# Harmonic Analysis, $\overline{\partial}$, and CR Geometry (15w5074)

## Organizers

Tatyana Barron (University of Western Ontario)

Siqi Fu (Rutgers University-Camden)

Malabika Pramanik (University of British Columbia, Vancouver)

Andrew Raich (University of Arkansas)

Emil Straube (Texas A&M University)

Alexander Tumanov (University of Illinois at Urbana-Champaign)

## Objectives

**1. $L^2$-theory of $\bar\partial$ and $\bar\partial_b$.** The $\overline{\partial}$--complex induces the $\overline{\partial}_{b}$--complex on the boundary of a domain in $\mathbb{C}^{n}$, or, more generally, the $\overline{\partial}_{M}$--complex on a CR submanifold. This complex can also be defined abstractly on CR manifolds not necessarily embedded. Although much is known (especially for $\overline{\partial}$), some basic questions remain unanswered. Many of these questions involve a delicate interplay of analysis with CR geometry and complex potential theory.

Straube (2008) has developed an approach to global regularity that unifies the ones via compactness and via "good vector fields" (Boas and Straube [1991, 1993]). Harrington (2011) did so unifying using "good vector fields", property (P), and Kohn's quantitative approach (Kohn [1999]). While it is clear that there is a big overlap between the two methods, it is not understood what their precise relationship is, or how one could unify the two.

Munasinghe and Straube (2006) developed a CR geometric approach to compactness, via certain flows induced by complex tangential vector fields. How their strategy compares to Catlin's more analytic method is basic to understanding compactness, yet nothing of substance is known.

The available evidence hints at an intriguing relationship between regularity properties of the $\overline{\partial}$--Neumann operator and the existence of a Stein neighborhood basis for the closure of the domain (Boas and Straube [1993], Zeytuncu). When the $\overline{\partial}$--Neumann operator $N$ is compact, the question can be (re)formulated in terms of a stability property of the spectrum of $N$, and so has close connections to Section 2 below.

While the basic $L^{2}$--theory for $\overline{\partial}_{b}$ (i.e., closed range) on the boundary of a pseudoconvex domain, along with global regularity on large classes of domains, has been known for quite some time (see, e.g., Boas, Kohn, Shaw, Straube), this is not the case for $\overline{\partial}_{M}$ on a CR submanifold of hypersurface type; here, these issues have been resolved only surprisingly recently (Nicoara [2006], Baracco [Invent. Math., 2012]). This did, however, trigger a flurry of activity, so that now much of the theory for $\overline{\partial}$ has analogues for $\overline{\partial}_{M}$; compactness (Raich, Straube), the weighted theory (Harrington and Raich [2010]), regularity equivalence for the complex Green operator and Szegő (or Szegő-like) projections (Harrington, Peloso, and Raich), and global regularity (Straube and Zeytuncu). In the latter, methods and results from CR geometry turned out to be crucial. In contrast to the hypersurface type situation, almost nothing is known when the CR submanifold is not of hypersurface type. The Levi form now becomes vector valued, with the attendant complications. It is expected that here too, CR geometry will play an essential role.

For an abstract CR manifold, the question of embeddability is intimately related to closed range (Kohn [1985]) and stronger regularity properties of $\overline{\partial}_{M}$ (Boutet de Monvel [1975], Burns [1979]). In fact, it is tempting to speculate that compactness estimates for $\overline{\partial}$ may be good enough to get embeddability. For more on embeddability, see Section 4.

**2. Spectral theory of complex Laplacians.** Spectral behavior of the $\bar\partial$-Neumann-Laplacian is more sensitive to the geometry of the underlying manifold than the usual Dirichlet or Neumann Laplacian (Fu (2008)). It follows from the fundamental work of Hőrmander on $L^2$-estimates of the $\bar\partial$-operator that for a bounded domain in ${\mathbb C}^n$, pseudoconvexity implies positivity of the $\bar\partial$-Neumann Laplacian. The converse is also true; this is a consequence of sheaf cohomology theory dating back to H. Cartan and J.-P. Serre. Thus one can determine pseudoconvexity from positivity of the $\bar\partial$-Neumann Laplacian. Catlin showed that property ($P$), a potential theoretic property, is a sufficient condition for pure discreteness of the spectrum of the $\bar\partial$-Neumann Laplacian. For convex domains or Hartogs domains in ${\mathbb C}^2$, property ($P$) is also a necessary condition (Fu and Straube (1998); Christ and Fu (2005)). Whether or not this is the case for a general pseudoconvex domain in ${\mathbb C}^2$ is a challenging open problem. It was discovered that pure discreteness of the spectrum on Hartogs domains in ${\mathbb C}^2$ is intimately related to diamagnetism and paramagnetism in semi-classical analysis of certain magnetic Schrődinger operators. How this connection evolves when no symmetry is presented remains to be seen.

The underpinnings of Witten's approach (1982) to the Morse inequality and Demailly's Holomorphic Morse inequality (1985) are spectral analyses of real and complex Laplacians. $L^2$-theory for the $\bar\partial$-operator has been used to prove spectacular results in complex geometry by Berndtsson (2009), Blocki (Inventiones), Demailly, Paun (Acta), Siu (2007), and others in recent years. The Ohsawa-Takegoshi extension theorem has found important applications in the study of invariance of plurigenera and the Fujita conjecture. $L^2$-estimates have also played an important role in establishing non-existence of Levi-flat hypersurfaces in complex projective spaces in the work of Siu and Cao-Shaw.

**3. Harmonic analysis and estimates of kernels.** Balls and metrics form an integral part of the study of convex bodies and harmonic analysis on domains in $\mathbb C^n$, $n > 1$. The issues here have their genesis in the classical function-theoretic problem of determining the relationship between the values of a holomorphic function in the interior and its trace on the boundary of a domain. Given a small set $\Gamma \subseteq \mathbb C^n$, one seeks to describe a domain $\Omega$ such that $\Gamma$ is in a sense the boundary of $\Omega$ and any nice function on $\Gamma$ can be extended to a function in $\Omega$ with prescribed analytical properties. For instance, does the boundary function admit a holomorphic extension to the "interior" of the domain? Note that if $\Gamma$ has high codimension, the notion of interior has to be made precise. Conversely, if a function is well-behaved in the interior of a domain, does it extend to the boundary and beyond, while retaining some of its desirable features? A fundamental issue in the analysis of several complex variables is whether a biholomorphic function between the interiors of two smoothly bounded domains extends smoothly between their boundaries.

Well-known tools in this study are the Bergman and Szegő kernels, which are reproducing kernels on natural Hilbert spaces of holomorphic functions. Singular integral operators with these kernels represent projections onto other function spaces of interest. For instance, given a domain $\Omega \subseteq \mathbb C^n$ with smooth boundary, the Bergman projection maps $L^2(\Omega)$ onto the space of $L^2$-holomorphic functions on $\Omega$. Given a function $f in L^2(\partial \Omega)$ occurring as the restriction of a holomorphic function, its Szegő projection produces the holomorphic extension of $f$ in $\Omega$. On one hand, the blow-up rates of these kernels and their derivatives near the boundary encapsulate important information about the geometry of the domain such as pseudoconvexity. On the other hand these singularities dictate analytical properties such as Lebesgue, Sobolev and Hőlder regularity of the corresponding projection operators, tying in with areas of central importance in harmonic analysis, such as Calder'on-Zygmund theory.

The precise relation between growth rates of these kernels, the geometry of the domain and the corresponding singular integral operators is understood in full generality for domains in $\mathbb C^2$ of finite type. See, for example, the work of Christ, Fefferman, Kohn, McNeal, Nagel, Stein, Wainger, etc. In particular, it is known that the sizes of the kernels are specified in terms of the volume of certain balls given by natural vector fields associated to the domains. Such geometric characterizations are widely sought in higher dimensions, but the direct generalizations of the bivariate metric are known to fail (Machedon (1988)). In fact, the definition of a correct metric whose balls would govern the Bergman or Szegő kernels of weakly pseudoconvex domains of finite type remains the overarching challenge in the field to this day, despite partial results under special assumptions such as convexity or strong pseudoconvexity (Machedon (1988), Nagel and Stein (Ann. Math., 2006), Charpentier and Dupain (2006)). Moreover, there are examples like the "cross of iron", suggesting that the problem is not merely technical (Herbort (1983), Nagel and Pramanik); weakly pseudoconvex domains in general possess truly distinct features from the subclass of domains hitherto considered.

**4. CR Geometry.** CR Geometry is an active developing area closely related to complex geometry and analysis of $\bar\partial$ equations. During recent decades fundamental results have been obtained, which in turn raised important open problems. Mapping problems and moduli spaces of CR structures have been subjects of recent research of many mathematicians including Baouendi, Ebenfelt, D'Angelo, Gong, Huang, Isaev, Kim, Lamel, Mir, Rothschild, Webster, Zaitsev, etc. An important direction is the local equivalence problem and normal forms of CR manifolds, that goes back to classical work of Cartan (1932), Tanaka (1962, 1967), and Chern--Moser (1974) on strictly pseudoconvex hypersurfaces. Recent research focus on CR structures of higher codimension and/or with degenerate Levi form. Huang and Yin (Invent. Math., 2009) constructed a complete set of invariants for Bishop surfaces with vanishing Bishop invariant. Isaev and Zaitsev (2013) constructed an absolute parallelism on 5-dimensional Levi-degenerate hypersurface. For general CR manifolds of higher codimension the problem is largely open.

Closely related are finite jet determination, regularity, and rigidity problems of CR mappings. Lamel and Mir (JAMS, 2007) obtained fundamental results on parametrization of local CR automorphisms by finite jets. Kim and Zaitsev (Invent. Math., 2013) prove the rigidity (linearity) of CR map between Shilov boundaries of bounded symmetric domains. A popular rigidity problem is global analytic continuation of a germ of a CR mapping, that goes back to Poincar'e (1907). Since the profound work of Baouendi, Ebenfelt, and Rothschild (Acta Math., 1996) there have been an impressive number of publications on the problem in the real *algebraic* case. Scalari and Tumanov (2007) proved a result on analytic continuation of CR maps between real *analytic* manifolds, in which the target manifold is a product of strongly pseudoconvex domains, however, for general real analytic CR manifolds of higher codimension the problem is largely open. Much progress has been achieved on the rigidity and classification problems of CR mappings in which the CR manifolds have different dimensions, especially when the manifolds or only the target manifold are hyperquadrics. However, even the theory of CR maps between spheres of different dimensions is far from completion.

We also focus on the fundamental difficult problem on embedding of abstract CR structures. The global problem for strongly pseudoconvex compact CR manifolds of hypersurface type of dimension at least 5 was solved by Boutet de Monvel (1975). Recently Chanillo, Chiu, and Yang (Duke, 2012) have made important progress. They prove the embeddability of 3-dimensional CR manifolds with additional positivity assumption on Webster's curvature and CR Paneitz operator. Solutions of the local version of the problem were due to Kuranishi (1982), Akahori (1987), and Webster (1989) for strongly pseudoconvex CR manifolds of dimension at least 7. Catlin (1994) extended this result to manifolds with Levi form of certain signatures. In a recent work, Gong and Webster (2012) significantly improve the smoothness assumptions. For CR structures of higher codimension the problem is largely open.