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The $$\overline\partial_b$$-complex on decoupled boundaries in $$\mathbb C^n$$. (English) Zbl 1126.32031
The authors obtain optimal estimates for the solutions of the Kohn Laplacian for the class of decoupled domains in $$\mathbb{C}^n$$. More precisely, a domain $$\Omega\subset \mathbb{C}^{n+1}$$ and its boundary $$M$$ are said to be decoupled if there are sub-harmonic, nonharmonic polynomials $$P_j$$ such that $$\Omega=\{(z_1,\dots,z_n, z_{n+1})\in \mathbb{C}^{n+1}\mid\operatorname{Im}|z_{n+1}|>\sum^n_{i=1}P_j(z_j)\}$$ and $$M= \{(z, z_{n+1})\in\mathbb{C}^{n+1}\mid\operatorname{Im}|z_{n+1}|=\sum^n_{i=1}P_j (z_j)\}$$. $$M$$ can be identified with $$\mathbb{C}^n\times\mathbb{R}$$ so that the point $$(z,t+i(\sum_jPj(z_j)))\in M$$ corresponds to $$(z,t)\in\mathbb{C}^n\times \mathbb{R}$$. Let $$\Omega_j=\{(z_j,w_j)\in \mathbb{C}^2\mid\operatorname{Im}|w_j|>P_j(z_j)\}$$ and $$M_j$$ be the corresponding boundary, and let $$\widetilde M=M_1 \times\cdots\times M_n\subset\mathbb{C}^{2n}$$. The linear holomorphic mapping $$\pi:\mathbb{C}^{2n}\to\mathbb{C}^{n+1}$$ given by $$\pi(z_1,\dots,z_n, w_1,\dots,w_n)=(z_1,\dots,z_n,w_1+\cdots+w_n)$$ induces a mapping from $$\widetilde M$$ to $$M$$ and this mapping allows to transfer functions from $$\widetilde M$$ to $$M$$, i.e.; for $$\varphi\in{\mathcal C}_0^\infty$$ $$(\mathbb{C}^n\times \mathbb{R}^n)$$, one can define $$\varphi^\#(z,t)\in{\mathcal C}_0^\infty(\mathbb{C}^n\times \mathbb{R})$$ by $$\varphi^\#(z,t)=\int_{\mathbb{R}^{n-1}}\varphi(z,r_1,\dots,r_{n-1},t-\sum^{n-1}_{j=1}r_j) \,dr_1\dots dr_{n-1}\equiv\int_{r\in\Sigma(t)}\varphi(z,r)\,d \widetilde r$$ where $$\Sigma (t)$$, the sphere (in $$\mathbb{R}^n)$$ of radius $$t$$, and $$d\overline r$$ is the $$(n-1)$$-dimensional Lebesgue measure on $$\Sigma(t)$$. On the other side $$\pi$$ induces a mapping from functions on $$M$$ to functions on $$\widetilde M$$, and hence induces a mapping $$d\pi$$ from tangent vectors on $$\widetilde M$$ to tangent vectors on $$M$$.
A basis for Cauchy-Riemann operators of type $$(1,0)$$, respectively, $$(0,1)$$, on $$M$$ is given by the operators $$Z_j=\frac{\partial}{\partial z_j}+i\frac{\partial P_j} {\partial z_j}(z_j)\frac{\partial}{\partial t}=X_j-iX_nj$$ $$(1\leq j\leq n)$$, respectively $$\overline Z_j$$, where $$\{X_1,\dots,X_{2n}\}$$ are real vector fields. Let $$\overline\partial_b[f]= \sum^n_{j=1}\overline Z_j[f]\,d \overline z$$, if $$f$$ is a function. For $$(0,q)$$ forms one has an analogue formula. Let $$\square_b$$ be the Kohn Laplacian which, for the decoupled boundary $$M$$, acts as follows: for $$1\leq j\leq n$$, denote $$\square_j^{(+)} =-\overline Z_jZ_j$$, $$\square_j^{(-)}=-Z_j\overline Z_j$$ and if $$J$$ is an increasing set of $$q$$ integers between 1 and $$n$$, set $$J(k)=(+)$$ if $$k\in J$$, $$(-)$$ if $$k\notin J$$.
Then $$\square_b(\Sigma \varphi_Jd\overline z_j)=\sum_J\square_j(\varphi_J)d\overline z_j$$ where $$\square_J= \sum^n_{k=1}\square_k^{J(k)}$$. The $$\overline\partial_b$$ complex is also considered on the product submanifold $$\widetilde M$$ and is defined with the aid of the vector fields $$Z_j=\frac{\partial}{\partial z_j}+i\frac {\partial P_j}{\partial z_j}(z_j)\frac{\partial}{\partial tj},\overline Z_j$$. Thus the study of the $$\overline\partial_b$$-complex on $$M$$ on $$(0,q)$$ forms is reduced to the study of operators $$\square_J$$.
Now, to construct the relative inverse operator $$K$$ for $$\square_b$$ (i.e., such that $$\square_bK=K\square_b=I-S$$ when $$S$$ is the Szegö projection) consider the operators $$\square_j=W_j^*W_j=-\overline W_j$$, where $$W_j$$ denotes either $$Z_j$$ or $$\overline Z_j$$ and depends only on $$Z_j$$ and $$t$$, the corresponding self-adjoint second-order differential operators on $$L^2(M)$$, and let $$S_j$$ be the orthogonal projection of $$L^2(M)$$ on $$\text{Ker}\square_j$$. If $$\{e^{-s\square_j} \}$$ is the semi-group of contractions on $$L^2(M)$$ with infinitesimal generator $$\square_j$$, then $$\square =\sum\square_j$$ is one of the $$\square_J$$.
If $$M_j$$ is the boundary in $$\mathbb{C}^2$$ defined above, $$\square_j$$, and $$e^{-s\square_j}$$ act on $$L^2(M_j)$$. But now one can use the theory of domains of finite type in $$\mathbb{C}^2$$: the $$S_j$$ and the heat kernel $$H_j$$ of the semi-group $$\{e^{-s\square_j}\}$$ are given explicitly, with distribution kernel $$S_j$$, respectivly, $$H_j$$. Set now $$S=\prod^n_{j=1}S_j$$, an operator on $$L^2(M)$$, which is the projection on $$\bigcap^n_{j=1}\text{Ker}\square_j$$ which is the same as the projection onto the null space of $$\square=\square_1+\cdots+ \square_n$$; moreover, the $$\square_j$$ commute and $$\chi=\exp[-s\sum \square_j\}$$ is just a product $$\prod^n_{j=1}\chi_j$$.
The spectral theorem ensures that $$e^{-s[\sum \square_j]}$$ converges strongly to $$S$$ as $$s\to\infty$$ and $$K=\int^\infty_0 [e^{-s[\sum\square_j]}-S]\,ds$$ is a relative fundamental solution for $$\square$$ on $$M$$. Finally, the authors obtain explicit formulas for $$K$$ and $${\mathcal N}= \int^\infty_0 \prod^n_{j=1}(e^{-s\square_j}-S_j)\,ds$$.
The authors consider
$\widetilde K(z,w,r)=\int^\infty_0[\prod H_j(s,z_j,w_j,r_j)-\prod S_j(z_j, w_j, r_j)]\,ds \quad\text{and}\quad \widetilde N=\int^\infty_0\prod(H_j-S_j)\,ds$
which are kernels of operators acting not on $$M$$ but on $$\widetilde M=M_1\times \cdots\times M_n$$, and are relative fundamental solutions for $$\sum \square_j$$. Integrating over $$\sum (t)$$ one obtains the desired relative fundamental solution on $$M$$. Using this method, the authors can take advantage of the product structure and use this to establish $$L^p$$ regularity. For each of the $$2^n$$ possible operators $$\{\square_J\}$$, the authors consider the distribution $$K_J$$ on $$M\times M$$ such that $${\mathcal K}_J[\varphi](j)=\int_M\varphi(q)K_J(p,q)\,dq$$ verifies $${\mathcal K}_j \square=\square{\mathcal K}_J=I-S_0$$ if $$\square_j$$ acts on functions, $$=I-S_n$$ if $$\square_J$$ acts on $$(0,n)$$ forms, and $$=I$$ (the identity) if $$\square_j$$ acts on $$(0,r)$$ forms, $$1\leq r\leq n-1$$. As M. Derridji [J. Differ. Geom. 13, 559–576 (1978; Zbl 0435.35057)] has shown, maximal hypoelliptic estimates can hold only if the eigenvalues of the Levi forms degenerate at the same rate. In particular, for decoupled domains in $$\mathbb{C}^{n+1}$$, $$n>1$$, $$\square_b$$ fails to be maximally subelliptic near $$p\in M$$ whenever $$\Delta P_j(p)=0$$ for some $$j$$. The main results of the authors are the following:
If $$\square_J=W_1 \overline W_1+ \cdots+W_n\overline W_n$$ where each $$W_j$$ is one of $$Z_j$$, or $$\overline Z_j$$ and $$\overline W_j$$ the other, and $${\mathcal K}_J$$ the operator constructed above then:
(1) When $$1\leq k$$, $$l\leq n$$, $$l\neq k$$, the operator $$W_k\overline W_k{\mathcal K}_J=-\square_k{\mathcal K}_J$$, $$\overline W_k\overline W {\mathcal K}_J$$ and $$\overline W_l\overline W_k {\mathcal K}_J$$ extends to bounded linear operators on $$L^p(M)$$ for $$1<p< \infty$$;
(2) If $$B_k$$ is a bounded smooth function on $$M$$ and there exist constants $$C_{k,l}$$ so that for all $$p=(z_1, \dots,z_n,t)\in M$$ $|B_k(p)|\Delta P_k(z_k)\leq C_{k,l}\Delta P_l(z_l)$ for $$1\leq k\leq n$$, then $$B_k\overline W_kW_k{\mathcal K}_J=-B\overline\square_kK_j$$ extends to a bounded linear operator in $$L^p(1<p<\infty)$$;
(3) If $$B_k$$ is as above and there exist constants $$C_k$$ so that for all $$p\in M$$ $|B_k(p)|\leq C_k\inf_{l\neq k}\Delta P_l(z_l),$ then $$B_kW_kW_k{\mathcal K}_J$$ extends to a bounded linear operator on $$L^p(M)$$ $$(1<p<\infty)$$.
For the study of Hölder regularity it is necessary to introduce various metrics on the space $$M$$. However, one has a global result involving just the standard isotropic metric. Let $$m_j=2+$$ degree of $$(\Delta P_j)$$ and let $$m=\max\{m_j\}$$ (the type of boundary $$M)$$. If $$m>2$$ for $$f$$ a function bounded and supported on a ball of radius 1 in $$M$$, for all $$J$$ there is a constant $$C$$ so that if $$h\in\mathbb{C}^n \times \mathbb{R}\simeq M$$ then $$|K_J[f](p,h)-K_J[f](p)|\leq C|h|^{\frac{2}{2m}}$$.
The analysis of the singularities of $$K_J$$ is very delicate and is based on a detailed study of the geometry and analysis on $$M_j$$ and $$M_1 \times\cdots\times M_n$$, and uses results of M. Christ [J. Am. Math. Soc. 1, No. 3, 587–646 (1988; Zbl 0671.35017), J. Geom. Anal. 1, No. 3, 193–230 (1991; Zbl 0737.35011); Proc. Symp. Pure Math. 52, No. 3, 63–82 (1991; Zbl 0747.32009)], D.-C. Chang, A. Nagel and E. M. Stein [Acta. Math. 169, No. 3–4, 153–228 (1992; Zbl 0821.32011)], A. Nagel, J.-P. Rosay E. M. Stein and S. Wainger [Bull. Am. Math. Soc., New Ser. 18, 55–59 (1988; Zbl 0642.32014)], A. Nagel and E. M. Stein [Math. Z. 238, 37–88 (2001; Zbl 1039.32051)].
The constructed operators $$\widetilde{\mathcal K}$$, and $$\widetilde {\mathcal N}$$ are not pseudo-local, but, in fact $${\mathcal K}$$ is pseudo-local. The singularities of the corresponding kernels of the relative fundamental solutions are expressed on terms of two metrics. The first one, the sum of square metric is an $$n$$-isotropic metric on $$M$$, denoted by $$d_\Sigma$$. The second one, called the Szegö pseudo-metric, denoted by $$d_S$$, is not isotropic in the complex directions. From the scaling arguments of J. D. McNeal [Duke Math. J. 58, No. 2, 499–512 (1989; Zbl 0675.32020)] it follows that the Szegö kernel behaves like a singular integral operator relative to $$d_S$$. Transfer from $$\widetilde M$$ to $$M$$ implies formal integration over nonintegrable singularities which is possible due to the fact that derivatives of the Szegö projection are high derivatives in $$t$$ of a bounded function, which allows integration by parts.
Finally, in the last section, the authors provide examples that show that the regularity results obtained are optimal.

##### MSC:
 32W10 $$\overline\partial_b$$ and $$\overline\partial_b$$-Neumann operators
##### Keywords:
Kohn Laplacian; decoupled domains
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