Strong Pseudoconvexity in Banach Spaces

Strong pseudoconvexity in Banach spaces Sofia Ortega Castillo Abstract. Having been unclear how to define that a domain is strongly, or strictly pseudoconvex in the infinite-dimensional setting, we first focus on finite dimension and eliminate the need of two degrees of differentiability of the boundary of a domain, since differentiable functions are difficult to find in infinite dimension. We introduce ℓ-strict pseudoconvexity for ℓ ≥ 1 and ℓ- uniform pseudoconvexity for ℓ ≥ 0. Defining ℓ-strict pseudoconvexity and ℓ-uniform pseudoconvexity for ℓ < 2 depends on extending a notion of strict plurisubharmonicity to cases lacking C2-smoothness, first studying it in the sense of distribution and then considering it in infinite dimension. Examples of strictly plurisubharmonic functions as well as ℓ-uniformly pseudoconvex domains for ℓ< 2 are presented. Keywords: differentiable; strictly plurisubharmonic; strictly plurisubharmonic on average continuously; 2-uniformly PL-convex; uniformly plurisubharmonic on average; strongly pseudoconvex; ℓ-strictly pseudoconvex; 1-strictly pseudoconvex at the boundary; ℓ-uniformly pseudoconvex. 1. Introduction Classically, strongly pseudoconvex domains provide concrete examples of domains of holomorphy in several complex variables that have additional structure. Domains of holomorphy are those domains Ω whose boundary points are each a singularity for a holomorphic function on Ω [12]. Basic examples of domains of holomorphy include any domain in the complex plane, domains of convergence of multivariable power series and any convex domain in a Banach space. Meanwhile, arXiv:1701.03823v3 [math.CV] 5 Dec 2020 the well known Hartogs’ domain is not a domain of holomorphy because it allows analytic continuation to a strictly larger domain [13, Ch. II, §1.1]. In the Euclidean complex space Cn, when the boundary of a given domain U has two degrees of smoothness we have that U being a domain of holomorphy is characterized by a complex-differential property at boundary points which cor- responds to a complex analogue of a well-known diffferential condition satisfied by convex domains, which is usually introduced as (Levi) pseudoconvexity. Thus 2010 Mathematics Subject Classification. Primary 32T15, 46B25; Secondary 46F10, 31C10. This author was under a CONACYT fellowship during part of the writing period of this manuscript. Additionally, the author was supported by a UUKi Rutherford Strategic Partner Grant to do a research visit at the University of Bath, and would also like to thank the Erwin Schrödinger International Institute for Mathematics and Physics for welcoming support during a workshop visit. 1 2 SOFIA ORTEGA CASTILLO strong pseudoconvexity is presented using the complex analogue of the differential condition that defines strict convexity [13, Ch. II, §2.6]. A simpler equivalent condition, that reduces to the strict plurisubharmonicity of a C2 defining function of the boundary, has become common too [13, Ch. II, §2.8], [7, §1.1]. Strictly plurisubharmonic functions are those whose complex Hessian is positive definite, where the complex Hessian is a complex analogue of the Hessian which is also called the Levi form. Still in finite dimension, pseudoconvexity was later extended to domains with- out C2 boundary as admitting a plurisubharmonic exhaustion function on the whole domain [13, Ch. II, §5.4], where an exhaustion function is one that has relatively compact sublevel sets. Once in an infinite-dimensional Banach space X, the notion of pseudoconvexity of an open set U has been extended by the plurisubharmonicity of logdU , where dU denotes the distance to the boundary of U. A list of other equivalent− ways to define pseudoconvexity in infinite dimension is in [12, §37]. In particular, pseudoconvexity can be characterized by its behavior on finite- dimensional spaces, that is, U is pseudoconvex if and only if U M is pseudoconvex for each finite-dimensional subspace M of X. In infinite dimension∩ still every domain of holomorphy is pseudoconvex, but there is a nonseparable Banach space for which the converse is false, while for general separable Banach spaces this problem remains open [12, Ch. VIII, §37]. In separable Banach spaces with the bounded approximation property, indeed pseudoconvex domains are domains of holomorphy [12, Ch. X, §45]. In this article we look at generalizations of strong pseudoconvexity to the infinite dimensional setting, for which we first extend this notion in the case that the boundary does not have two degrees of smoothness. This is a crucial work for the study of complex analysis in Banach spaces, such as the study of Cauchy-Riemann equations as in [10], and the study of boundary behavior of bounded holomorphic functions on certain domains as in [11] and [1]. Since this manuscript aims to present suitable notions of strict pseudoconvexity, Section 2 commences recalling plurisubharmonicity and setting notation for derivatives in arbitrary dimension, and then describes strict plurisubharmonicity. We then provide a natural definition of a strictly plurisubharmonic distribution that leads to a notion of strict plurisubharmonicity independent of having two degrees of smoothness, that we justify using even in the infinite-dimensional case and provide examples of. In Section 3 we discuss strict pseudoconvexity, and we focus on presenting 1- strict pseudoconvexity, 1-strict pseudoconvexity at the boundary, 1-uniform pseudoconvexity and 0-uniform pseudoconvexity, which are suitable notions of strict pseudoconvexity for domains whose boundary lacks two degrees of smoothness. As expected, we obtain that these strongly pseudoconvex domains continue being pseudoconvex domains, and thus domains of holomorphy in nice enough settings. We also discuss properties and examples of ℓ-uniformly pseudoconvex domains, and in particular we see a characterization of ℓ-uniform pseudoconvexity as a certain limit of (ℓ + 1)-uniformly pseudoconvex domains for ℓ equal to 0 and 1. Foundations on plurisubharmonicity and pseudoconvexity in Banach spaces, as well as a basic treatment of distributions, can be found in [12]. The reader interested in a deep study of pseudoconvexity in Cn will find it in [14]. STRONG PSEUDOCONVEXITY 3 2. Strict plurisubharmonicity From now on, let X denote a complex Banach space with open unit ball BX and norm , let U denote an open subset of X with boundary bU, and let dU denote the distancek·k function to bU. We will also denote with m the Lebesgue measure in Cn seen as R2n. Let us recall that a function f : U [ , ) is called plurisubharmonic if f → −∞ ∞ is upper semicontinuous and for each a U and b X such that a + D b U we have that ∈ ∈ · ⊂ 1 2π f(a) f(a + eiθb)dθ. ≤ 2π Z0 Given a differentiable mapping f : U R and a U, we will write Df(a) for the Fréchet derivative of f at a, and in→ turn its complex-linear∈ and complex- antilinear parts will be denoted by D′f(a) and D′′f(a), respectively, which are given by D′f(a)(b)=1/2[Df(a)(b) iDf(a)(ib)], − D′′f(a)(b)=1/2[Df(a)(b)+ iDf(a)(ib)], for every b X. ∈ It is known that a function f C2(U, R) is plurisubharmonic if and only of its complex Hessian is positive semi-definite,∈ i.e. for each a U and b X we have that ∈ ∈ (2.1) D′D′′f(a)(b,b) 0. ≥ Because of that, a function f C2(U, R) is called strictly plurisubhamonic when the complex Hessian of f is positive∈ definite, i.e. when a proper inequality in (2.1) for b = 0 is satisfied [12, §35]. 6 When we are in the Euclidean complex space Cn, we aim to understand a suitable extension of strict plurisubharmonicity to distributions, using that in finite dimension a function f C2(U) is strictly plurisubharmonic if and only if there exists ψ C(U) positive∈ such that ∈ D′D′′f(a)(b,b) ψ(a) b 2 for all a U and b Cn. ≥ k k ∈ ∈ Due to the previous observation, in the arbitrary Banach space setting we will say that f C2(U, R) is strictly plurisubhamonic continuously when there exists ψ C(U) positive∈ such that D′D′′f(a)(b,b) ψ(a) b 2 for all a U and b X. ∈ ≥ k k ∈ ∈ If U is an open subset of Cn, we will denote the real-valued test functions on U by (U). A distribution on U is known to be a continuous functional on (U). We shallD denote by ′(U) the vector space of all distributions on U. D D Given f L1(U, loc), we say that f is (strictly) plurisubharmonic in distribution if the distribution∈ it induces is (strictly) plurisubharmonic. At the same time, a distribution T ′(U) is called plurisubharmonic if ∈ D n 2 ′ ′′ ∂ T n D D T (φ)(w, w) := (φ)wj wk 0, for all φ 0 in (U) and w C . ∂zj∂zk ≥ ≥ D ∈ j,kX=1 4 SOFIA ORTEGA CASTILLO And we will say that T ′(U) is strictly plurisubharmonic if there exists ψ C(U) positive such that ∈ D ∈ (2.2) D′D′′T (φ)(w, w) ( ψ φ dm) w 2, for all φ 0 in (U) and w Cn. ≥ · k k ≥ D ∈ ZU It has been proved, e. g. in [6, §3.2 and 4.1], that plurisubharmonicity is equivalent to plurisubharmonicity in distribution in the following sense: Suppose that U is a connected domain in Cn. If f = is plurisubharmonic on U, then f L1(U, loc) and f is plurisubharmonic in6 distribution.−∞ Conversely, if T ′(U) is∈ plurisubharmonic then there exists f L1(U, loc) plurisubharmonic such∈ D that f induces the distribution T . As a corollary,∈ if f L1(U, loc) is plurisubharmonic in distribution then there exists g L1(U, loc)∈ plurisubharmonic such that f = g m-a.e.

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