Uniform estimates for the Cauchy-Riemann equation on q -convex wedges

Christine Laurent-Thiébaut; Jurgen Leiterer

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 2, page 383-436
  • ISSN: 0373-0956

Abstract

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We study the -equation with Hölder estimates in q -convex wedges of n by means of integral formulas. If D n is defined by some inequalities { ρ i 0 } , where the real hypersurfaces { ρ i = 0 } are transversal and any nonzero linear combination with nonnegative coefficients of the Levi form of the ρ i ’s have at least ( q + 1 ) positive eigenvalues, we solve the equation f = g for each continuous ( n , r ) -closed form g in D , n - q r n , with the following estimates: if d denotes the distance to the boundary of D and if d β g is bounded, then for all ϵ > 0 , f is Hölder continuous with exponent 1 / 2 - β - ϵ if 0 β < 1 / 2 and d β + ϵ - 1 / 2 f is bounded if 1 / 2 β < 1 .

How to cite

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Laurent-Thiébaut, Christine, and Leiterer, Jurgen. "Uniform estimates for the Cauchy-Riemann equation on $q$-convex wedges." Annales de l'institut Fourier 43.2 (1993): 383-436. <http://eudml.org/doc/75002>.

@article{Laurent1993,
abstract = {We study the $\overline\{\partial \}$-equation with Hölder estimates in $q$-convex wedges of $\{\Bbb C\}^n$ by means of integral formulas. If $D \subset \{\Bbb C\}^ n$ is defined by some inequalities $\lbrace \rho _i \le 0\rbrace $, where the real hypersurfaces $\lbrace \rho _i=0\rbrace $ are transversal and any nonzero linear combination with nonnegative coefficients of the Levi form of the $\rho _ i$’s have at least $(q+1)$ positive eigenvalues, we solve the equation $\overline\{\partial \}f=g$ for each continuous $(n,r)$-closed form $g$ in $D$, $n-q \le r \le n$, with the following estimates: if $d$ denotes the distance to the boundary of $D$ and if $d^\beta g$ is bounded, then for all $\varepsilon &gt;0$, $f$ is Hölder continuous with exponent $1/2-\beta -\varepsilon $ if $0 \le \beta &lt; 1/2$ and $d^\{\beta +\varepsilon -1/2\}f$ is bounded if $1/2 \le \beta &lt; 1$.},
author = {Laurent-Thiébaut, Christine, Leiterer, Jurgen},
journal = {Annales de l'institut Fourier},
keywords = {integral formula with uniform estimates; piecewise smooth -convex domains; tangential Cauchy-Riemann equations},
language = {eng},
number = {2},
pages = {383-436},
publisher = {Association des Annales de l'Institut Fourier},
title = {Uniform estimates for the Cauchy-Riemann equation on $q$-convex wedges},
url = {http://eudml.org/doc/75002},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Laurent-Thiébaut, Christine
AU - Leiterer, Jurgen
TI - Uniform estimates for the Cauchy-Riemann equation on $q$-convex wedges
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 2
SP - 383
EP - 436
AB - We study the $\overline{\partial }$-equation with Hölder estimates in $q$-convex wedges of ${\Bbb C}^n$ by means of integral formulas. If $D \subset {\Bbb C}^ n$ is defined by some inequalities $\lbrace \rho _i \le 0\rbrace $, where the real hypersurfaces $\lbrace \rho _i=0\rbrace $ are transversal and any nonzero linear combination with nonnegative coefficients of the Levi form of the $\rho _ i$’s have at least $(q+1)$ positive eigenvalues, we solve the equation $\overline{\partial }f=g$ for each continuous $(n,r)$-closed form $g$ in $D$, $n-q \le r \le n$, with the following estimates: if $d$ denotes the distance to the boundary of $D$ and if $d^\beta g$ is bounded, then for all $\varepsilon &gt;0$, $f$ is Hölder continuous with exponent $1/2-\beta -\varepsilon $ if $0 \le \beta &lt; 1/2$ and $d^{\beta +\varepsilon -1/2}f$ is bounded if $1/2 \le \beta &lt; 1$.
LA - eng
KW - integral formula with uniform estimates; piecewise smooth -convex domains; tangential Cauchy-Riemann equations
UR - http://eudml.org/doc/75002
ER -

References

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  1. [AiHe] R.A. AIRAPETJAN, G.M. HENKIN, Integral representations of differential forms on Cauchy-Riemann manifolds and the theory of CR-functions, Usp. Mat. Nauk, 39 (1984), 39-106, [Engl. trans. Russ. Math. Surv., 39 (1984), 41-118 and : Integral representations of differential forms on Cauchy-Riemann manifolds and the theory of CR-functions II, Matem. Sbornik, 127 (169) (1985), 1, [Engl. trans. Math. USSR Sbornik, 55 (1986), 1, 91-111]. Zbl0593.32015
  2. [AnG] A. ANDREOTTI, H. GRAUERT, Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193-259. Zbl0106.05501MR27 #343
  3. [AnHi1] A. ANDREOTTI, C.D. HILL, E. E. Levi convexity and the Hans Lewy problem. Part I : Reduction to vanishing theorems, Ann. Scuola Norm. Sup. Pisa, 26 (1972), 325-363. Zbl0256.32007MR57 #718
  4. [AnHi2] A. ANDREOTTI, C.D. HILL, E. E. Levi convexity and the Hans Lewy problem. Part II : Vanishing theorems, Ann. Scuola Norm. Sup. Pisa, 26 (1972), 747-806. Zbl0283.32013MR57 #16693
  5. [BFi] B. FISCHER, The Cauchy-Riemann equation in spaces with uniform weights, Math. Nachr., 156 (1992), 45-55. Zbl0793.32007MR94h:32026
  6. [FiLi] W. FISCHER, I. LIEB, Lokale Kerne und beschränkte Lösungen für den ∂-Operator auf q-konvexen Gebieten, Math. Ann., 208 (1974) 249-265. Zbl0277.35045MR49 #7477
  7. [He1] G.M. HENKIN, The Lewy equation and analysis on pseudoconvex manifolds (russ.), Usp. Mat. Nauk, 32 (1977), 57-118, [Engl. trans. Russ. Math. Surv., 32 (1977), 59-130]. Zbl0382.35038MR56 #12318
  8. [He2] G.M. HENKIN, Solution des équations de Cauchy-Riemann tangentielles sur des variétés de Cauchy-Riemann q-convexes, C. R. Acad. Sci. Paris, Sér. I Math, 292 (1981), 27-30. Zbl0472.32014MR82b:32031
  9. [He3] G.M. HENKIN, Analytic representation for CR-functions on submanifolds of codimension 2 in ℂn, Lecture Notes in Math. Springer, 798 (1980), 169-191. Zbl0431.32007MR81k:32006
  10. [He4] G.M. HENKIN, The method of integral representations in complex analysis (russ.). In : Sovremennge problemy matematiki, Fundamentalnye napravlenija, Moscow Viniti, 7 (1985), 23-124, [Engl. trans. in : Encyclopedia of Math. Sci., Several complex variables I, Springer-Verlag, 7 (1990), 19-116]. Zbl0781.32007
  11. [HeLe1] G.M. HENKIN, J. LEITERER, Theory of functions on complex manifolds, Akademie-Verlag Berlin and Birkhäuser-Verlag Boston, 1984. Zbl0726.32001
  12. [HeLe2] G.M. HENKIN, J. LEITERER, Andreotti-Grauert theory by integral formulas, Akademie-Verlag Berlin and Birkhäuser-Verlag Boston (Progress in Math. 74) (1988). Zbl0654.32002MR90h:32002b
  13. [LiR] I. LIEB, R.M. RANGE, Estimates for a class of integral operators and applications to the ∂-Neumann problem, Invent. Math., 85 (1986), 415-438. Zbl0569.32008MR88j:32022b
  14. [M] J. MICHEL, Randregularität des ∂-Problems für stückweise streng pseudokonvexe Gebeite in ℂn, Math. Ann., 280 (1988) 46-68. Zbl0617.32032MR89f:32033
  15. [N1] M. NACINOVICH, On strict Levi q-convexity and q-concavity on domains with piecewise smooth boundaries, Math. Ann., 281 (1988), 459-482. Zbl0628.32021MR90b:32033
  16. [N2] M. NACINOVICH, On a theorem of Airapetjan and Henkin, Seminari di Geometria 1988-1991, Univ. Bologna (1991), 99-135. Zbl0747.32011
  17. [RS] R.M. RANGE, Y.T. SIU, Uniform estimates for the ∂-equation on domains with piecewise smooth strictly pseudoconvex boundaries, Math. Ann., 206 (1973), 325-354. Zbl0248.32015MR49 #3214
  18. [T] F. TREVES, Homotopy formulas in the tangential Cauchy-Riemann complex, Mem. Ann. Math. Soc, 434 (1990). Zbl0707.35105MR90m:32012

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