Superlogarithmic estimates on pseudoconvex domains and CR manifolds. (English) Zbl 1037.32032

Let \(M\) be a pseudoconvex CR manifold of dimension \(n\) and \(U\) be a neighborhood of some \(x_0\in M\).
The aim of the paper is to obtain a superlogarithmic estimate: for each \(\delta> 0\) there exists \(C_\delta\) such that \[ \|(\log\Lambda)\varphi\|^2\leq \delta^2(\square_b \varphi,\varphi)+ C_\delta\|\varphi\|^2,{(\text{SL})_q} \] for all \(C^\infty(0,q)\)-form with support in \(U\) (where \(\log\Lambda\) is defined by \(\log\widehat\Lambda(u)(\xi)=\) \({1\over 2}(\log(1+ |\xi|^2)\widehat u(\xi)\)).
Using a microlocal decomposition the author defines two ideals \({\mathcal I}^+_q(x_0)\) and \({\mathcal I}^-_q(x_0)\) of subelliptic multipliers and proves that if \(S\) is a submanifold of \(M\) with holomophic dimension less than or equal to \(\min\{q-1, n-q-2\}\) and there exists \(\rho\in C^\infty(U)\cap{\mathcal I}^+_q(x_0)\cap{\mathcal I}^-_q(x_0)\) such that \(\lim_{x\to S}\,d(x, S)\log\rho(x)= 0\) then the estimate \((\text{SL})_q\) is valid.
He then proves that \((\text{SL})_q\) implies the hypoellipticity of \(\square_b\) for the square integrable \((p,q)\)-forms.
Finally, using his result when \(M\) is the boundary of a relatively compact domain \(\Omega\) in a complex Hermitian manifold, he obtains hypoellipticity results for the \(\overline\partial\)-Neumann problem on \(\Omega\).


32W10 \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators
32V15 CR manifolds as boundaries of domains
35H10 Hypoelliptic equations
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