Strong Pseudoconvexity in Banach Spaces

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Subject Mathematics uiompedcneiyfor pseudoconvexity -uniform togpedcneiyi aahspaces Banach in pseudoconvexity Strong ieetal;srcl lrsbamnc tityplurisubharmonic strictly plurisubharmonic; strictly differentiable; ℓ uiomypseudoconvex. -uniformly aigbe nla o odfieta oani togy or strongly, is domain a that define to how unclear been Having < ℓ r presented. are 2 oaOtg Castillo Ortega Sofia .Introduction 1. ℓ ℓ srcl suoovx -titypseudoconvex 1-strictly pseudoconvex; -strictly < ℓ ≥ C ℓ .Defining 0. eed netniganto fstrict of notion a extending on depends 2 src suoovxt for pseudoconvexity -strict n rmr 21,4B5 eodr 61,31C10. 46F10, Secondary 46B25; 32T15, Primary C 1 hntebudr fagvndomain given a of boundary the when , 2 sotns,fis tdigi nthe in it studying first -smoothness, 12 U hsc o ecmn upr uiga during support welcoming for Physics .Bsceape fdmisof domains of examples Basic ]. ℓ 13 src suoovxt and pseudoconvexity -strict en oano holomorphy of domain a being ℓ h II, Ch. , uiomypseudoconvex -uniformly efis ou nfinite on focus first we , das iet hn h Erwin the thank to like also ld uhrodSrtgcPartner Strategic Rutherford i nin Examples ension. § utt n in find to cult 1.1]. blt fthe of ability ℓ rtn eido this of period writing ≥ it hc cor- which oints and 1 hroi on bharmonic xt.Thus exity. structure. l lso do- of ples satisfied n tallows it e ℓ eec a each re eanwhile, - ec of gence 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 without 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. ∈ To prove an analogous version of such result for strict plurisubharmonicity, we will say that an upper semicontinuous function g : U X [ , ) is strictly plurisubharmonic on average (continuously) if there exists⊂ ϕ→ C−∞(U)∞ positive such that for all a U and b X of small norm (with size depending∈ lower semicontinuously on a) we∈ have that,∈ 1 2π (2.3) ϕ(a) b 2 + g(a) g(a + eiθb)dθ. k k ≤ 2π Z0 Proposition 2.1. Suppose that U is a connected domain in Cn. If f : U [ , ), with f = , is strictly plurisubharmonic on average then f L1(U, loc→) and−∞f∞is strictly6 plurisubharmonic−∞ in distribution. Conversely, if T∈ ′(U) is strictly plurisubharmonic then there exists f L1(U, loc) strictly plurisubhar-∈ D monic on average such that f induces the distribution∈ T . As a consequence, if f L1(U, loc) is strictly plurisubharmonic in distribution then there exists g L1∈(U, loc) strictly plurisubharmonic on average such that f = g m-a.e. ∈ Proof. If f = is strictly plurisubharmonic on average, then f is in particular plurisubharmonic,6 −∞ so we can use the relationship to plurisubharmonicity in distribution to deduce that f L1(U, loc). Moreover, since f is strictly plurisubharmonic on average in U C∈n, there exists a positive function ψ C(U) such that ⊂ ∈ 1 2π ψ(a) b 2 + f(a) f(a + eiθb)dθ, k k ≤ 2π Z0 for all a U and b Cn of norm less than δ (a) (δ > 0 lower semicontinuous). ∈ ∈ f f Consider the test function ρ : Cn R given by → 2 k e−1/(1−kxk ), if x < 1 ρ(x)= · k k 0, if x 1 ( k k≥ where the constant k is chosen so that Cn ρdm = 1. More generally, for each n −n n δ > 0 let ρδ (C ) be deﬁned by ρδ(x) = δ ρ(x/δ) for every x C , so that ∈ D ¯ R ∈ Cn ρδdm = 1 and supp(ρδ)= B(0,δ). n R Consequently, for δ > 0, a Uδ := z U : dU (z) > δ and b C of small ∈ { ∈ } ∈ norm (namely, less than infB(a,δ) δf > 0), STRONG PSEUDOCONVEXITY 5

2 2 ψ(a) b + f ρδ(a)= (ψ(a) b + f(a ζ))ρδ (ζ)dm(ζ) k k ∗ ¯ k k − ZB(0,δ) 2π 1 iθ ( f(a ζ + e b)dθ)ρδ(ζ)dm(ζ) ≤ ¯ 2π − ZB(0,δ) Z0 2π 1 iθ = ( f(a ζ + e b)ρδ(ζ)dm(ζ))dθ 2π ¯ − Z0 ZB(0,δ) where the last equality follows from Fubini’s theorem because f L1(U, loc). There- fore ∈ 1 2π ψ(a) b 2 + f ρ (a) f ρ (a + eiθb)dθ. k k ∗ δ ≤ 2π ∗ δ Z0 ∞ That is, f ρδ C (Uδ) is strictly plurisubharmonic on average for each δ > 0, and from the∗ proof∈ of Proposition 2.4 below, we obtain

n 2 ∂ (f ρδ)(a) 2 n ∗ bjbk ψ(a) b , a Uδ and b C . ∂zj ∂zk ≥ k k ∀ ∈ ∈ j,kX=1 Consequently, given w Cn and φ (U) positive, say with supp(φ) U , and ∈ ∈ D ⊂ δ0 taking δm 0 with δm < δ0, we have that f ρδm converges uniformly on Uδ0 to f, so by the→ dominated convergence theorem and∗ then integration by parts, n ∂2φ(z) n ∂2φ(z) f(z) wj wk dm(z) = lim f ρδm (z) wj wk dm(z) ∂z ∂z δm→0 ∗ · ∂z ∂z Uδ0 j k Uδ0 j k Z j,kX=1 Z j,kX=1 n 2 ∂ (f ρδm )(z) = lim ∗ wj wk φ(z)dm(z) δm→0 ∂z ∂z Uδ0 j k Z j,kX=1 ψ(z) w 2φ(z)dm(z) ≥ U k k Z δ0 i.e. f is strictly plurisubharmonic in distribution. Now suppose that T ′(U) is a strictly plurisubharmonic distribution, where ∈ D ∞ ψ is a positive continuous function satisfying equation (2.2). Then T ρδ C (Uδ) and for all z U and b Cn, ∗ ∈ ∈ δ ∈ n 2 n 2 ∂ (T ρδ)(z) ∂ T ∗ bj bk = ρδ(z)bjbk ∂zj ∂zk ∂zj∂zk ∗ j,kX=1 j,kX=1 n ∂2T = [ρδ(z )]bj bk ∂zj∂zk − · j,kX=1 ψ(w) b 2ρ (z w)dm(w) ≥ k k δ − ZU = ψ ρ (z) b 2. ∗ δ k k

2 1 2π iθ Due to Proposition 2.4, ψ ρδ(z) b /2+ T ρδ(z) 2π 0 T ρδ(z + e b)dθ when b has small norm depending∗ k lowerk semicontinuously∗ ≤ on z ∗(namely, when R b < sup r> 0 : B(z, r) (ψ ρ )−1(B(ψ ρ (z), ψ ρ (z)/2)) ). k k { ⊂ ∗ δ ∗ δ ∗ δ } 6 SOFIA ORTEGA CASTILLO

Since T is in particular a plurisubharmonic distribution, we get that T ρδ decreases to f L1(U, loc) plurisubharmonic that induces T . Consequently,∗ due to the dominated∈ and monotone convergence theorems applied respectively to the n positive and negative parts of each T ρδ, for all z U and b C of small norm (namely, when b < sup r> 0 : B(z,∗ r) ψ−1(B(ψ∈(z), ψ(z)/2))∈ ), k k { ⊂ } 2π 2π 1 iθ 1 iθ f(z + e b)dθ = lim T ρδm (z + e b)dθ 2π 0 δm→0 2π 0 ∗ Z Z 2 lim (T ρδm (z)+ ψ ρδm (z) b /2) ≥ δm→0 ∗ ∗ k k = f(z)+ ψ(z) b 2/2. k k The proposition is now clear.

Due to Proposition 2.1 we have that, in Cn, a C2 function is strictly plurisubharmonic if and only if it is strictly plurisubharmonic on average, and we will see in Proposition 2.4 below that in infinite dimension, a C2 function is strictly plurisubharmonic continuously if and only if it is strictly plurisubharmonic on average continuously. Meanwhile, it had been known that in finite dimension a C2 function satisfies strict plurisubharmonicity if and only if in a neighborhood of each point in the domain there exists an ǫ> 0 such that F ǫ 2 is plurisubharmonic, but this is no longer true in infinite dimension. We constructed− k·k the following example with the help of Santillán Zerón: N Example 2.2. Let X be the Hilbert space of complex sequences (zn) C such that ⊂ ∞ (z ) := z 2/k2 < . k n k v | k| ∞ uk=1 uX t∞ 2 3 Then the function on X, F (z) = k=1 zk /k , has positive definite complex Hessian at z, | | P ∞ D′D′′F (z)(w, w)= w 2/k3, for all w X; | k| ∈ Xk=1 however F does not admit ǫ> 0 such that, for z near to z0, we have plurisubharmonicity of the function ∞ ∞ ∞ F (z) ǫ z 2 = z 2/k3 ǫ z 2/k2 = z 2(1/k3 ǫ/k2), − k k | k| − | k| | k| − Xk=1 Xk=1 kX=1 since its complex Hessian at each z0 is given by ∞ D′D′′(F ǫ 2)(z )(w, w)= w 2(1/k3 ǫ/k2) for w X, − k·k 0 | k| − ∈ kX=1 that eventually has eigenvalues which are negative. Conversely, if X is infinite dimensional, a function F : U X [ , ) that locally admits an ǫ> 0 such that F ǫ 2 is plurisubharmonic⊂ may→ not−∞ be∞ strictly plurisubharmonic on average, as exhibited− k·k by the next example. STRONG PSEUDOCONVEXITY 7

2 Example 2.3. The squared norm in ℓ∞, ∞, is not strictly plurisubharmonic on average since for a = (1, 0, 0, ) and b =k·k (0, 1, 0, 0, ), ··· ··· 1 2π ( a + eiθb 2 a 2 )dθ =0, 2π k k∞ −k k∞ Z0 however, for ǫ (0, 1), (1 ǫ) 2 is known to be plurisubharmonic. ∈ − k·k∞ Le us now present the main proposition of this section, as we have referred to it and we will use it to show further examples. Proposition 2.4. Let U be an open domain in a Banach space X. A function f C2(U; R) is strictly plurisubharmonic continuously if and only if there exists a strictly∈ positive function ϕ C(U; R) such that, for all a U and b X of small norm (with size lower semicontinously∈ depending on a), ∈ ∈ 1 2π (2.4) (f(a + eiθb) f(a))dθ ϕ(a) b 2. 2π − ≥ k k Z0 Proof. Suppose that there exists a positive function ϕ C(U; R) satisfying (2.4) for a U and b X of small norm. Given a U,∈ fix b X of small ∈ ∈ ∈ ∈ enough norm so that in particular we have a + Db U, and consider the function u(z)= f(a + z b), which is defined on D. ⊂ · Then, for all r (0, 1), ∈ 1 2π ϕ(a) b 2 r2 (u(r eiθ) u(0))dθ k k · ≤ 2π · − Z0 Consequently, by Lemma 5 in [15], and exercises 35.B and 35.D in [12], ∂2u (2.5) ϕ(a) b 2 (0) = D′D′′f(a)(b,b). k k ≤ ∂z∂z¯ Now suppose that there exists a positive function ϕ C(U; R) such that (2.5) holds for all a U and b X. Fix a U. Since ϕ is continuous∈ at a, there exists an upper bound δ∈(a) > 0 for∈ the norm of∈ b to make ϕ(a) ϕ(a + b) < ϕ(a)/2 hold (δ is lower semicontinuous for δ(a) = sup r > 0 : B| (a, r)− ϕ−1(B|(ϕ(a), ϕ(a)/2)) ). { ⊂ } Fix b as before, and define M(r)= 1 2π[f(a + reiθb) f(a)]dθ, for all r (0, 1]. 2π 0 − ∈ Consider also the function u(ζ)= f(a + ζb) defined on a disk D(0, R) D. Then, R for all ζ D(0, R), ⊃ ∈ ∂2u ∂2u ∂2u (ζ)+ (ζ)=4 (ζ)=4 D′D′′f(a + ζb)(b,b) 4 ϕ(a + ζb) b 2. ∂x2 ∂y2 ∂ζ∂ζ¯ · ≥ · k k

∂2u ∂2u ∂2u 1 ∂u 1 ∂2u Since 2 + 2 = 2 + + 2 2 , then for r (0, 1), ∂x ∂y ∂r r ∂r r ∂θ ∈ 1 2π ∂2 1 ∂ 1 ∂2 1 2π ( + + )u(reiθ)dθ 4 ϕ(a+reiθb) b 2dθ 2 ϕ(a) b 2, 2π ∂r2 r ∂r r2 ∂θ2 ≥ ·2π k k ≥ · k k Z0 Z0 i.e. M ′′(r)+ 1 M ′(r) 2 ϕ(a) b 2, r (0, 1). r ≥ · k k ∀ ∈ 2 Thus (rM ′(r) 2 ϕ(a) b 2 r )′ = rM ′′(r)+ M ′(r) 2 ϕ(a) b 2r 0 for all − · k k 2 − · k k ≥ r (0, 1), so r(M ′(r) ϕ(a) b 2r) is an increasing function of r. Since clearly r(M∈ ′(r) ϕ(a) b 2r) − 0 askr k 0 (because M ′ is a bounded function on (0,ǫ) for some−ǫ > 0),k wek conclude→ that→ r(M ′(r) ϕ(a) b 2r) 0 for every r (0, 1). − k k ≥ ∈ 8 SOFIA ORTEGA CASTILLO

2 r2 ′ 2 r2 Hence (M(r) ϕ(a) b 2 ) 0 for every r > 0, so M(r) ϕ(a) b 2 is an − k k ≥ 2 r2 − k k increasing function of r. Since clearly M(r) ϕ(a) b 2 0 as r 0 then 2 − k k → → M(r) ϕ(a) b 2 r for each r (0, 1). ≥ k k 2 ∈ Since M is continuous on (0, 1], we conclude that M(1) ϕ(a) b 2, so indeed ≥ 2 k k 1 2π ϕ(a) [f(a + eiθb) f(a)]dθ b 2. 2π − ≥ 2 k k Z0

An important remark about the proof of Proposition 2.4 is that, when we have f C2(U; R) satisfying D′D′′f(a)(b,b) L b 2 for some L > 0 we can conclude ∈ ≥ k k that 1 2π(f(a+eiθb) f(a))dθ L b 2 for all a U and b X with b < d (a). 2π 0 − ≥ k k ∈ ∈ k k U LetR us finish this section exhibiting functions strictly plurisubharmonic on average continuously, without two degrees of differentiability. For that let us discuss a family of Banach spaces X having a norm that is strictly plurisubharmonic k·k on average continuously on 2BX , of which some lack two degrees of differentiability [3]. The following notion of uniform convexity for complex quasi-normed spaces found in [2] generalizes uniform c-convexity as defined by Globevnik [5]. They pass from the real to the complex concept of uniform convexity by replacing norms of midpoints of segments in the space by average norms of complex discs in the space. As it turns out, while the real modulus of convexity measures uniformly the convexity of the ball of a normed space, its complex analog measures subharmonicity instead of convexity. Definition 2.5. If 0 0 such that k·k 1 2π ( a + eiθb qdθ)1/q ( a r + λ b r)1/r 2π k k ≥ k k k k Z0 for all a and b in X; we shall denote the largest possible value of λ by Ir,q(X). It is known that the previous definition does not depend on q. Let us recall that a quasi-normed space (X, ) is continuously quasi-normed if is uniformly continuous on the bounded setsk·k of X. Banach spaces are obviouslyk·k continuously quasi-normed.

Davis, Garling and Tomczak-Jaegermann proved that for p [1, 2], Lp(Σ, Ω,µ) is 2-uniformly PL-convex ([2, Cor. 4.2]), and they obtain that I∈ (L )= I (C) 2,p p 2,p ≥ 1 > 1/2 = I2,1(C). Other examples of 2-uniformly PL-convex spaces include the dual of any C∗-algebra ([2, Thm. 4.3]), the complexiﬁcation of a Banach lattice with 2-concavity constant 1 ([8, Cor. 4.2]) and the non commutative Lp(M), 1 p 2, where M is a von Neumann algebra acting on a separable Hilbert space ([4, Thm.≤ ≤ 4].

The following proposition gives us in particular that an Lp(Σ, Ω,µ) space, for p [1, 2], has strictly plurisubharmonic norm. ∈ Proposition 2.6. If (X, ) is a 2-uniformly PL-convex Banach space then the norm is strictly plurisubharmonick·k on average continuously on 2B . k·k X STRONG PSEUDOCONVEXITY 9

Proof. Let a 2BX and b X so that a+Db 2BX . In particular b 2BX . Also, ∈ ∈ ⊂ ∈ 1 2π ( a 2 + I (X) b 2)1/2 a + eiθb dθ k k 2,1 k k ≤ 2π k k Z0 Let λ = I (X)/4+1/4 1/2 > 0. Observe that 0 2,1 − p a + λ b 2 ( a 2 + I (X) b 2)1/2 k k 0k k ≤ k k 2,1 k k because ( a + λ b 2)2 = a 2 +2λ a b 2 + λ2 b 4 k k 0k k k k 0k kk k 0k k a 2 + (4λ +4λ2) b 2 ≤k k 0 0 k k = a 2 + 4((λ +1/2)2 1/4) b 2 k k 0 − k k = a 2 + I (X) b 2. k k 2,1 k k Thus, 1 2π λ b 2 + a a + eiθb dθ. 0k k k k≤ 2π k k Z0 as desired

The proof of Proposition 2.6 ensures that the norm of a 2-uniformly PL-convex Banach space is not only strictly plurisubharmonic on average continuously on 2BX but moreover uniformly plurisubharmonic on average, where an upper semicontinuous function g : U X [ , ) is called uniformly plurisubharmonic on average if there exists⊂ a constant→ −∞L>∞0 such that for all a U and b X of small norm (with size depending lower semicontinuously on a), ∈ ∈ 1 2π L b 2 + g(a) g(a + eiθb)dθ. k k ≤ 2π Z0 If g : U X [ , ) is as just described above, let us speciﬁcally call it L-uniformly⊂ plurisubharmonic→ −∞ ∞ on average. The reader can correspondingly deﬁne uniform plurisubharmonicity in distribution and prove a relationship to uniform plurisubharmonicity on average analogous to the one shown in Proposition 2.1. The reader can also check that for any M > 0, the norm of a 2-uniformly PL-convex Banach space is uniformly plurisubharmonic on average on MBX by restricting the circles of integration to have radius less than one. A last important remark about functions strictly plurisubharmonic on average continuously is that if we add to one of those a plurisubharmonic function, we preserve a function strictly plurisubharmonic on average continuously.

3. Strict pseudoconvexity Let us step back to finite dimension to discuss strict pseudoconvexity even in the case when the boundary of a given domain lacks two degrees of smoothness. Recall that a domain U Cn with C2 boundary is strictly pseudoconvex when the strict Levi condition is satisfied⊂ by a defining function r of the boundary, i. e. 10 SOFIA ORTEGA CASTILLO when r is a C2 real-valued function defined on a neighborhood V of bU such that U V = z V : r(z) < 0 and Dr(w) = 0 for w bU, it holds that: ∩ { ∈ } 6 ∈ (3.1) D′D′′r(w)(b,b) > 0 when w bU and b Cn satisfy D′r(w)(b)=0. ∈ ∈ Equivalently, according to [13, Ch. II, §2.8], a domain U in Cn with C2 boundary is strongly pseudoconvex when there is a C2 defining function of bU, r : V R, admitting a positive constant L such that D′D′′r(z)(b,b) L b 2 for all z V→and b Cn. ≥ k k ∈ ∈ Actually, it is well-known that U is convex if and only if log d is convex, − U while U is pseudoconvex if and only if log dU is plurisubharmonic on U. The following is a similar result for bounded strictly− pseudoconvex domains in Cn with C2 boundary, that we proved with the help of Ramos Peón. Proposition 3.1. Let U be a bounded open domain in Cn with C2 boundary. Then U is strictly pseudoconvex if and only if there exist a positive constant L, a neighborhood V of bU and ρ C2(V ) a defining function of bU such that, ∈ L (3.2) D′D′′( log ρ )(a)(b,b) b 2 for all a U V and b Cn. − | | ≥ ρ(a) k k ∈ ∩ ∈ | | Equivalently, V above can be replaced by a neighborhood of U. Proof. Suppose that there exist a positive constant L, a neighborhood V of bU and ρ C2(V ) a defining function of bU such that D′D′′( log ρ )(a)(b,b) ∈ − | | ≥ L b 2 for every a U V and b Cn. Since for a U V and b Cn arbitrary |ρ(a)| k k ∈ ∩ ∈ ∈ ∩ ∈ we have 1 1 D′D′′( log ρ )(a)(b,b)= D′D′′ρ(a)(b,b)+ D′ρ(a)(b) 2 − | | ρ(a) ρ(a)2 · | | | | we obtain that D′D′′ρ(a)(b,b) L b 2 when a U V and b Cn satisfy D′ρ(a)(b)=0. ≥ k k ∈ ∩ ∈ A passage to the limit shows that on the boundary we have what we desired: D′D′′ρ(w)(b,b) > 0 when w bU and b = 0 satisfy D′ρ(w)(b)=0. ∈ 6 Now suppose that U is strictly pseudoconvex. Then we can find a positive constant L, a neighborbood V of bU and ρ C2(V ) a defining function of the boundary of U such that ∈ D′D′′ρ(a)(b,b) L b 2 for all a V and b Cn. ≥ k k ∈ ∈ Then for a U V and b Cn arbitrary, ∈ ∩ ∈ 1 1 D′D′′( log ρ )(a)(b,b)= D′D′′ρ(a)(b,b)+ D′ρ(a)(b) 2 − | | ρ(a) ρ(a) 2 | | | | | | L b 2, ≥ ρ(a) k k | | as desired. Because of the previous proposition and the developments in Section 2, given ℓ 1 we will say that a bounded domain U with Cℓ boundary, in a Banach space X≥, is ℓ-strictly pseudoconvex if there exist a positive constant L, a neighborhood STRONGPSEUDOCONVEXITY 11

V of bU and ρ Cℓ(V ) a defining function of bU such that for all a U V and b X of small norm∈ (with size lower semicontinuously depending on ∈a), ∩ ∈ 1 2π L (3.3) log ρ (a + eiθb)dθ log ρ (a)+ b 2. 2π − | | ≥− | | ρ(a) k k Z0 | | As a consequence, an ℓ-strictly pseudoconvex domain U admits a plurisubharmonic function σ defined on all of U such that σ(z) as z bU (see [13, Ch.II, §2.7] and [12, Ch. VIII, §34]). → ∞ → Clearly, a bounded domain in Cn with C2 boundary is strictly pseudoconvex if and only if it is 2-strictly pseudoconvex, and all 2-strictly pseudoconvex domains are 1-strictly pseudoconvex. Moreover, it is clear that an ℓ-strictly pseudoconvex domain in Cn admits an associated function σ as a plurisubharmonic exhaustion function, hence ℓ-strictly pseudoconvex domains in Cn are pseudoconvex. Likewise, an ℓ-strictly pseudoconvex domain in a Banach space X is pseudoconvex because its restriction to each finite-dimensional subspace admits a restricted associated function σ as a plurisubharmonic exhaustion function. Let us now look at the following necessary condition for ℓ-strict pseudoconvexity in Cn. Proposition 3.2. If a domain U Cn is ℓ-strictly pseudoconvex for some ℓ 1 then there exist a positive constant⊂L, a neighborhood V of bU and a defining function≥ ρ Cℓ(V ) of bU such that for all a U V and b Cn with D′ρ(a)(b)=0 and b <∈ d (a)we have, ∈ ∩ ∈ k k U∩V 1 2π ρ(a + eiθb) ρ(a)+ L b 2. 2π ≥ k k Z0 Proof. Since U is ℓ-strictly pseudoconvex, there exist a positive constant L, a neighborhood V of bU and a defining function ρ Cℓ(V ) of bU such that for all a U and b Cn of small norm (with size lower∈ semicontinuously depending on a),∈ ∈ 1 2π L log ρ (a + eiθb)dθ log ρ (a)+ b 2. 2π − | | ≥− | | ρ(a) k k Z0 | | For each n N define the function ρ1/n as in Proposition 2.1, and take σn = ( log( ρ)) ρ ∈ , which is defined and smooth on (U V ) . Then, as in the − − ∗ 1/n ∩ 1/n proof of Proposition 2.1, σn is strictly plurisubharmonic on average on (U V )1/n with respect to the function L/ ρ , and because of the proof of proposition∩ 2.4 we have that for all a (U V ) | and| b Cn, ∈ ∩ 1/n ∈ L D′D′′σ (a)(b,b) b 2. n ≥ ρ (a)k k | | Because of distribution theory it is clear that σm converges uniformly on each ′ (U V ) 2 to log( ρ), and likewise the vector-valued function D σm converges ∩ n − − ′ −1 ′ −σm uniformly on each (U V ) 2 to D ( log( ρ)) = D ρ. Define rm = e on ∩ n − − ρ − (U V )1/m, which again converges uniformly on each (U V ) 2 to ρ. ∩ ∩ n L Pick n N. Now choose ǫn > 0 such that ǫn < min( , 1), where Mn = ∈ Mn+1 sup (L/ ρ(a) + ρ(a) ). Without loss of generality, for all m n we have a∈(U∩V )2/n | | | | ≥ 12 SOFIA ORTEGA CASTILLO

−σm ′ 2 1 ′ 2 that e ρ ǫn on (U V ) 2 , and that D σm(a)(b) D ρ(a)(b) | −| ||≤ ∩ n | | | −| |ρ(a)| | |≤ 2 n ǫn b when a (U V ) 2 and b C . k k ∈ ∩ n ∈ Then, when a (U V ) and D′ρ(a)(b) = 0, we have that for m n, ∈ ∩ 2/n ≥ D′D′′r (a)(b,b)= e−σm(a)D′D′′σ (a)(b,b) e−σm(a) D′σ (a)(b) 2 m m − | m | L e−σm(a) b 2 e−σm(a) D′σ (a)(b) 2 ≥ ρ(a) k k − | m | | | ρ(a) ǫ 1 2 | |− n L b 2 ( ρ(a) + ǫ )( D′ρ(a)(b) + ǫ b 2) ≥ ρ(a) k k − | | n ρ(a) nk k | | L = (L ǫ ( + ρ(a) ) ǫ2 ) b 2 − n ρ(a) | | − n k k | | where L ǫ ( L + ρ(a) ) ǫ2 L ǫ (M + 1) > 0.Then, following the proof of − n |ρ(a)| | | − n ≥ − n n Proposition 2.4, each r satisfies for a (U V ) as well as b with D′ρ(a)(b)=0 m ∈ ∩ 2/n and b < d (a), k k (U∩V )2/n 1 2π (r (a + eiθb) r (a))dθ (L ǫ (M + 1)) b 2. 2π m − m ≥ − n n k k Z0 so taking the limit as m , → ∞ 1 2π (ρ(a + eiθb) ρ(a))dθ (L ǫ (M + 1)) b 2. 2π − ≥ − n n k k Z0 As ǫ 0 and then n , we conclude what we desired. n → → ∞ Proposition 3.2 is significant since it confirms a very expected behavior of a defining function of an ℓ-strictly pseudoconvex domain. However, if ℓ = 1, it does not recover a condition at the boundary resembling equation (3.1), so we consider it separately. Given a domain U X with C1 boundary, we will say that U is 1-strictly pseudoconvex at the boundary⊂ if it is pseudoconvex and there exist a neighborhood V of bU, ρ C1(V ) a defining function of bU and ϕ C(bU) a positive function, such that for∈ all for all w bU and b Cn with D′ρ(w∈)(b) = 0 and b small (lower semicontinuously depending∈ on w) we∈ have k k 1 2π (3.4) ρ(w + eiθb)dθ ρ(w)+ ϕ(w) b 2. 2π ≥ k k Z0 We will soon explore examples of domains 1-strictly pseudoconvex at the boundary. But first let us see that the following is a sufficient condition for ℓ-strict pseudoconvexity when ℓ 1. ≥ Proposition 3.3. Let ℓ 1. If U in a Banach space X is a bounded open domain that admits a Cℓ defining≥ function of bU, r : V bU R, which is uniformly plurisubharmonic on average on U V then U is ℓ-strictly⊃ → pseudoconvex. ∩ Proof. Let r : V bU R be a Cℓ defining function of bU which is uniformly plurisubharmonic on average⊃ → on U V . Then we can find L> 0 such that, for all ∩ STRONGPSEUDOCONVEXITY 13 a U V and b Cn of small norm (with size depending lower semicontinuously ∈ ∩ ∈ on a), we have that a + Db is in U V and ∩ 1 2π (3.5) r(a + eiθb)dθ r(a)+ L b 2. 2π ≥ k k Z0 Fix a U V and b X of small norm as before. Since a + D b U V , we have that ∈r(a +∩eiθb) < 0∈ for each θ [0, 2π]. · ⊂ ∩ ∈ Since x log( x) is convex on ( , 0), we obtain by Jensen’s inequality that 7→ − − −∞ 1 2π 1 2π (3.6) log( r)(a + eiθb)dθ log( r(a + eiθb)dθ). 2π − − ≥− −2π Z0 Z0 Meanwhile, due to equation (3.5) we have that 1 2π 0 < − r(a + eiθb)dθ (r(a)+ L b 2), 2π ≤− k k Z0 so using that log is decreasing on (0, ) we obtain that − ∞ 1 2π (3.7) log(− r(a + eiθb)dθ) log( (r(a)+ L b 2)) − 2π ≥− − k k Z0 As a consequence of equations (3.6) and (3.7) we obtain that

1 2π log( r)(a + eiθb)dθ log( (r(a)+ L b 2)) 2π − − ≥− − k k Z0 L b 2 = log( r(a)(1 k k )) − − − [ r(a)] − L b 2 = log( r(a)) log(1 k k ) − − − − [ r(a)] − log( r(a)) + L b 2/[ r(a)] ≥− − k k − as desired. Note that if U bounded admits a C1 deﬁning function ρ : V bU R which is uniformly plurisubharmonic on average on U V then there exists⊃ δ >→0 such that ∩ for all c ( δ, 0), Uρ,c := (U V ) z V : ρ(z)

Domains U in a Banach space X that are strongly C-linearly convex are weakly C linearly convex, since given w bU we have that w w+Tw (bU), and for all z U, there exists C > 0 such that ∈ ∈ ∈ 1 C inf w+b z inf D′ρ(w)(w+b z) w z > 0. b∈T C (bU) k − k≥ b∈T C (bU) D′ρ(w) ·| − |≥ D′ρ(w) k − k w w k k k k It is well known that domains weakly linearly convex are holomorphically convex (see [6, Ch. IV, §4.6], whose proof still works in infinte dimension), thus if X is a separable Banach space with the bounded approximation property, its domains weakly linearly convex are pseudoconvex. Let us now show that domains strongly C-linearly convex U Cn admit a C1 defining function ρ : V bU R satisfying equation (3.4) for⊂ all w bU and b Cn with D′ρ(w)(b) =⊃ 0 and→b < min(d (w), 1). ∈ ∈ k k V Consider the function d (z), if z U ρ(z)= − U ∈ . d (z), if z U c ( U ∈ It is well known that ρ is a C1 defining function in a neighborhood V of bU. C We now fix w bU and b Tw (bU) with b < min(dV (w), 1). For every ∈ ∈ iθ k k iθ ′ θ [0, 2π], we know that dU (w + e b) = infw′∈bU w + e b w , and for each w′∈ bU let us consider two cases: k − k ∈ If w w′ b /2 then w+eiθb w′ b w w′ b /2 b 2/2 • If kw− wk≤k′ kb /2 thenk as we did− beforek≥k wek−k obtain− thatk≥k k ≥k k • k − k≥k k C C w + eiθb w′ inf w + b′ w′ w w′ b 2/4 k − k≥ b′∈T C (bU) k − k≥ D′ρ(w) k − k≥ D′ρ(w) k k w k k k k C iθ 2 then taking C = min(1/2, ′ ), we obtain that d (w + e b) C b . w 4kD ρ(w)k U ≥ wk k Thus, for C/4 L = min(1/2, ) > 0, sup D′ρ(w) w∈bU k k we have that, since w + eiθb / U for every θ [0, 2π], ρ(w + eiθb) ρ(w) = d (w + eiθb) L b 2, hence ∈ ∈ − U ≥ k k 1 2π (ρ(w + eiθb) ρ(w))dθ L b 2. 2π − ≥ k k Z0 We conclude that domains strongly C-linearly convex U Cn are 1-strictly pseudoconvex at the boundary. ⊂ We will come back to domains 1-strictly pseudoconvex at the boundary, but for now let us discuss domains U that are ℓ-strictly pseudoconvex at the boundary as well as ℓ-strictly pseudoconvex. Obviously, this will happen when a defining function of bU is uniformly plurisubharmonic on average on all of its domain. Thus we will be interested in the following notion. If ℓ 1 and U in a Banach space X is a bounded open domain that admits a Cℓ defining≥ function of bU which is uniformly plurisubharmonic on average then we will say that U is ℓ-uniformly pseudoconvex. STRONGPSEUDOCONVEXITY 15

It is clear that a domain in Cn with C2 boundary is strongly pseudoconvex if and only if it is 2-uniformly pseudoconvex, and all 2-uniformly pseudoconvex domains are 1-uniformly pseudoconvex. And as before, it is obvious that ℓ-uniformly pseudoconvex domains are pseudoconvex. Let us see nontrivial examples of 1-uniformly pseudoconvex domains, of which at least Bℓ2 is 2-uniformly pseudoconvex.

Proposition 3.5. If n N and p (1, 2), Bℓn is 1-uniformly pseudoconvex. ∈ ∈ p Proof. Let n N and p (1, 2). We have seen that the norm p of n ∈ ∈ k·k ℓp is uniformly plurisubharmonic on average, consequently so is also the function r = 1. Moreover, as shown in [3], the norm is of class C1, and hence so k·kp − k·kp n is r. To complete showing that bBℓp has 1 degree of diﬀerentiability, observe that r satisﬁes that

z (2Bℓn ) (1/2B¯ℓn ): r(z) < 0 = Bℓn (1/2B¯ℓn ) { ∈ p \ p } p \ p n and the gradient of r is not null at every element of bBℓp , since for each point z in bBℓn there is i 1, ,n so that zi = 0, and for such i we have, p ∈{ ··· } 6 p p/2 ∂ zi ∂(zizi) p/2−1 p−2 | | = = p/2(zizi) zi = p/2 zi zi =0 ∂zi z ∂zi z | | 6 so ( z 1) = 0. ∇ k kp − 6 Proposition 3.6. If p (1, 2] then B is 1-uniformly pseudoconvex. ∈ ℓp Proof. Let p (1, 2]. As in the previous proposition, we know that the norm ∈ p of ℓp is uniformly plurisubharmonic on average, and hence so is r = p 1. k·k 1 1k·k − Again, as shown in [3], the norm p is of class C , and thus r is C as well. Moreover, r satisﬁes that k·k z (2B ) (1/2B¯ ): r(z) < 0 = B (1/2B¯ ) { ∈ ℓp \ ℓp } ℓp \ ℓp and the derivative of r is not null at points z in bBℓp , because for each z in bBℓp there exists n N such that zn = 0, so if en denotes the n-th element of the canonical basis∈ of ℓ and θ = arg(z6 ) [0, 2π), we have that p n ∈ iθ iθ z + he en z Dr(z)(e en) = lim k k−k k h→0+ h (1 z p + ( z + h)p)1/p 1 = lim − | n| | n| − h→0+ h z p−1h + o( h ) = lim | n| | | , h→0+ h where the last equality has been obtained with Taylor series, leading us to obtain iθ p−1 that Dr(z) Dr(z)(e en) = zn > 0. We conclude that Bℓp is 1-uniformly pseudoconvexk k ≥ . | | | |

Suitable modiﬁcations to the deﬁning functions p 1 by plurisubharmonic functions can lead to examples of domains that are atk·k least− 1-strictly pseudoconvex at the boundary, as in the next examples. n Proposition 3.7. If p (1, 2], then z C : Re(z1)+ z p < 1 is 1-strictly pseudoconvex at the boundary.∈ { ∈ − k k } 16 SOFIA ORTEGA CASTILLO

Proof. The function r(z)= Re(z )+ z 1 is still uniformly plurisubhar- − 1 k kp − n monic on average when restricted to balls MBℓp . Then we can use a continuous n C n n n n partition of unity on , subordinated to the open cover Bℓp , 2Bℓp 1/2Bℓp , 3Bℓp n { \ n\ (1+1/2)Bℓn , , to get a positive function φ C(C ) such that, for all w C p ···} ∈ ∈ and any b Cn of norm less than one, ∈ 1 2π r(w + eiθb)dθ r(w)+ φ(w) b 2. 2π ≥ k k Z0 Also, since the function r(z) is in particular a continuous plurisubharmonic function, z Cn : r(z) < 0 is pseudoconvex. { ∈ } ′ Cn ′ ′ Finally, the gradient of r is not null at each z b z : Re(z1)+ z p < 1 because, if z = 0 for some j > 1 then ∈ { ∈ − k k } j 6 ∂r 1 1−p p−2 (z)= z p zj zj =0, ∂zj 2k k | | 6 or if Im(z ) = 0 then ﬁnding Taylor series one gets that 1 6 ∂r 1−p p−2 (z)= z p z1 Im(z1) = 0; ∂y1 k k | | 6 otherwise Im(z1)= z2 = = zn = 0, and since Re(z1)+ z p = 1, we have that Re(z )= 1/2, where ··· − k k 1 − ∂r 1 1−p p−2 1 1−p 1 p−2 1 ( e1)= 1+ z p z1 Re(z1) (− 1 e ) = 1 + ( ) ( ) ( )= 2. ∂x1 −2 − k k | | 2 1 − 2 2 −2 −

Proposition 3.8. If p (1, 2], then z ℓp : Re(z1)+ z p < 1 is 1-strictly pseudoconvex at the boundary.∈ { ∈ − k k } The proof is analogous to the ﬁnite-dimensional one, where the appropiate continuous partition of unity can be found because ℓp is separable (such result can be proved by just adapting the ﬁndings in [12, §15]). Due to distribution theory, given ℓ 1, a bounded open domain U Cn ≥ ⊂ is ℓ-uniformly pseudoconvex if and only if there exist open domains V1 and V2 ′ such that V2 V1 bU, there exist positive constants L, M and M and a ⊃ ⊃ ℓ+1 sequence of pointwise bounded C functions on V2, rm , which are L-uniformly {′ } plurisubharmonic on average on V2 and such that M Drm(z) M for all ℓ ≥ k k ≥ z V1, and there exists a C function, r : V2 R, satisfying U V2 = z V2 : r(∈z) < 0 and whenever 0 α ℓ we have that→ the following holds∩ uniformly{ ∈ on 2n} ≤ | |≤ V1 R : ⊂ ∂αr ∂αr = lim m . ∂xα m→∞ ∂xα Equivalently, r above can be replaced by a family of C∞ functions. { m} Based on the previous remark, let us call U ⋐ Cn a 0-uniformly pseudoconvex domain when there exist positive constants L, M and M ′, bounded neighborhoods V1 and V2 of bU with V2 V1 and a function r : V2 R such that U V2 = z ⊃ → ∩1 { ∈ V2 : r(z) < 0 and r is the limit of a sequence of pointwise bounded C functions given on V , }r , such that each r is L-uniformly plurisubharmonic on average 2 { m} m on V and M ′ Dr (z) M for all z V . 2 ≥k m k≥ ∈ 1 STRONGPSEUDOCONVEXITY 17

As before, it is not hard to check that 0-uniformly pseudoconvex domains are pseudoconvex, and that 1-uniformly pseudoconvex domains are 0-uniformly pseudoconvex. Let us now show that the ball of a ﬁnite-dimensional ℓ1 space is a 0-uniformly pseudoconvex domain.

n n Theorem 3.9. B n = z C : z < 1 is 0-uniformly pseudoconvex. ℓ1 { ∈ j=1 | j | } n Proof. The function r(z) = P z 1 on 2B n 1/2B n satisﬁes that j=1 | j|− ℓ1 \ ℓ1 B n 1/2B n = z 2B n 1/2B n : r(z) < 0 . Note that r is the pointwise limit ℓ1 \ ℓ1 { ∈ ℓ1 \ ℓ1 P } of 1 for 2 > p 1, where each 1 is ( 1 +4 2)-uniformly k·kpn − n → k·kpn − e − plurisubharmonic on average because for p (1, 2), I2,p(Lp)q 1 and we can argue ∈ ≥ as in the proofs of [2, Thm. 2.4] and 2.6, indeed, for z 2B n 1/2B n and b B n , ∈ ℓ1 \ ℓ1 ∈ ℓ1

1 2π 1 2π 1 z + eiθb dθ ( z + eiθb 2)1/2 2π k kp ≥ 2π k √e kp Z0 Z0 1 2π 1 ( z + eiθb p)1/p ≥ 2π k √e kp Z0 1 ( z 2 + b 2)1/2 ≥ k kp e k kp

1 2 z p + ( +4 2) b p. ≥k k r e − k k It remains to ﬁnd pointwise upper bounds for the family of C1 functions {|k · 1 on 2B n (1/2B n ) and to get common upper and lower bounds for kp − |}p∈(1,2) ℓ1 \ ℓ1 the gradients of the functions 1 on 2B n (1/2B n ) with p (1, 2). k·kp − ℓ1 \ ℓ1 ∈ The functions 1 are clearly bounded by 1 on 2B n (1/2B n ). {|k · kp − |}p∈(1,2) ℓ1 \ ℓ1 To bound the norm of the gradients of the functions in 1 , it is {k·kp − }p∈(1,2) easy to get that, for p (1, 2) and z 2B n (1/2B n ), ∈ ∈ ℓ1 \ ℓ1 n n ( z p 1) z p 1/p−1 p ( z p 1) 2 ∇ k k − · = ( zj ) ( zi )/ z 2, k∇ k k − k ≥ z 2 | | | | k k j=1 i=1 k k X X where

(3.8) z z 2 when z 2B n (1/2B n ) and r> 1, k kr ≤k k1 ≤ ∈ ℓ1 \ ℓ1 hence using equation (3.8) for r = 2 and using H¨older’s inequality,

1 z 1 ( z p 1) 2 z p/2 k k k∇ k k − k ≥k k ≥ 2 −→1 k kq 1 1 . ≥ 4 n1−1/p ≥ 4√n · Meanwhile, if again p (1, 2) and z 2B n (1/2B n ), we have that ∈ ∈ ℓ1 \ ℓ1 (3.9) n 2(p−1) 1/2 p−1 p−1 ( z ) ( z ) 2 when z 2B n (1/2B n ) and p (1, 2), | i | ≤ | i| ≤ ∈ ℓ1 \ ℓ1 ∈ i=1 X X 18 SOFIA ORTEGA CASTILLO and due to Hölder’s inequality as before n z (1/2)p n (3.10) z p ( k k1 )p = , | j| ≥ ≥ (n1−1/p)p (2n)p j=1 −→1 q X k k so using equation (3.8) for r = p along with equations (3.9) and (3.10), n n ( z 1) = ( z p)1/p−1( z 2(p−1))1/2 k∇ k kp − k2 | j| | i| j=1 i=1 X X 2 n ( z 2(p−1))1/2 ≤ ( n z p) | i| j=1 | j | i=1 p X P 2 ≤ ( n z p) j=1 | j | 2p(2n)p/n 16n, ≤ P ≤ so we have found nontrivial upper and lower bounds for the gradients of the functions 1 on 2B n (1/2B n ) with p (1, 2), as desired. k·kp − ℓ1 \ ℓ1 ∈ We continue our discussion of ℓ-uniform pseudoconvexity with a characterization of 0-uniformly pseudoconvex domains as a certain limit of 1-uniformly pseudoconvex domains. Proposition 3.10. In Cn, a domain U is 0-uniformly pseudoconvex if and only if it is exhausted by an increasing sequence U of 1-uniformly pseudoconvex { m} domains given by a respective family of C1 defining functions r : V ′ V bU m ⊃ ⊃ m ∪ bU R such that > lim sup rm(z) lim inf rm(z) > 0 for each z V bU, and→ such that there∞ exist common| positive| ≥ bounds L| , M and| M ′ satisfying∈ that\ each r is L-uniformly plurisubharmonic on average on V and M ′ Dr (z) M m ≥ k m k ≥ for all z V compact and w Cn. ∈ ∈ Proof. Suppose that we can exhaust U with an increasing sequence Um of 1 { } 1-uniformly pseudoconvex domains, where each bUm is given by the C defining function r : V ′ V bU bU R so that the family r satisfies > m ⊃ ⊃ m ∪ → { m} ∞ lim sup r (z) lim inf r (z) > 0 for each z V bU, each r is L-uniformly | m | ≥ | m | ∈ \ m plurisubharmonic on average on V , and for all z in the compact set V as well as any w Cn, it holds that M ′ Dr (z) M. Then we can use Arzelà-Ascoli ∈ ≥ k m k ≥ theorem to find a subsequence r of r as restricted to V that converges { mk } { m} uniformly to a function r : V R, and it will also hold that U V = z V : r(z) < 0 . → ∩ { ∈ } Conversely, suppose that r is the uniform limit on V of a sequence of C1 ′ functions on V V , rm , that are L-uniformly plurisubharmonic on average and ′ ⊃ { } ′ such that M Drm(z) M for all z V , and where r : V V bU R satisfies U V≥′ = k z Vk′ ≥: r(z) < 0 .∈ Let U be a neighborhood⊃ of⊃ bU →with ∩ { ∈ } 0 V V . For each m N take c = min r , and set U = (U V ) z V : 0 ⊂ ∈ m V \U m m \ 0 ∪{ ∈ rm(z) cm < 0 . It is clear that cm 0 and that Um = U. By substracting small positive− numbers} d to each c with→ d 0 and∪ then passing to a subsequence, m m m → we may assume that Um is an increasing sequence of open sets whose union is U, 1 { } where bUm has C defining function rm (cm dm) defined on V0. We conclude that U satisfies the desired conditions. − − STRONGPSEUDOCONVEXITY 19

For ℓ 1, the reader can similarly show a characterization of ℓ-uniformly pseudoconvex≥ domains in Cn as a certain limit of (ℓ + 1)-uniformly pseudoconvex domains. In particular, the following holds. Proposition 3.11. A domain U in Cn is 1-uniformly pseudoconvex if and only if it is exhausted by an increasing sequence Um of strongly pseudoconvex domains 2 { } ′ given by a respective family of C defining functions rm : V V bUm bU R such that > lim sup r (z) lim inf r (z) > 0 for each⊃ z⊃ V ∪bU,→ that ∞ | m | ≥ | m | ∈ \ each family ∂rm is equicontinuous on V for i = 1, , 2n, and that there exist ∂xi { } ′ ′ ···′′ 2 common positive bounds L, M and M such that D D rm(z)(w, w) L w and ′ n ≥ k k M Drm(z) M for all z V compact and w C . Equivalently,≥k rk≥ above can be∈ replaced by a family∈ of C∞ functions. { m} Let us use the previous approximation result to show the following invariance property of 1-uniformly pseudoconvex domains in Cn, which in particular yields that the image of a 1-uniformly pseudoconvex domain in Cn under an invertible linear map remains 1-uniformly pseudoconvex. Of course, this extends our family of examples of 1-uniformly pseudoconvex domains. Proposition 3.12. The biholomorphic image of a 1-uniformly pseudoconvex domain U in Cn is still 1-uniformly pseudoconvex, when the domain of the biholo- morphism contains U. Proof. Suppose that U is a 1-uniformly pseudoconvex domain which is exhausted by the increasing sequence U of strongly pseudoconvex domains given { m} by the respective family of C2 defining functions r : V ′ V bU bU R such m ⊃ ⊃ m ∪ → that > lim sup rm(z) lim inf rm(z) > 0 for each z V bU, for which there exist∞ common positive| bounds|≥ L, M| and|M ′ such that D∈′D′′r\ (z)(w, w) L w 2 m ≥ k k and M ′ Dr (z) M for all z in the compact set V and w Cn, and such ≥ k m k ≥ ∈ that each family ∂rm is equicontinuous, i =1, , 2n. { ∂xi } ··· Then, as in [13, Ch. II, §2.6], if F : W ′ W U Cn is a biholomorphic ⊃ ⊃ → map, then F (U) is exhausted by the increasing sequence of domains F (Um) given by the respective family of defining functions ρ = r F −1 : F (W{′ V ′) } m m ◦ ∩ ⊃ F (W V ) bF (U ) bF (U) R, which satisfy for z W V and w Cn, ∩ ⊃ m ∪ → ∈ ∩ ∈ i) > lim sup ρ F (z) lim inf ρ F (z) > 0 when z / bU, ∞ | m ◦ |≥ | m ◦ | ∈ ′ ′′ ′ ′′ ′ ′ ii)D D rm(z)(w, w)= D D ρm(F (z))(F (z)w, F (z)w), ′ iii)Drm(z)(w)= Dρm(F (z))(F (z)w); where each F ′(z) is a nonsingular C-linear map, so it defines an isomorphism on n C . Hence each domain F (Um) is still strongly pseudoconvex with

′ ′ ′′ ′ ′ ′ L ′ 2 i )D D ρm(z )(w , w ) ′ −1 2 w , ≥ max F (F (z1)) k k z1∈F (W ∩V ) k k ′ ′ −1 ′ ′ M ii )M max (F ) (z1) Dρm(z ) ′ −1 · z1∈F (W ∩V ) k k≥k k≥ max F (F (z1)) z1∈F (W ∩V ) k k for all z′ F (W V ) and w′ Cn. It is also easy to check that each family ∈ ∩ ∈ ∂ρm (i = 1, , 2n) is equicontinuous on F (W V ). Thus F (U) is strongly { ∂xi } ··· ∩ pseudoconvex with boundary F (bU). 20 SOFIA ORTEGA CASTILLO

Let us finish this section stating an invariance property of 0-uniformly pseudoconvex domains U under biholomorphisms whose domain contain U; its proof is an easy consequence of the corresponding result for 1-uniformly pseudoconvex domains. As with 1-uniformly pseudoconvex domains, we have extended our family of examples of 0-uniformly pseudoconvex domains. Proposition 3.13. The biholomorphic image of a 0-uniformly pseudoconvex domain U in Cn is still 0-uniformly pseudoconvex, when the domain of the biholo- morphism contains U. 4. Acknowledgement The author thanks William B. Johnson, Harold P. Boas, Maite Fernández Un- zueta, César Octavio Pérez Regalado and Veronique Fischer for several helpful discussions. I also wish to thank Xavier Gómez-Mont Avalos,´ Eduardo Santillán Zerón and Loredana Lanzani for helpful suggestions to improve the material in this article. References

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