Abstract
An intrinsic definition of Lipschitz classes in terms of vector fields on man-ifolds is provided and it is shown that it is locally equivalent with a more classical definition. A finer result is then proved for strongly pseudo-convex CR manifolds and applications of the theorems are given to smoothness of holomorphic functions and estimates for the\(\bar \partial \) and\(\bar \partial _b \). equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
PAUL BUTZER and HUBERT BERENS,Semi Groups of Operators and Approximation, Springer, New York, 1967.
HERBERT FEDERER,Geometric Measure Theory, Springer, New York, 1969.
G. B. FOLLAND, Subelliptic Estimates and Function Spaces On Nilpotent Lie Groups,Ark. Mat., 13 (1975), 161–207.
C. B. FOLLAND and J. J. KOHN,The Neumann Problem for the Cauchy-Riemann Complex, Princeton University Press, Princeton, 1972.
C. B. FOLLAND and E. M. STEIN, Estimates For The\(\bar \partial _b \) Complex And Analysis On The Heisenberg Group, Comm.Pure Appl. Math., 27 (1974), 429–522.
P. C. GREINER and E. M. STEIN,Estimates For The \(\bar \partial \)-Neuman Problem, Princeton University Press, Princeton, 1977.
R. HARVEY and B. LAWSON, On Boundiness Of Complex Analytic Varieties I,Annals of Math., 102 (1975), 223–90.
S. HELGASON,Differential Geometry And Symmetric Spaces, Academic Press, New York, 1962.
L. HÖRMANDER,An Introduction To Complex Analysis In Several Variables, Van Nostrand, Princeton, 1966.
N. KERZMAN, Holder and Lp Estimates For Solutions Of\(\bar \partial u = f\) On Strongly Pseudo-Convex Domains,Comm. Pure Appl. Math., 24 (1971), 301–79.
S. KRANTZ, Optimal Lipschitz and Lp Regularity For The Equation\(\bar \partial u = f\) On Strongly Pseudo-Convex Domains,Math. Annalen, 219 (1976), 233–60.
S. KRANTZ, Structure And Interpolation Theorems For Certain Lipschitz Classes And Estimates For The\(\bar \partial \) Equation,Duke Math. Journal, 2 (1976), 417–39.
—, Characterizations Of Certain Lipschitz Classes And Applications To Interpolation And Function Theory On Nilpotent Lie Groups, preprint.
—, Characterizations Of Various Domains Of Holomorphy Via\(\bar \partial \) Estimates And Applications To A Problem Of Kohn, preprint.
S. M. NIKOL'SKII, Approximation Of Functions Of Several Variables And Imbedding Theorems, Springer, New York, 1975.
L. P. ROTHSCHILD and E. M. STEIN, Hypoelliptic Differential Operators And Nilpotent Groups,Acta. Mathematica, 137 (1977), 247–320.
W. RUDIN, Holomorphic Lipschitz Functions In Balls, preprint.
J. P. SERRE, Lie Algebras And Lie Groups, Benjamin, New York, 1965.
E. M. STEIN, Singular Integrals And Estimates For The Cauchy-Riemann Equations,Bull. Amer. Math. Soc., 79 (1973), 440–45.
—,Singular Integrals And Differentiability Properties Of Functions, Princeton University Press, Princeton, 1970.
E. M. STEIN,Boundary Behavior of Holomorphic Functions Of Several Complex Variables, Princeton University Press, Princeton, 1972.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Krantz, S.G. Intrinsic Lipschitz classes on manifolds with applications to complex function theory and estimates for the\(\bar \partial \) and\(\bar \partial _b \) equations. Manuscripta Math 24, 351–378 (1978). https://doi.org/10.1007/BF01168882
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01168882