The \(\overline\partial_b\)-complex on decoupled boundaries in \(\mathbb C^n\).

*(English)*Zbl 1126.32031The 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.

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.

Reviewer: Gheorghe Gussi (Bucureşti)

##### MSC:

32W10 | \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators |