example [4,5,7,8,9, 10] and the most relevant to our work papers of Stout [15], Palm [11] and Henkin and Leiterer [6].…”

mentioning

confidence: 99%

Integral representation formulas on analytic varieties

Hatziafratis¹

1986

Pacific J. Math.

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Integral representation formulas for holomorphic functions on analytic subvarieties of domains of C" are derived. These formulas generalize the Cauchy-Fantappie formula and the Weil formula for analytic polyhedra. The kernels we obtain are explicitly defined.Introduction. In recent years integral formulas and their applications have attracted a lot of attention in several complex variables; see for example [4,5,7,8,9, 10] and the most relevant to our work papers of Stout [15], Palm [11] and Henkin and Leiterer [6].In this paper we develop analogues of Cauchy-Fantappie Kernels for analytic subvarieties of domains of C". First let us recall the CauchyFantappie formula. Let ΰcC" be a bounded domain with smooth boundary and let γ: (dD) X D -> C" be a smooth function so that Some of the interesting features of Theorem I.I is on the one hand the explicit form of the kernel K^ξ, z) and on the other hand the fact that M is allowed to have, finitely many, singular points.The case m = 0 of Theorem LI is the formula (2) and the case m = 1 of it was obtained by Stout [15]. In fact Stout's paper was the starting point of this work. The kernel obtained by Stout coincides with ours in the case m = 1; this is not immediate however and in §11 we show that they are indeed the same.The proof of Theorem I.I (given in this paper) is an extension of Stout's proof from the case m = 1 to the general case. A second proof of Theorem I.I is contained in Hatziafratis [3].In §111 we develop a Weil type integral formula for analytic polyhedra on analytic varieties. The main result of this section is Theorem III.l which generalizes the Weil integral formula for analytic polyhedra in C" (see [14, 16]). To obtain this result we combine results from §1 together with some standard techniques contained for example in Range-Siu [12].As we pointed out before the results in this paper are related to those of Palm [11] and Henkin and Leiterer [6]. The setting of Henkin and Leiterer [6] is more general than ours (we allow, however, finitely many singular points on M). On the other hand our results are more explicit than theirs. In fact we do not know the relation between our results and those of Henkin-Leiterer and Palm.Acknowledgment. I would like to express my warmest thanks to Professor Alexander Nagel for very helpful discussions.

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Boundary values of integral operators

Hefer

2003

Mathematische Nachrichten

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Consider a differential form u in the image of an integral operator K on a smooth domain D in a Riemannian manifold, i.e. u(y) = Kf(y) = x∈D f (x) ∧ K(x, y) for y ∈ D. If the kernel K of the integral operator is sufficiently regular and defined also for y in the boundary bD of D, one may define two different kinds of boundary values of u on bD. Firstly, u may have boundary values in the sense of distributions, i.e. boundary values satisfying a Stokes' formula in a suitably weak sense. Secondly, one can simply restrict (pull back) y in the kernel K(x, y) to the boundary of D, then integrate with respect to x in the above formula. It is interesting to know under which hypotheses on f and K both types of boundary values agree, because the boundary values defined by restricting the kernel can often be estimated by direct methods, whereas the abstractly given distributional boundary values are less tractable but analytically interesting objects linked to u. We will give counter-examples to show that even in quite regular situations the two notions do not necessarily agree. Then we study conditions implying equality. We also mention some interesting applications, e.g. generalizations of Stokes' formula and applications in the theory of integral representations in several complex variables.

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On integral representations and a priori Lipschitz estimates for the canonical solution of the $$\bar \partial $$ -equation

Cited by 22 publications

References 16 publications

Optimal 𝐿^{𝑝} and Hölder estimates for the Kohn solution of the \overline∂-equation on strongly pseudoconvex domains

Optimal 𝐿^{𝑝} and Hölder estimates for the Kohn solution of the \overline∂-equation on strongly pseudoconvex domains

Integral representation formulas on analytic varieties

Boundary values of integral operators

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