% Run under latex2e \documentstyle[12pt,amssymb]{amsart} \textheight=574pt \textwidth=432pt \oddsidemargin=18.88pt \evensidemargin=18.88pt \topmargin=14.21pt \begin{document} {\makeatletter \gdef\eqalign#1{\null\,\vcenter{\openup\jot\m@th \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil \crcr#1\crcr}}\,} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Macros defined by S. Fu and others % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\ale{\mathrel{\mathop{<}\limits_{\sim}}} \def\age{\mathrel{\mathop{>}\limits_{\sim}}} \def\gs{\mathrel{\mathop{>}\limits_{\sim}}} \def\qed{{\bf q.e.d.}\par\medbreak} \def\qedn{\thinspace\null\nobreak\hfill\hbox{\vbox{\kern-.2pt\hrule height.2pt depth.2pt\kern-.2pt\kern-.2pt \hbox to2.5mm{\kern-.2pt\vrule width.4pt \kern-.2pt\raise2.5mm\vbox to.2pt{}\lower0pt\vtop to.2pt{}\hfil\kern-.2pt \vrule width.4pt\kern-.2pt}\kern-.2pt\kern-.2pt\hrule height.2pt depth.2pt \kern-.2pt}}\par\medbreak} \def\pf{ \noindent{\sl Proof\/}.\enspace} \let\emptyset\varnothing \let\de=\partial \def\eps{\varepsilon} \def\phe{\varphi} \def\Hol{\mathop{\mathrm{Hol}}\nolimits} \def\Aut{\mathop{\mathrm{Aut}}\nolimits} \def\Re{\mathop{\mathrm{Re}}\nolimits} \def\Im{\mathop{\mathrm{Im}}\nolimits} \def\Fix{\mathop{\mathrm{Fix}}\nolimits} \def\id{\mathop{\mathrm{id}}\nolimits} \def\cancel#1#2{\ooalign{$\hfil#1/\hfil$\crcr$#1#2$}} \def\R{{\Bbb R}} \def\P{{\Bbb P}} \def\N{{\Bbb N}} \def\Z{{\Bbb Z}} \def\Q{{\Bbb Q}} \def\C{{\Bbb C}} \def\T{\hbox{\bf T}} \newbox\bibliobox \def\setref #1{\setbox\bibliobox=\hbox{[#1]\enspace} \parindent=\wd\bibliobox} \def\biblap#1{\noindent\hang\rlap{[#1]\enspace}\indent\ignorespaces} \def\art#1 #2: #3! #4! #5 #6 #7-#8 \par{\biblap{#1}#2: {\sl #3\/}. #4 {\bf #5}~(#6)\if.#7\else, \hbox{#7--#8}\fi.\par\smallskip} \def\book#1 #2: #3! #4 \par{\biblap{#1}#2: {\rm #3.} #4.\par\smallskip} \def\atanh{\mathop{\rm atanh}\nolimits} \overfullrule=0pt \let\smb=\smbar \let\j=\jmath \let\i=\imath \lineskiplimit=1pt \def\ca{Cara\-th\'{e}o\-dory~} \def\ko{Ko\-ba\-ya\-shi~} \def\psc{pseu\-do\-con\-vex~} \def\psh{pluri\-sub\-har\-monic~} \def\ngbd{neigh\-bor\-hood~} \def\bdry{boundary~} \def\O{\Omega} \def\CC{{\cal C}} \def\CH{{\cal H}} \def\CN{{\cal N}} \def\CO{{\cal O}} \def\CS{{\cal S}} \def\CF{{\cal F}} \def\CZ{{\cal Z}} \def\ls{\mathrel{\mathop{<}\limits_{\sim}}} \def\cl{{\rm cl}} \def\dist{{\rm dist}} \def\Lip{{\rm Lip}} \def\aut{{\rm Aut}} %%%%%%%%%%%%%%%%%%%%%%% % End of Macros % %%%%%%%%%%%%%%%%%%%%%%% \title{Finite Type Conditions on Reinhardt Domains} \subjclass{Primary 32F25, Secondary 52A50.} \keywords{Reinhardt domain, pseudoconvex, finite type.} \author[Siqi Fu, Alexander V. Isaev and Steven G. Krantz] {Siqi Fu\\ Alexander V. Isaev\\ Steven G. Krantz} \address{\hskip-\parindent Siqi Fu, Department of Mathematics, University of California, Riverside, CA 92521-0135, USA} \email{sfu@@math.ucr.edu} \address{\hskip-\parindent A. V. Isaev, Centre for Mathematics and Its Applications, The Australian National University, Canberra, ACT 0200, Australia} \email{Alexander.Isaev@@anu.edu.au} \address{\hskip-\parindent S. G. Krantz, Department of Mathematics, Washington University, St.~Louis, MO 63130, USA } \email{sk@@math.wustl.edu} \thanks{Research supported in part by a grant from the National Science Foundation. Research at MSRI supported in part by NSF grant \#DMS 9022140.} \maketitle \begin{abstract} In this paper we prove that, if $p$ is a boundary point of a smoothly bounded pseudoconvex Reinhardt domain in $\C^n$, then the variety type at $p$ is identical to the regular type. \end{abstract} \vskip 30pt In this paper we study the finite type conditions on pseudoconvex Reinhardt domain. We prove that, if $p$ is a boundary point of a smoothly bounded pseudoconvex Reinhardt domain in $\C^n$, then the variety type at $p$ is identical to the regular type. In a forthcoming paper, we will study the biholomorphically invariant objects ({\it e.g.}, the Bergman kernel and metric, the Kobayashi and Carath\'{e}odory metrics) on a pseudoconvex Reinhardt domain of finite type. \bigskip We first recall some definitions. A domain $\Omega\subset \C^n$ is Reinhardt if $$(e^{i\theta_1}z_1,\ldots, e^{i\theta_n}z_n)\in \O$$ whenever $(z_1,\ldots, z_n)\in \O$ and $0\le \theta_j\le 2\pi, 1\le j\le n$. Denote $$\CZ_j =\{ (z_1,\ldots, z_n)\in \C^n ; \quad z_j=0\},$$ for $j = 1,\dots,n$. Let $\CZ = \bigcup_{j=1}^{n} \CZ_j$. Define $L\colon \C^n\setminus \CZ \to \R^n$ by $$ L(z_1, \ldots, z_n) =(\log |z_1|, \ldots, \log |z_n|) $$ and $$ L^* (z_1, \ldots, z_n)=(\log z_1, \ldots, \log z_n). $$ where, in the second case, the logarithm takes the principle branch, and $L^*$ is defined locally near every point of $\C^n\setminus \CZ$. %Does the principal branch really make sense here? %Steven: I think, you are right here, and we should say that $L^*$ is %only defined locally near every point of $\C^n\setminus \CZ$. I have %inserted a correction here. Let $\O=\{ z\in \C^n; \quad \rho (z) < 0\}$ be a bounded domain with smooth boundary. A boundary point $p\in b\O$ is of finite {\it variety type m} if $$ m=\sup \left \{ {v(\rho\circ f)\over v(f)} \right \} < \infty\leqno(1) $$ where the supremum is taken over all analytics disc $f\colon \Delta\to \C^n$ such that $f(0)=p$. We use $v(f)$ to denote the order of vanishing of $f$ at $0$ (see [K] for details on this notion). When the supremum in (1) is taken over all regular analytic discs $f$ ({\it i.e.}, $f'(0)\ne 0$), then $m$ is called the {\it regular type} of $p$; when the supremum in (1) is taken over all complex lines through $p$, then $m$ is called the {\it line type} of $p$. It has been proved by McNeal [M] that, for a boundary point of a smoothly bounded convex domain, the variety type is identical to the line type. This result was generalized by Boas and Straube [B-S] to a star-like boundary point (see Theorem 4 below for the exact statement). Meanwhile, J. Yu [Y] showed that the Catlin multi-type is identical to the D'Angelo q-type for a boundary point of a convex domain. In this paper, we shall prove that, for a boundary point of a pseudoconvex Reinhardt domain, the regular type is identical to the variety type. \medskip \proclaim{Lemma 1}. (Reinhardt) If $\O$ is a pseudoconvex Reinhardt domain, then $L(\O\setminus \CZ)$ is convex. Furthermore, %I think this has to be a Reinhardt domain. %Steven: I did not quite understand this comment. Which domain does have %to be Reinhardt? The original domain is indeed Reinhardt, this is given. if $\O\cap \CZ_j\ne\emptyset$, then $(z_1,\ldots, z_{j-1}, \lambda z_j, z_{j+1}, z_n)\in\O$ whenever $(z_1, z_2, \ldots, z_n)\in\O$ and $|\lambda|\le 1$. \par \smallskip \proclaim{Lemma 2}. Let $\O$ be a smoothly bounded pseudoconvex Reinhardt domain and let $p\in b\O$. Then there is a neighborhood $U$ of $p$ such that $b\O\cap U$ has a plurisubharmonic defining function $\rho (z)$. \par \pf If $p\in \O\setminus \CZ$, then $L(z)$ is well-defined in a neighborhood of $p$. Since $L(\O\setminus \CZ)$ is convex, there is a local convex defining function $\tilde\rho (u_1, \ldots, u_n)$ of $L(\O\setminus \CZ)$ near $L(p)$. Let $\rho =\tilde\rho (L(z))$. Then $\rho$ is a plurisubharmonic defining function of $b\O$ near $p$. If $p\in \O\cap \CZ$, then, without lose of generality, we may assume that $$p=(p_1, \ldots, p_k, 0,\ldots, 0),$$ with $p_j\ne 0$, $0\le j\le k$. Let $\sigma (z)$ be the Euclidean distance from $z$ to $b\O$. Then $\sigma (z)$ is smooth in a neighborhood of $b\O$ and satisfy $\sigma (z_1, \ldots, z_n) =\sigma (e^{i\theta_1}z_1, \ldots, e^{i\theta_n}z_n)$. It follows that ${\de \sigma \over \de z_j}(z_1, \ldots, z_n) = 0$ whenever $z_j =0$. Since ${\de \sigma\over \de z_j}(p) =0$, $k+1\le j \le n$, one of ${\de \sigma\over \de z_j} (p)$ , $1\le j\le k$, must not be zero. For simplicity of notation, we may assume that ${\de \sigma\over \de z_1} (p)\ne 0$. Therefore $b\O$ is locally defined by an equation of the form $|z_1| + f(|z_2|, \ldots, |z_n|)=0$ near $p$. This in turn implies that $L(b\O\setminus\CZ)$ is defined by $$ u_1 + g(u_2, \ldots, u_n)=0 $$ where $$ g(u_2, \ldots, u_n)=-\log(-f(e^{u_2}, e^{u_3}, \ldots, e^{u_n})) $$ for $(u_2, \ldots u_n)\in I_2\times \cdots I_k\times I^{n-k}$. Here $I_j$ is a neighborhood of $\log |p_j|$, $2\le j\le k$, and $I=(-\infty, -C)$ ($C >0$ is a large constant). The convexity of $L(b\O\setminus \CZ)$ implies that $g(u_2, \ldots, u_n)$ is a convex function. Therefore $$ \eqalign{\rho(z_1, \ldots, z_n)&= \log |z_1|+ g(\log |z_2|, \ldots, \log |z_n|)\cr &=\log |z_1| - \log (-f(|z_2|, \ldots, |z_n|))\cr } $$ is a smooth plurisubharmonic defining function for $b\O$ near $p$. \qedn \smallskip %\noindent{\bf Remark 3.}\enspace It seems that one should %be able to prove that, for a pseudoconvex Reinhardt domain, there %is a global smooth plurisubharmonic defining function. Notice that there is %a pseudoconvex Hartogs domain (the ``worm domain'' constructed %by Diederich-Fornaess) in $\C^2$ without a global plurisubharmonic %defining %function. %\smallskip %Steven: I do not know what to say here. The current proof gives just the %local answer. Would you like to work more on this, or possibly to formulate %your last comment as a conjecture? \noindent For $z=(z_1, \ldots, z_n)$, let $\hat z =(z_2,\ldots, z_n)$. \proclaim{Theorem 3}([B-S]). Suppose that a smooth hypersurface is defined by $\Re z_1+h(\hat z, \Im z_1)=0$ near the origin. If $h(\hat z, 0)\ge 0$, then the variety type of the hypersurface at the origin is obtained by analytic discs of form $\phi (\zeta)= (0, \phi_2(\zeta), \ldots, \phi_{n}(\zeta))$. Furthermore, if the hypersurface is star-like at the origin, {\it i.e.}, there exists a $\delta >0$ such that $t\to h(a_2t, \ldots, a_{n}t, 0)$ is an increasing function on $[0, \delta ]$ for all unit vectors $(a_2, \ldots, a_{n})$, then the variety type at the origin is obtained by a complex line. \par Now we can prove the main theorem of this paper: \proclaim{Theorem 4}. If $\O$ is a smoothly bounded pseudoconvex Reinhardt domain and if $p\in b\O$, then the variety type at $p$ is identical to the regular type. \par \pf It follows from the smoothness and the Reinhardt property of $b\O$ that $p$ cannot be the origin (this can be easily seen from the proof of Lemma 2 in this paper or from Lemma 1.3 in [FIK]). We divide the proof into three cases. If $p\in b\O\setminus \CZ$, then by Lemma 1, $L^* (b\O\setminus\CZ)$ is convex near $p$. Therefore, it follows from the results of McNeal [Mc] and Boas/Straube [B-S] that the variety type of $L^* (b\O\setminus Z)$ at $L^*(p)$ is obtained by a complex line of the form $L^*(p) +a\zeta$. Thus, the variety type of $b\O$ at $p$ is obtained by $$ \phi (\zeta) =(p_1e^{a_1 \zeta}, \ldots, p_ne^{a_n\zeta}). $$ Therefore the variety type is identical to the regular type at $p$. \smallskip\smallskip If $p\in b\O\cap \CZ$, then the mapping $L^*$ is not well-defined near $p$. Thus the above method fails. We first consider the case that only one coordinate of $p$ is non-zero. We shall show that, in this case, the domain $\O$ is star-like at $p$. After a simple dilation and rotation, we may assume that $p=(1, 0,\ldots, 0)$ and $\Re z_1$ is the outward normal direction at $p$. It follows from the proof of Lemma 2 that $b\O$ is defined near $p$ by $$ \log |z_1| + h(\hat z) =0 $$ where $h$ is a smooth function defined near $\hat p$ such that \smallskip \begin{enumerate} \item[(a)]$ h(\hat p) =0$ and $dh(\hat p)=0;$ \item[(b)] $h(e^{i\theta_2}z_2, \ldots, e^{i\theta_n}z_n)=h(z_2, \ldots, z_n);$ \item[(c)] The function $h$ is plurisubharmonic. Furthermore, $$h(z_2,\ldots, z_n) = g(\log |z_2|, \ldots, \log |z_n|)$$ for some convex function $g$ and $z_i\ne 0$, $2\le i\le n$. \end{enumerate} \smallskip\smallskip For $a=(a_2,\ldots, a_n)$ with $\| a\| =1$ and $a_i\ne 0$, $2\le i \le n$, define $$ \eqalign{ k_a(t)&=h(ta_2, \ldots, ta_n)\cr &=g(\log t + \log |a_2|, \ldots, \log t +\log |a_n|) .\cr} $$ It follows from the convexity of $g$ that $\tilde{g}(s)=g(s +\log |a_2|, \ldots, s+\log |a_n|)$ is a convex function of $s$. By (a), we have $$ \lim_{s\to -\infty}\tilde{g}'(s)=0. $$ Therefore, $\tilde{g}'(s) \ge 0$ for $s<<-1$. It then follows that $k_a(t)$ is an increasing function of $t$. Thus, by the (proof of the) theorem of Boas/Straube, the variety type at $p$ is obtained by a complex line of the form $\phi(\zeta) = (1, b_2 \zeta, \ldots, b_n\zeta).$ \smallskip Now we treat the general situation for the case when $p\in b\O\cap \CZ$. Without lost of generality, we may assume that the first $k$ coordinates of $p$ are 1's and the rest of the coordinates are 0's. Let $\sigma (z)$ be the Euclidean distance from $z$ to $b\O$. It follows from the proof of Lemma 2 that one of ${\de \sigma \over \de z_j}(p)$, $1\le j \le k$ is not zero. For simplicity of notation, we assume that $\partial \sigma (p)/\partial z_1 \ne 0$. Let $z'=\Phi (z)$ be defined by $$ \eqalign{z'_1& =z^{\alpha_1}_1\cdots z^{\alpha_k}_k;\cr z'_j&=z_j, \quad 2\le j\le n\cr} $$ where $\alpha_j =\partial \sigma(p)/\partial z_j, 1\le j\le k$. It is obvious that $z'=\Phi(z)$ is biholomorphic in a neigborhood of $p$ and $\Re z'_1$ is the outward normal direction to $b\O$ at $p$. This change of coordinates preserves the Reinhardt property of the domain near $p$. Changing coordinates if necessary, we may assume at the outset that $\Re z_1$ is the outward normal direction to $b\O$ at $p$. It follows from the proof of Lemma 2 that $b\O$ is defined near $p$ by $$ \log |z_1| + h(z_2, \ldots, z_n) = 0 , $$ where $h$ is a function satisfying properties (a)-(c). By a similar argument, we can prove that, for fixed $z_j$, $2\le j\le k$, near 1 and $a=(a_{k+1},\ldots, a_n)$ with $\|a\|=1$, $$ t \mapsto h(z_2,\ldots, z_k, ta) $$ is an increasing function of $t$ for small $t>0$. Thus we have $$ h(z_2, \ldots, z_n)\ge h(z_2, \ldots, z_k, 0, \ldots, 0).\leqno(2) $$ Let $\rho(w_2,\ldots, w_n)= h(e^{w_2}, \ldots, e^{w_k}, w_{k+1}, \ldots, w_n)$. Then $\rho(\hat w)$ is a plurisubharmonic function. Furthermore, for fixed $(w_{k+1}, \ldots, w_n)$, $\rho(\hat w)$ is a convex function. Let $k(t)=\rho(tw_2, \ldots, tw_{k}, 0, \ldots, 0)$. Then $k(t)$ is a convex function with $k(0)=k'(0)=0$. Therefore $k(1)=\rho(w_2, \ldots, w_k, 0,\ldots, 0)\ge 0$. Hence $h(\hat z)\ge 0$. \smallskip \proclaim{Claim}. The variety type of $b\O$ at $p$ is obtained by analytic discs of the form $\phi(\zeta)=(1, \hat \phi(\zeta))$. \par The proof of the claim comes directly from the proof of the theorem of Boas and Straube [B-S]. We provide details here for the reader's convenience. Let $r(z)=\log |z_1| + h(z_2, \ldots, z_n)$. Let $ \phi(\zeta)=(\phi_1(\zeta), \hat \phi(\zeta))\colon \Delta\to \C^n$ be an analytic disc such that $\phi(0)=p$. We may assume that $v(r\circ \phi)> v(\phi)$. If $\phi_1(\zeta)$ is not identically zero, then there exists a sequence $\{ \zeta_j \}\subset \Delta$, $\zeta_j\to 0$ such that $\{ \phi_1(\zeta_j) \}$ is a sequence of real numbers and $\phi_1(\zeta_j)\to 1^+$. Therefore $$ \log |\phi_1(\zeta_j)|=|\log \phi_1(\zeta_j)| \le r\circ \phi (\zeta_j). $$ Let $\psi(\zeta)=(1, \hat \phi(\zeta))$. Since the order of vanishing of a holomorphic function is determined by its order of vanishing along a sequence, we obtain that $$ v(\phi_1(\zeta)-1)=v(\log |\phi_1(\zeta)|)\ge v(r\circ \phi). $$ This in turn implies that $v(\psi)=v(\phi)$. It then follows from the equality $$ |r\circ\psi(\zeta) -r\circ\phi(\zeta) |=|\log |\phi_1(\zeta)| | $$ that $v(r\circ\psi)\ge v(r\circ \phi)$. This finishes the proof of the claim. \smallskip\smallskip We now return to the proof of the main theorem. Let $\tilde \phi(\zeta) =(\phi_2(\zeta),\ldots, \phi_k(\zeta))$ and let $m=v(h\circ \hat\phi)$. It follows from (2) that $$ v(h(\tilde \phi(\zeta),0, \ldots, 0))\ge m.\leqno(3) $$ Let $\beta_j =v(\phi_j)$, $k+1\le j\le n$ and $\psi_j(\zeta)=\phi_j(\zeta) /\zeta^{\beta_j}$. For $\delta>0$, we have $$ \eqalign{ \delta^m&\gs {1\over 2\pi\delta}\int_{|\zeta|=\delta} h\circ \hat\phi(\zeta) |d\zeta|\cr &={1\over 2\pi\delta}\int_{|\zeta|=\delta} h(\tilde\phi(\zeta),\delta^{\beta_{k+1}}\psi_{k+1}(\zeta), \ldots \delta^{\beta_n}\psi_n(\zeta))|d\zeta|\cr &\ge h(1, \ldots, 1, \delta^{\beta_{k+1}}\psi_{k+1}(0), \ldots \delta^{\beta_n}\psi_n(0)).\cr } $$ Therefore, $$ v(h(1, \ldots, 1, \zeta^{\beta_{k+1}}\psi_{k+1}(0), \ldots \zeta^{\beta_n}\psi_n(0)))\ge m.\leqno(4) $$ It follows from (3) and (4) that the variety type at $p$ can be realized by analytic discs of the form $$ \phi(\zeta)=(1, \phi_2(\zeta),\ldots, \phi_k(\zeta), 0,\ldots, 0) \quad {\rm or}\quad \phi(\zeta)=(1,\ldots, 1, \phi_{k+1}(\zeta),\ldots, \phi_n(\zeta)) . $$ However, it follows from the first and second cases of this proof that the variety type can actually be realized by analytic discs of form $$ \phi(\zeta)=(1, e^{a_2\zeta}, \ldots, e^{a_k\zeta}, 0, \ldots, 0),\leqno(5) $$ or $$ \phi(\zeta)=(1, \ldots, 1, a_{k+1}\zeta, \ldots, a_n\zeta).\leqno(6) $$ Indeed, to obtain the special form (5), consider the section of $\O$ by $\CZ_{k+1,\dots,n}=\bigcap_{j=k+1}^n\CZ_j$. Then, by Lemma 1.3 of [FIK], this section is a nonempty smoothly bounded set which is a finite collection of Reinhardt domains in $\CZ_{k+1,\dots,n}$. The point $p$ then becomes a point with all nonzero components, and therefore the first case of the proof applies. To obtain the special form (6), one needs to consider the section of $\O$ by a complex affine subspace of the form $\{z_{j_1}=\dots=z_{j_{k-1}}=1\}$, where $1\le j_i\le k$ for all $i$. Then we can use the argument from the second case of the proof. \qedn \noindent{\bf Remark 5.}\enspace {(1)} One certainly cannot expect that the line type at $p$ is equal to the variety type. For example, let $\O$ be a pseudoconvex Reinhardt domain in $\C^2$ such that $b\O$ is locally defined by $$ \rho(z_1, z_2)=\log |z_1| +\log |z_2| +(\log |z_1|-\log |z_2|)^4 $$ for $z=(z_1, z_2)$ near $p=(1, 1)$. Then the line type at $p$ is 2 while the regular type at $p$ is 4. \noindent{(2)} The result of Yu [Y] cannot be generalized to Reinhardt domains. Let $\Omega$ be a pseudoconvex Reinhardt domain defined by $$ \rho(z_1, z_2, z_3)=|z_1|^2 + |z_2|^6 + |z_3|^6 + |z_2 z_3|^2 - 1 . $$ Then the 3-tuple of the D'Angelo q-types at $p=(1, 0, 0)$ is $(\Delta_3, \Delta_2, \Delta_1)=(1, 4, 6)$ while the Catlin multi-type is $(1, 4, 4)$. We thank Professor Yu for pointing out this fact. \begin{thebibliography}{} \bibitem[B-S]{B-S} H. Boas and E. Straube: On equality of line type and variety type of real submanifolds of $\C^n$, J. of Geometric Anal. {\bf 2} (1992), 95--98. \bibitem[FIK]{FIK} S. Fu, A. Isaev and S. Krantz: Reinhardt domains with non-compact automorphism groups, preprint, 1995, 1--12. \bibitem[K]{K} S. Krantz: Function Theory of Several Complex Variables, Second Ed., Wadsworth, Belmont, 1992. \bibitem[M]{M} J. McNeal: Convex domains of finite type, J. of Functional Anal. {\bf 108} (1992), 361--373. \bibitem[Y]{Y} J. Yu: Multitypes of convex domains, Indiana Univ. Math. J., {\bf 41} (1992), 837--849. \end{thebibliography} \end{document}