## 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 $$\Omega$$.

### MSC:

 32W10 $$\overline\partial_b$$ and $$\overline\partial_b$$-Neumann operators 32V15 CR manifolds as boundaries of domains 35H10 Hypoelliptic equations
Full Text: