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 do- mains 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 Schr¨odinger 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 differen- tial 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 plurisubharmonic- ity 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 do- main 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 infi- nite 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 pseudoconvex- ity, 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 pseu- doconvexity 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 pseu- doconvex 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 Fr´echet 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 plurisub- harmonic in distribution then there exists g L1(U, loc)∈ plurisubharmonic such that f = g m-a.e.