Hölder and \(L^p\) estimates for \(\square_b\) on CR manifolds of arbitrary codimension
type: Item
instance of: scholarly article
title: Hölder and \(L^p\) estimates for \(\square_b\) on CR manifolds of arbitrary codimension
zbMATH Open document ID: 1076.32030
author: Mei-Chi Shaw
author: Lihe Wang
published in: Mathematische Annalen
publication date: 2005-04-15T00:00:00Z
review text: The authors adapt \(L^p\) and Hölder estimates methods, originally devised for elliptic and parabolic systems of PDE's, to the study of the tangential Cauchy-Riemann complex and of its Laplacian \(\square_b\). They introduce suitable nonisotropic Sobolev and Hölder spaces (the last ones following the original \textit{S. Campanato} construction [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18, 137--160 (1964; Zbl 0133.06801); Ann. Mat. Pura Appl., IV. Ser. 69, 321--381 (1965; Zbl 0145.36603)]). Let \(M\) be a connected CR manifold of CR dimension \(n\) and CR codimension \(k\), with analytic tangent space \(HM\) and partial complex structure \(J\). The nonisotropic Sobolev spaces are obtained by considering only derivatives along \(HM\)-valued vector fields. The Campanato's and Hölder spaces are defined by using the sub-Riemannian metric associated to a smooth metric on \(HM\). The distance of the points \(x\) and \(y\) of \(M\) is defined as the infimum of the length of the pice-wise smooth paths that join \(x\) to \(y\) and have at each regular point a tangent that belongs to \(HM\) [see e.g., \textit{M. Gromov}, Prog. Math. 144, 79--323 (1996; Zbl 0864.53025)]. The authors introduce at each point \(x\) of \(M\) the \(Y(q)\) condition, concerning the Levi form [for its definition see e.g., \textit{C. D. Hill} and \textit{M. Nacinovich}, Lect. Notes Pure Appl. Math. 173, 275--297 (1996; Zbl 0921.32004)]. They require that for each characteristic direction \(\xi\in H^0M\) (the annihilator bundle of \(HM\) in \(T^*M\)) the Levi form \(\mathcal L(\xi,\cdot)\) has at least either \((q+1)\) negative or \((n-q+1)\) positive eigenvalues. Under this condition they reprove the \((1/2)\)-estimate for \(\bar\partial_b\) and forms with compact support of type \((p,q)\) [see Hill and Nacinovich (1996; loc. cit.)] and give for \(\square_b\) the nonisotropic Sobolev estimate \(\| \phi\| ^2_{W_*^{m+2,2}}\leq C_m ( \| \square_b\phi\| ^2_{W_*^{m,2}}+\| \phi\| ^2)\) (the lowe…
reviewed by: Mauro Nacinovich
Mathematics Subject Classification ID: 32V20
Mathematics Subject Classification ID: 35H20
Mathematics Subject Classification ID: 35N99
zbMATH DE Number: 2156223
zbMATH Keywords: Campanato spaces
zbMATH Keywords: Cauchy-Riemann manifolds
MaRDI profile type: Publication
full work available at URL: s00208-004-0583-5
OpenAlex ID: W2095188979
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DOI: 10.1007/S00208-004-0583-5
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description: scientific article; zbMATH DE number 2156223