Hölder and \(L^p\) estimates for \(\square_b\) on CR manifolds of arbitrary codimension
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 lower \(*\) indicates the nonisotropic Sobolev space). Estimates of this sort, under weaker assumptions than \(Y(0)\), where obtained by \textit{C. D. Hill} and \textit{M. Nacinovich} [Invent. Math. 142, No. 2, 251--283 (2000; Zbl 0973.32018)] for forms of type \((p,0)\). From these estimates they deduce regularity for solutions of the equations \(\square_b u=f\) for \((p,q)\)-forms and finiteness and regularity results for the cohomology in the compact case. The regularity results in the \(W_*^{m,p}\) for all \(1<p<\infty\) and in the Hölder spaces \(\Gamma^{m,\alpha}\) for \(0<\alpha<1\) are of special interest, also for the techniques of the proofs that avoid micro-localization and parametrix methods. Finally, the authors obtain a regularity result for the Szegö orthogonal projection \(\mathcal S:L^2(M)\to L^2(M)\cap\ker(\overline\partial_b)\) for a compact \(M\) satisfying \(Y(1)\), namely they show that \(\text{sing\,supp}\,\mathcal S u\subset\text{sing\,supp}\, u\) and also the local \(W_*^{m,p}\) and \(\Gamma^{m,\alpha}\)-regularity is preserved by \(\mathcal S\).
scientific article; zbMATH DE number 2156223
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