Strong pseudoconvexity in Banach spaces

Abstract

Since it is unclear how to define a domain that is strictly pseudoconvex in the infinite-dimensional setting, we develop a general theory with Banach spaces in mind. We first focus on the finite dimensional case and eliminate the need for two degrees of differentiability of the boundary of a domain, since differentiable functions are difficult to find in infinite dimensions. We introduce $\ell$-strict pseudoconvexity for $\ell \ge 1$, 1-strict pseudoconvexity at the boundary, $\ell$-uniform pseudoconvexity for $\ell \ge 0$, and finally strong pseudoconvexity. Defining $\ell$-strict pseudoconvexity and $\ell$-uniform pseudoconvexity for $\ell <2$ depends on extending a notion of strict plurisubharmonicity to cases lacking $C^2$-smoothness, first studying it in the sense of distribution and then considering it in infinite dimensions. Examples of strictly plurisubharmonic functions as well as strongly pseudoconvex domains related to important classical Banach spaces are presented. Finally, some related solutions to the inhomogeneous Cauchy–Riemann equations for ${\overline{\partial }}$-closed (0, 1)-forms in infinite-dimensional domains are shown.

1 Introduction

Classically, strongly pseudoconvex domains provide concrete examples of domains of holomorphy in several complex variables that have additional structures. Domains of holomorphy are those domains $\Omega$ whose boundary points are each a singularity for a holomorphic function on $\Omega$ [15]. 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. On the other hand, the well-known Hartogs figure is not a domain of holomorphy because it allows analytic continuation to a strictly larger domain [17, Ch. II, §1.1].

In the Euclidean complex space ${\mathbb {C}}^n$, when the boundary of a given domain U has two degrees of smoothness, we have the fact that U is a domain of holomorphy is characterized by a complex-differential property at boundary points that corresponds to a complex analog of a well-known differential condition satisfied by convex domains, which is usually referred to as (Levi) pseudoconvexity. Thus, strong pseudoconvexity is presented using the complex analog of the differential condition that defines strict convexity [17, Ch. II, §2.6]. A simpler equivalent condition that reduces to the strict plurisubharmonicity of a $C^2$ function defined in a neighborhood of the boundary has also become common [17, Ch. II, §2.8], [8, §1.1]. Strictly plurisubharmonic functions are those $C^2$ functions whose complex Hessian is positive definite, where the complex Hessian is a complex analog of the Hessian, which is also called the Levi form.

Still in finite dimension, pseudoconvexity was subsequently extended to domains without $C^2$ boundaries by admitting a plurisubharmonic exhaustion function over the entire domain [17, 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 $-log d_U$, where $d_U$ denotes the distance to the boundary of U. A list of other equivalent ways to define pseudoconvexity in infinite dimension is provided in [15, Ch. VIII, §37]. In particular, pseudoconvexity can be characterized by its behavior on finite-dimensional spaces; that is, U is pseudoconvex if and only if $U\cap M$ is pseudoconvex for each finite-dimensional subspace M of X. In infinite dimensions, every domain of holomorphy is still pseudoconvex, but there is a nonseparable Banach space for which the converse is false, while for general separable Banach spaces this problem remains open [15, Ch. VIII, §37]. In separable Banach spaces with the bounded approximation property, pseudoconvex domains are indeed domains of holomorphy [15, Ch. X, §45].

In this article, we look at a generalization of strong pseudoconvexity to the infinite dimensional setting, for which we first consider related notions in the case that the boundary of a given domain does not have two degrees of smoothness. We were initially interested in determining whether the unit ball of $\ell _1$ has enough structure to be considered strongly pseudoconvex since it was shown in [12] that such ball admits solutions to the (inhomogeneous) Cauchy–Riemann equations with Lipschitz conditions, although the Cauchy–Riemann equations with simply bounded conditions are not solvable, not even locally. Meanwhile, the Cauchy–Riemann equations with bounded conditions on certain finite-dimensional strongly pseudoconvex domains admit solutions that extend continuously to the boundary [8]. Having solutions that extend continuously to the boundary to the bounded Cauchy–Riemann equations is a helpful tool for studying the boundary behavior of bounded holomorphic functions on a ball of a Banach space [14]. The study of the boundary behavior of bounded holomorphic functions on the ball of a separable Banach space, in a sense, reduces to studying the boundary behavior of bounded holomorphic functions on the ball of Banach spaces that are $\ell _1$-sums of finite-dimensional spaces [7]. Thus, we focus on proving the strong pseudoconvexity of $B_{\ell _1}$ and $B_{\ell _1^n}$, yet we will see that further examples can be obtained as the image of affine isomorphisms of $B_{\ell _1}$ and $B_{\ell _1^n}$ and that we can solve the Cauchy–Riemann equations for ${\overline{\partial }}$-closed (0, 1)-forms on such domains. Other examples of strongly pseudoconvex domains exhibited in this article include $B_{\ell _p}$ and $B_{\ell _p^n}$ ($1<p \le 2$, $n\in {\mathbb {N}}$). It remains an open problem to find further examples of strongly pseudoconvex domains and to determine whether we can solve the Cauchy–Riemann equations for ${\overline{\partial }}$-closed (0, 1)-forms on arbitrary strongly pseudoconvex domains in a Banach space, such as $B_{\ell _p}$ and $B_{\ell _p^n}$ ($1<p \le 2$, $n\in {\mathbb {N}}$).

Foundations on plurisubharmonicity and pseudoconvexity in Banach spaces, as well as a basic treatment of distributions, can be found in [15]. The reader interested in an in-depth study of pseudoconvexity in ${\mathbb {C}}^n$ will find it in [18].

2 Strict plurisubharmonicity

Hereafter, let X denote a complex Banach space with open unit ball $B_X$ and norm $\Vert \cdot \Vert$, let U denote an open subset of X with boundary bU, and let $d_U$ denote the distance function to bU. We will denote by m the Lebesgue measure in ${\mathbb {C}}^n$ seen as ${\mathbb {R}}^{2n}$.

Definition 1

A function $f:U\rightarrow [-\infty ,\infty )$ is called plurisubharmonic if f is upper semicontinuous, and for each $a\in U$ and $b\in X$ such that $a+\overline{{\mathbb {D}}}\cdot b\subset U$, we have that

$$\begin{aligned} f(a)\le \frac{1}{2\pi } \int _0^{2\pi } f(a+e^{i\theta }b) d\theta . \end{aligned}$$

Given a differentiable mapping $f:U\rightarrow {\mathbb {R}}$ and $a\in U$, 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

$$\begin{aligned} & D'f(a)(b)=1/2[Df(a)(b)-iDf(a)(ib)], \\ & D''f(a)(b)=1/2[Df(a)(b)+iDf(a)(ib)], \end{aligned}$$

for every $b \in X$.

It is known that a function $f \in C^2(U,{\mathbb {R}})$ is plurisubharmonic if and only if its complex Hessian is positive semi-definite, i.e., for each $a \in U$ and $b\in X$, we have that

$$\begin{aligned} D'D''f(a)(b,b)\ge 0. \end{aligned}$$

(2.1)

Definition 2

A function $f\in C^2(U,{\mathbb {R}})$ is called strictly plurisubharmonic when the complex Hessian of f is positive definite, i.e., when the inequality in (2.1) for $b\ne 0$ is strict [15, §35].

When we are in the Euclidean complex space ${\mathbb {C}}^n$, we aim to understand a suitable extension of strict plurisubharmonicity to distributions, using that in finite dimension, a function $f\in C^2(U)$ is strictly plurisubharmonic if and only if there exists $\psi \in C(U)$ positive such that

$$\begin{aligned} D'D''f(a)(b,b)\ge \psi (a) \Vert b\Vert ^2 \text{ for } \text{ all } a\in U \text{ and } b\in {\mathbb {C}}^n. \end{aligned}$$

Due to this observation, we introduce the following notion.

Definition 3

In the arbitrary Banach space setting, we will say that $f\in C^2(U,{\mathbb {R}})$ is strictly plurisubharmonic continuously when there exists $\psi \in C(U)$ positive such that $D'D''f(a)(b,b)\ge \psi (a) \Vert b\Vert ^2 \text{ for } \text{ all } a\in U \text{ and } b\in X.$

If U is an open subset of ${\mathbb {C}}^n$, we will denote the real-valued test functions on U by ${\mathcal {D}}(U)$. A distribution on U is known to be a continuous functional on ${\mathcal {D}}(U)$. We shall denote by ${\mathcal {D}}'(U)$ the vector space of all distributions on U.

Definition 4

If $U\subset {\mathbb {C}}^n$, given $f \in L^1(U,\textit{loc})$, we say that f is (strictly) plurisubharmonic in distribution if the distribution it induces is (strictly) plurisubharmonic. At the same time, a distribution $T \in {\mathcal {D}}'(U)$ is called plurisubharmonic if

$$\begin{aligned} & D'D''T(\phi )(w,w):=\sum _{j,k=1}^{n} \frac{\partial ^2 T}{\partial z_j \partial \overline{z_k}}(\phi )w_j \overline{w_k}\ge 0, \\ & \quad \text{ for } \text{ all } \phi \ge 0 \text{ in } {\mathcal {D}}(U) \text{ and } w \in {\mathbb {C}}^n. \end{aligned}$$

And we will say that $T \in {\mathcal {D}}'(U)$ is strictly plurisubharmonic if there exists $\psi \in C(U)$ positive such that

$$\begin{aligned} & D'D''T(\phi )(w,w)\ge \Big (\int _U \psi \cdot \phi \; dm\Big ) \Vert w\Vert ^2, \nonumber \\ & \quad \text{ for } \text{ all } \phi \ge 0 \text{ in } {\mathcal {D}}(U) \text{ and } w \in {\mathbb {C}}^n. \end{aligned}$$

(2.2)

It has been proven, 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 ${\mathbb {C}}^n$. If $f \not \equiv -\infty$ is plurisubharmonic on U, then $f \in L^1(U,\textit{loc})$ and f is plurisubharmonic in distribution. Conversely, if $T \in {\mathcal {D}}'(U)$ is plurisubharmonic then there exists a $f \in L^1(U,\textit{loc})$ plurisubharmonic such that f induces the distribution T. As a corollary, if $f \in L^1(U,\textit{loc})$ is plurisubharmonic in distribution then there exists a $g \in L^1(U,\textit{loc})$ plurisubharmonic such that $f=g$ m-a.e.

To prove an analogous version of such a result for strict plurisubharmonicity, let us introduce the following concept. Let us clarify that below, without the word “continuously” in parentheses, we are only introducing the notion in finite dimension.

Definition 5

We will say that an upper semicontinuous function $g: U\subset X \rightarrow [-\infty ,\infty )$ is strictly plurisubharmonic on average (continuously) if there exists $\varphi \in C(U)$ positive such that for all $a \in U$ and $b \in X$ of small norm with size depending lower semicontinuously on a, we have

$$\begin{aligned} \varphi (a)\Vert b\Vert ^2 +g(a) \le \frac{1}{2\pi }\int _0^{2\pi } g(a+e^{i\theta }b)d\theta . \end{aligned}$$

(2.3)

If g is as above, let us specifically call it strictly plurisubharmonic on average (continuously) with respect to the function $\varphi$.

Proposition 2.1

Suppose that U is a bounded connected domain in ${\mathbb {C}}^n$. If $f:U\rightarrow [-\infty ,\infty )$, with $f\not \equiv -\infty$, is strictly plurisubharmonic on average, then $f \in L^1(U,\textit{loc})$ and f is strictly plurisubharmonic in distribution. Conversely, if $T \in {\mathcal {D}}'(U)$ is strictly plurisubharmonic, then there exists $f \in L^1(U,\textit{loc})$ strictly plurisubharmonic on average such that f induces the distribution T. As a consequence, if $f \in L^1(U,\textit{loc})$ is strictly plurisubharmonic in distribution, then there exists $g \in L^1(U,\textit{loc})$ strictly plurisubharmonic on average such that $f=g$ m-a.e.

Proof

If $f\not \equiv -\infty$ is strictly plurisubharmonic on average, then f is in particular plurisubharmonic, so we can use the relationship to plurisubharmonicity in distribution to deduce that $f \in L^1(U,\textit{loc})$. Moreover, since f is strictly plurisubharmonic on average in $U\subset {\mathbb {C}}^n$, there exists a positive function $\psi \in C(U)$ such that

$$\begin{aligned} \psi (a)\Vert b\Vert ^2+f(a) \le \frac{1}{2\pi }\int _0^{2\pi } f(a+e^{i\theta }b)d\theta , \end{aligned}$$

for all $a \in U \text{ and } b \in {\mathbb {C}}^n$ of norm less than $\delta _{f}(a)$ ($\delta _{f}>0$ lower semicontinuous).

Consider the test function $\rho : {\mathbb {C}}^n \rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} \rho (x)={\left\{ \begin{array}{ll} k \cdot e^{-1/(1-\Vert x\Vert ^2)}, & \text{ if } \Vert x\Vert <1 \\ 0, & \text{ if } \Vert x\Vert \ge 1 \end{array}\right. } \end{aligned}$$

where the constant k is chosen so that $\int _{{\mathbb {C}}^n} \rho dm=1$. More generally, for each $\delta >0$, let $\rho _{\delta } \in {\mathcal {D}}({\mathbb {C}}^n)$ be defined by $\rho _{\delta }(x)=\delta ^{-2n} \rho (x/\delta )$ for every $x \in {\mathbb {C}}^n$, so that $\int _{{\mathbb {C}}^n} \rho _{\delta } dm=1$ and $\text {supp}(\rho _{\delta })={\bar{B}}(0,\delta )$.

Fix $\delta _0>0$, and let $U_{\delta _0}:=\{z\in U: d_U(z)> \delta _0\}$. Since $\psi$ is positive and uniformly continuous on $\overline{U_{\delta _0/2}}$, there is $\delta _1\in (0,\delta _0/2)$ such that $|\psi (a)-\psi (a-\zeta )|<\inf _{\overline{U_{\delta _0}}}\psi /2$ when $\zeta \in {\bar{B}}(0,\delta _1)$ and $a\in U_{\delta _0}$. Given $a\in U_{\delta _0}$, choose $b\in {\mathbb {C}}^n$ of small norm, namely, less than $\inf _{{\overline{B}}(a,\delta _2)} \delta _f>0$, where $\delta _2\in (0,\delta _1)$ is small enough, so for $\delta \in (0,\delta _2)$,

$$\begin{aligned}&\psi (a)/2\Vert b\Vert ^2+f*\rho _{\delta }(a)\\&\quad =\int _{{\bar{B}}(0,\delta )} (\psi (a)/2\Vert b\Vert ^2 +f(a-\zeta ))\rho _{\delta }(\zeta )dm(\zeta )\\&\quad \le \int _{{\bar{B}}(0,\delta )} (\psi (a-\zeta )\Vert b\Vert ^2+f(a-\zeta ))\rho _{\delta }(\zeta )dm(\zeta )\\&\quad \le \int _{{\bar{B}}(0,\delta )} \Big (\frac{1}{2\pi }\int _0^{2\pi } f(a-\zeta +e^{i\theta }b)d\theta \Big )\rho _{\delta }(\zeta )dm(\zeta )\\&\quad =\frac{1}{2\pi }\int _0^{2\pi } \Big (\int _{{\bar{B}}(0,\delta )} f(a-\zeta +e^{i\theta }b)\rho _{\delta }(\zeta )dm(\zeta )\Big )d\theta \end{aligned}$$

where the last equality follows from Fubini’s theorem because $f \in L^1(U, \textit{loc})$. Therefore,

$$\begin{aligned} \psi (a)/2\Vert b\Vert ^2+f*\rho _{\delta }(a)\le \frac{1}{2\pi } \int _0^{2\pi } f*\rho _{\delta }(a+e^{i\theta }b)d\theta . \end{aligned}$$

That is, $f*\rho _{\delta } \in C^{\infty }(U_{\delta })$ is strictly plurisubharmonic on average on $U_{\delta _0}$ for $\delta \in (0,\delta _2)$, and from the proof of Proposition 2.2, we obtain

$$\begin{aligned} & \sum _{j,k=1}^n \frac{\partial ^2 (f*\rho _{\delta })(a)}{\partial z_j \partial \overline{z_k}}b_j \overline{b_k}\ge \psi (a)/2 \Vert b\Vert ^2, \\ & \quad \text { for all }a\in U_{\delta _0}, b\in {\mathbb {C}}^n \text { and }\delta \in (0,\delta _2). \end{aligned}$$

Consequently, given $w\in {\mathbb {C}}^n$ and $\phi \in {\mathcal {D}}(U)$ positive, say with $\text {supp}(\phi )\subset U_{\delta _0}$, and taking $\delta _m\rightarrow 0$ with $\delta _m<\delta _2$ for $m\ge 3$, we have that $f*\rho _{\delta _m}$ converges uniformly on $\overline{U_{\delta _0}}$ to f, so by the dominated convergence theorem and then integration by parts twice,

$$\begin{aligned}&\int _{U_{\delta _0}} f(z) \Big (\sum _{j,k=1}^n \frac{\partial ^2 \phi (z)}{\partial z_j \partial \overline{z_k}}w_j \overline{w_k}\Big )dm(z)\\&\quad =\lim _{\delta _m\rightarrow 0} \int _{U_{\delta _0}} f*\rho _{\delta _m}(z)\cdot \Big (\sum _{j,k=1}^n \frac{\partial ^2 \phi (z)}{\partial z_j \partial \overline{z_k}}w_j \overline{w_k}\Big ) dm(z)\\&\quad =\lim _{\delta _m\rightarrow 0} \int _{U_{\delta _0}} \Big (\sum _{j,k=1}^n \frac{\partial ^2 (f*\rho _{\delta _m})(z)}{\partial z_j \partial \overline{z_k}}w_j \overline{w_k}\Big )\phi (z) dm(z)\\&\quad \ge \int _{U_{\delta _0}} \psi (z)/2 \Vert w\Vert ^2 \phi (z) dm(z), \end{aligned}$$

that is, f is strictly plurisubharmonic in distribution.

Now, suppose that $T \in {\mathcal {D}}'(U)$ is a strictly plurisubharmonic distribution, where $\psi$ is a positive continuous function satisfying equation (2.2). Then, $T*\rho _{\delta } \in C^{\infty }(U_{\delta })$ and for all $z \in U_{\delta }$ and $b \in {\mathbb {C}}^n$,

$$\begin{aligned} \sum _{j,k=1}^n \frac{\partial ^2 (T*\rho _{\delta })(z)}{\partial z_j \partial \overline{z_k}} b_j \overline{b_k}&=\sum _{j,k=1}^n \frac{\partial ^2 T}{\partial z_j \partial \overline{z_k}} *\rho _{\delta }(z) b_j \overline{b_k}\\&=\sum _{j,k=1}^n \frac{\partial ^2 T}{\partial z_j \partial \overline{z_k}}[ \rho _{\delta }(z-\cdot )] b_j \overline{b_k}\\&\ge \int _U \psi (w)\Vert b\Vert ^2 \rho _{\delta }(z-w) dm(w)\\&=\psi *\rho _{\delta }(z)\Vert b\Vert ^2. \end{aligned}$$

According to Proposition 2.2, $\psi *\rho _{\delta }(z)\Vert b\Vert ^2/2+T*\rho _{\delta }(z)\le \frac{1}{2\pi } \int _0^{2\pi } T*\rho _{\delta }(z+e^{i\theta }b)d\theta$ when b has small norm depending lower semicontinuously on z (namely, when $\Vert b\Vert < \sup \{r>0: B(z,r)\subset (\psi *\rho _{\delta })^{-1}(B(\psi *\rho _{\delta }(z),\psi *\rho _{\delta }(z)/2))\}$).

Since T is in particular a plurisubharmonic distribution, we get that $T*\rho _{\delta }$ decreases to a $f \in L^1(U,\textit{loc})$ plurisubharmonic that induces T. Consequently, due to the dominated and monotone convergence theorems applied to the positive and negative parts of each $T*\rho _{\delta }$, respectively, for all $z\in U$ and $b\in {\mathbb {C}}^n$ of small norm (namely, when $\Vert b\Vert <\sup \{r>0: B(z,r)\subset \psi ^{-1}(B(\psi (z),\psi (z)/2))\}$),

$$\begin{aligned}&\frac{1}{2\pi }\int _0^{2\pi } f(z+e^{i\theta }b)d\theta \\&\quad =\lim _{\delta _m\rightarrow 0} \frac{1}{2\pi }\int _0^{2\pi } T*\rho _{\delta _m}(z+e^{i\theta }b)d\theta \\&\quad \ge \lim _{\delta _m \rightarrow 0} (T*\rho _{\delta _m}(z)+\psi *\rho _{\delta _m}(z) \Vert b\Vert ^2/2)\\&\quad =f(z)+\psi (z)\Vert b\Vert ^2/2. \end{aligned}$$

The proposition is now clear. $\square$

Due to Proposition 2.1, in ${\mathbb {C}}^n$, a $C^2$ function is strictly plurisubharmonic if and only if it is strictly plurisubharmonic on average, and we will see in Proposition 2.2 that in infinite dimension, a $C^2$ function is strictly plurisubharmonic continuously if and only if it is strictly plurisubharmonic on average continuously. Meanwhile, in the finite dimension, a $C^2$ function is known to satisfy strict plurisubharmonicity if and only if in a neighborhood of each point in the domain there exists an $\epsilon >0$ such that $F-\epsilon \Vert \cdot \Vert ^2$ is plurisubharmonic. However, this is not true in infinite dimensions. We constructed the following example with the help of Santillán Zerón:

Example 1

Let X be the Hilbert space of complex sequences $(z_n)\subset {\mathbb {C}}^{{\mathbb {N}}}$ such that

$$\begin{aligned} \Vert (z_n)\Vert :=\sqrt{\sum _{k=1}^{\infty }|z_k|^2/k^2}<\infty . \end{aligned}$$

Then, the $C^2$ function on $X\setminus \{0\}$, $F(z)=\sum _{k=1}^{\infty } |z_k|^2/k^3$, has a positive definite complex Hessian at $z\ne 0$,

$$\begin{aligned} D'D''F(z)(w,w)=\sum _{k=1}^{\infty }|w_k|^2/k^3, \text{ for } \text{ all } w\in X. \end{aligned}$$

However, F does not admit an $\epsilon >0$ such that for z near $z_0\ne 0$, we have plurisubharmonicity of the function

$$\begin{aligned} & F(z)-\epsilon \Vert z\Vert ^2=\sum _{k=1}^{\infty }|z_k|^2/k^3-\epsilon \sum _{k=1}^{\infty }|z_k|^2/k^2\\ & \quad =\sum _{k=1}^{\infty }|z_k|^2(1/k^3-\epsilon /k^2), \end{aligned}$$

since its complex Hessian at each $z_0\ne 0$ is given by

$$\begin{aligned} & D'D''(F-\epsilon \Vert \cdot \Vert ^2)(z_0)(w,w)\\ & \quad =\sum _{k=1}^{\infty }|w_k|^2(1/k^3-\epsilon /k^2) \text{ for } w\in X, \end{aligned}$$

that eventually has eigenvalues that are negative.

Conversely, if X is infinite-dimensional, a function $F:U\subset X\rightarrow [-\infty ,\infty )$ that locally admits an $\epsilon >0$ such that $F-\epsilon \Vert \cdot \Vert ^2$ is plurisubharmonic may not be strictly plurisubharmonic on average continuously, as exhibited by the next example.

Example 2

The squared norm in $\ell _{\infty }$, $\Vert \cdot \Vert ^2_{\infty }$, is not strictly plurisubharmonic on average continuously since for $a=(1,0,0,\cdots )$ and $b=(0,1,0,0,\cdots )$,

$$\begin{aligned} \frac{1}{2\pi }\int _{0}^{2\pi } (\Vert a+e^{i\theta }b\Vert _{\infty }^2-\Vert a\Vert _{\infty }^2)d\theta =0. \end{aligned}$$

However, for $\epsilon \in (0,1)$, $(1-\epsilon )\Vert \cdot \Vert _{\infty }^2$ is known to be plurisubharmonic.

The proof of the next result has been simplified using arguments in the old work of Takeuchi [19], yet we believe that our remaining ideas still deserve attention.

Proposition 2.2

Let U be an open domain in a Banach space X. A function $f \in C^2(U; {\mathbb {R}})$ is strictly plurisubharmonic continuously if and only if it is strictly plurisubharmonic on average continuously.

Proof

Suppose that there exists a positive function $\varphi \in C(U)$ satisfying

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi } (f(a+e^{i\theta }b)-f(a))d\theta \ge \varphi (a)\Vert b\Vert ^2. \end{aligned}$$

(2.4)

for $a\in U$ and $b\in X$ of small norm (with size lower semicontinously depending on a). Given $a \in U$, fix $b \in X$ of small enough norm so that (2.4) holds and that we have $a+\overline{{\mathbb {D}}} b \subset U$, and consider the function $u(z)=f(a+z \cdot b)$, which is defined on ${\mathbb {D}}$.

Then, for all $r\in (0,1)$,

$$\begin{aligned} \varphi (a)\Vert b\Vert ^2\cdot r^2 \le \frac{1}{2\pi }\int _0^{2\pi } (u(r\cdot e^{i\theta })-u(0))d\theta \end{aligned}$$

Consequently, by Lemma 5 in [19], and exercises 35.B and 35.D in [15],

$$\begin{aligned} \varphi (a)\Vert b\Vert ^2 \le \frac{\partial ^2 u}{\partial z \partial {\bar{z}}}(0)=D'D''f(a)(b,b). \end{aligned}$$

(2.5)

Now, suppose that there exists a positive function $\varphi \in C(U)$ such that (2.5) holds for all $a \in U$ and $b\in X$. Fix $a\in U$. Since $\varphi$ is continuous at a, there exists an upper bound $\delta (a)>0$ for the norm of b to make $|\varphi (a)-\varphi (a+b)|< \varphi (a)/2$ hold ($\delta$ is lower semicontinuous for $\delta (a)=\sup \{r>0: B(a,r)\subset \varphi ^{-1}(B(\varphi (a),\varphi (a)/2))\}$). Fix b bounded as before, and define $M(r)=\frac{1}{2\pi }\int _0^{2\pi }[f(a+re^{i\theta }b)-f(a)]d\theta$, for all $r \in (0,1]$. Consider also the function $u(\zeta )=f(a+\zeta b)$ defined on a disk $D(0,R)\supset \overline{{\mathbb {D}}}$. Then, for all $\zeta \in D(0,R)$,

$$\begin{aligned} \frac{\partial ^2 u}{\partial x^2}(\zeta )+\frac{\partial ^2 u}{\partial y^2}(\zeta )= & 4\frac{\partial ^2 u}{\partial \zeta \partial {\bar{\zeta }}}(\zeta )=4 \cdot D'D''f(a{+}\zeta b)(b,b)\\\ge & 4 \cdot \varphi (a+\zeta b) \Vert b\Vert ^2. \end{aligned}$$

Since $\frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}=\frac{\partial ^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{\partial ^2 u}{\partial \theta ^2}$, then for $r\in (0,1)$,

$$\begin{aligned} & \frac{1}{2\pi }\int _0^{2\pi }\Big (\frac{\partial ^2}{\partial r^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial ^2}{\partial \theta ^2}\Big )u(r e^{i\theta })d\theta \\ & \quad \ge 4\cdot \frac{1}{2\pi }\int _0^{2\pi } \varphi (a+re^{i\theta } b)\Vert b\Vert ^2d\theta \ge 2\cdot \varphi (a)\Vert b\Vert ^2, \end{aligned}$$

i.e., $M''(r)+\frac{1}{r}M'(r)\ge 2\cdot \varphi (a)\Vert b\Vert ^2, \; \forall r \in (0,1)$.

Thus, $\Big (rM'(r)-2\cdot \varphi (a)\Vert b\Vert ^2\frac{r^2}{2}\Big )'=rM''(r)+M'(r)-2\cdot \varphi (a)\Vert b\Vert ^2 r\ge 0$ for all $r \in (0,1)$, so $r\Big (M'(r)-\varphi (a)\Vert b\Vert ^2 r\Big )$ is an increasing function of r. Since clearly $r\Big (M'(r)-\varphi (a)\Vert b\Vert ^2 r\Big )\rightarrow 0$ as $r\rightarrow 0$ (because $M'$ is a bounded function on $(0,\epsilon )$ for some $\epsilon >0$), we conclude that $r\Big (M'(r)-\varphi (a)\Vert b\Vert ^2 r\Big )\ge 0$ for every $r \in (0,1)$. Hence, $\Big (M(r)-\varphi (a)\Vert b\Vert ^2 \frac{r^2}{2}\Big )' \ge 0$ for every $r>0$, so $M(r)-\varphi (a)\Vert b\Vert ^2 \frac{r^2}{2}$ is an increasing function of r. Since clearly $M(r)-\varphi (a)\Vert b\Vert ^2 \frac{r^2}{2}\rightarrow 0$ as $r\rightarrow 0$ then $M(r)\ge \varphi (a)\Vert b\Vert ^2 \frac{r^2}{2}$ for each $r \in (0,1)$.

Since M is continuous on (0, 1], we conclude that $M(1)\ge \frac{\varphi (a)}{2}\Vert b\Vert ^2$, so indeed

$$\begin{aligned} \frac{1}{2\pi } \int _0^{2\pi } [f(a+e^{i\theta }b)-f(a)]d\theta \ge \frac{\varphi (a)}{2}\Vert b\Vert ^2. \end{aligned}$$

$\square$

An important remark about the proof of Proposition 2.2 is that when we have $f\in C^2(U;{\mathbb {R}})$ satisfying $D'D''f(a)(b,b)\ge L\Vert b\Vert ^2$ for some $L>0$, we can conclude that $\frac{1}{2\pi }\int _0^{2\pi }(f(a+e^{i\theta }b)-f(a))d\theta \ge L\Vert b\Vert ^2$ for all $a\in U$ and $b\in X$ with $\Vert b\Vert <d_U(a)$.

We will finish this section exhibiting functions that are strictly plurisubharmonic on average continuously, without two degrees of differentiability. For that, let us discuss a family of Banach spaces X having a norm $\Vert \cdot \Vert$ that is strictly plurisubharmonic on average continuously, some of which 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].

Definition 6

If $0<q<\infty$ and $2\le r <\infty$, a continuously quasi-normed space $(X, \Vert \cdot \Vert )$ is called r-uniformly PL-convex if there exists $\alpha >0$ such that

$$\begin{aligned} \Big (\frac{1}{2\pi }\int _{0}^{2\pi } \Vert a+e^{i\theta }b\Vert ^q d\theta \Big )^{1/q}\ge (\Vert a\Vert ^r+\alpha \Vert b\Vert ^r)^{1/r} \end{aligned}$$

for all a and b in X; we shall denote the largest possible value of $\alpha$ by $I_{r,q}(X)$.

It is known that the previous definition does not depend on q. Let us recall that a quasi-normed space $(X, \Vert \cdot \Vert )$ is continuously quasi-normed if $\Vert \cdot \Vert$ is uniformly continuous on the bounded sets of X. Banach spaces are obviously continuously quasi-normed.

Davis, Garling, and Tomczak-Jaegermann proved that for $p \in [1,2]$, $L_p(\Sigma , \Omega , \mu )$ is 2-uniformly PL-convex ( [2, Cor. 4.2]) with $I_{2,p}(L_p)=I_{2,p}({\mathbb {C}})$ and $I_{2,2}({\mathbb {C}})\ge 1>1/2=I_{2,1}({\mathbb {C}})$. 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 a 2-concavity constant of 1 ( [10, Cor. 4.2]) and the noncommutative $L^p(M)$, $1\le p \le 2$, where M is a von Neumann algebra acting on a separable Hilbert space ( [4, Thm. 4]. It is moreover proven in [13] that $I_{2,p}({\mathbb {C}})=p/2$ for $0<p<2$, which we will use later.

The following proposition gives us in particular that an $L_p(\Sigma , \Omega , \mu )$ space, for $p \in [1,2]$, has a strictly plurisubharmonic norm.

Proposition 2.3

If $(X, \Vert \cdot \Vert )$ is a 2-uniformly PL-convex Banach space, then the norm $\Vert \cdot \Vert$ is strictly plurisubharmonic on average continuously.

Proof

Let $a \in X$ and $b\in B_X$. Then,

$$\begin{aligned} (\Vert a\Vert ^2+I_{2,1}(X) \Vert b\Vert ^2)^{1/2}\le \frac{1}{2\pi }\int _{0}^{2\pi } \Vert a+e^{i\theta }b\Vert d\theta \end{aligned}$$

Let $\varphi (a)=\sqrt{I_{2,1}(X)+\Vert a\Vert ^2}-\Vert a\Vert >0$. Since $\Vert b\Vert <1$,

$$\begin{aligned} \Vert a\Vert +\varphi (a) \Vert b\Vert ^2 \le (\Vert a\Vert ^2+I_{2,1}(X) \Vert b\Vert ^2)^{1/2}. \end{aligned}$$

Thus,

$$\begin{aligned} \varphi (a) \Vert b\Vert ^2+\Vert a\Vert \le \frac{1}{2\pi }\int _{0}^{2\pi } \Vert a+e^{i\theta }b\Vert d\theta . \end{aligned}$$

as desired. $\square$

Clearly, the norm of a 2-uniformly PL-convex Banach space is not only strictly plurisubharmonic on average continuously, but also uniformly plurisubharmonic on average on bounded balls.

Definition 7

An upper semicontinuous function $g:U\subset X \rightarrow [-\infty ,\infty )$ will be called uniformly plurisubharmonic on average if there exists a constant $L>0$ such that for all $a\in U$ and $b\in X$ of small norm (with size depending lower semicontinuously on a),

$$\begin{aligned} L\Vert b\Vert ^2+g(a)\le \frac{1}{2\pi }\int _{0}^{2\pi } g(a+e^{i\theta }b)d\theta . \end{aligned}$$

If $g:U\subset X \rightarrow [-\infty ,\infty )$ 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.

A last remark about functions strictly plurisubharmonic on average continuously is that if we add to one of those a plurisubharmonic function, we of course preserve a function strictly plurisubharmonic on average continuously.

3 Strict pseudoconvexity

We return to the finite dimension to discuss strict pseudoconvexity even in the case when the boundary of a given domain lacks two degrees of smoothness.

Definition 8

A domain $U\subset {\mathbb {C}}^n$ with a $C^2$ boundary is called strictly pseudoconvex when the strict Levi condition is satisfied by a $C^2$ defining function r of the boundary, i.e., when r is a $C^2$ real-valued function defined on a neighborhood V of bU such that $U\cap V=\{z\in V: r(z)<0\}$ and $Dr(w)\ne 0$ for $w\in bU$, it follows that

$$\begin{aligned} & D'D''r(w)(b,b)>0 \text{ when } w\in bU \text{ and } b\text { nonzero in } \nonumber \\ & \quad {\mathbb {C}}^n \text{ satisfy } D'r(w)(b)=0. \end{aligned}$$

(3.1)

According to [17, Ch. II, §2.8], a bounded domain U in ${\mathbb {C}}^n$ with a $C^2$ boundary is strictly pseudoconvex if and only if there is a $C^2$ defining function of bU, $r:V \rightarrow {\mathbb {R}}$, admitting a positive constant L such that $D'D''r(z)(b,b)\ge L \Vert b\Vert ^2$ for all $z \in V$ and $b\in {\mathbb {C}}^n$, where V can be taken as a neighborhood of ${\overline{U}}$.

It is well known that U is convex if and only if $-\log d_U$ is convex, while U is pseudoconvex if and only if $-\log d_U$ is plurisubharmonic on U. Note that $d_U$ is the absolute value of the signed distance function, which is a defining function of bU when U has $C^2$ boundary. Thus, the following is a similar result for bounded strictly pseudoconvex domains in ${\mathbb {C}}^n$ with a $C^2$ boundary, which we prove with the help of Ramos Peón.

Proposition 3.1

Let U be a bounded open domain in ${\mathbb {C}}^n$ with a $C^2$ boundary. Then, U is strictly pseudoconvex if and only if there exist a positive constant L, a neighborhood V of bU, and $\rho \in C^2(V)$ a defining function of bU such that

$$\begin{aligned} & D'D''(-\log |\rho |)(a)(b,b)\ge \frac{L}{|\rho (a)|} \Vert b\Vert ^2 \nonumber \\ & \quad \text { for all } a\in U\cap V \text { and } b\in {\mathbb {C}}^n. \end{aligned}$$

(3.2)

Equivalently, V above can be replaced by a neighborhood of ${\overline{U}}$.

Proof

Suppose that there exist a positive constant L, a neighborhood V of bU and $\rho \in C^2(V)$ a defining function of bU such that $D'D''(-\log |\rho |)(a)(b,b)\ge \frac{L}{|\rho (a)|} \Vert b\Vert ^2 \text { for every } a\in U\cap V \text { and } b\in {\mathbb {C}}^n$. Since for $a\in U\cap V$ and $b \in {\mathbb {C}}^n$ arbitrary, we have

$$\begin{aligned} D'D'' (-\log |\rho |)(a)(b,b)= & \frac{1}{|\rho (a)|}D'D'' \rho (a)(b,b)\\ & +\frac{1}{\rho (a)^2}\cdot |D'\rho (a)(b)|^2 \end{aligned}$$

we obtain that

$$\begin{aligned} & D'D''\rho (a)(b,b)\ge L\Vert b\Vert ^2 \text { when } a\in U\cap V \text { and }\\ & \quad b\in {\mathbb {C}}^n \text{ satisfy } D'\rho (a)(b)=0. \end{aligned}$$

A passage to the limit shows that on the boundary, we have what we desire:

$$\begin{aligned} & D'D'' \rho (w)(b,b)>0 \text { when } w\in bU \text { and } b\ne 0\\ & \quad \text { satisfy } D'\rho (w)(b)=0. \end{aligned}$$

Now, suppose that U is strictly pseudoconvex. Then, we can find a positive constant L, a neighborhood V of bU, and $\rho \in C^2(V)$ a defining function of the boundary of U such that

$$\begin{aligned} D'D''\rho (a)(b,b)\ge L \Vert b\Vert ^2 \text { for all } a\in V \text { and } b\in {\mathbb {C}}^n. \end{aligned}$$

Then, for $a \in U\cap V$ and $b \in {\mathbb {C}}^n$ arbitrary,

$$\begin{aligned}&D'D''(-\log |\rho |)(a)(b,b)\\&\quad =\frac{1}{|\rho (a)|}D'D''\rho (a)(b,b)+\frac{1}{|\rho (a)|^2}|D'\rho (a)(b)|^2\\&\quad \ge \frac{L}{|\rho (a)|}\Vert b\Vert ^2, \end{aligned}$$

as desired. $\square$

Because of the previous proposition and the developments in Sect. 2, we consider the following concept:

Definition 9

Given $\ell \ge 1$, we will say that a bounded domain U with a $C^{\ell }$ boundary, in a Banach space X, is $\ell$-strictly pseudoconvex if there exist a positive constant L, a neighborhood V of bU, and $\rho \in C^{\ell }(V)$ a defining function of bU such that for all $a\in U\cap V$ and $b\in X$ of small norm (with size lower semicontinuously depending on a),

$$\begin{aligned} & \frac{1}{2\pi }\int _0^{2\pi }-\log |\rho |(a+e^{i\theta }b) d\theta \nonumber \\ & \quad \ge -\log |\rho |(a)+\frac{L}{|\rho (a)|}\Vert b\Vert ^2. \end{aligned}$$

(3.3)

Clearly, a bounded domain in ${\mathbb {C}}^n$ with a $C^2$ 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 easy to check that an $\ell$-strictly pseudoconvex domain U in a Banach space X admits a plurisubharmonic function $\sigma$ defined on all U such that $\sigma (z)\rightarrow \infty$ as $z\rightarrow bU$ (see [17, Ch.II, §2.7] and [15, Ch. VIII, §34]). Thus, it is clear that an $\ell$-strictly pseudoconvex domain in ${\mathbb {C}}^n$ admits an associated function $\sigma$ as a plurisubharmonic exhaustion function; hence, $\ell$-strictly pseudoconvex domains in ${\mathbb {C}}^n$ are pseudoconvex. Likewise, an $\ell$-strictly pseudoconvex domain in a Banach space X is pseudoconvex because its restriction to each finite-dimensional subspace admits a restricted associated function $\sigma$ as a plurisubharmonic exhaustion function.

Let us now look at the following necessary condition for $\ell$-strict pseudoconvexity in ${\mathbb {C}}^n$.

Proposition 3.2

If a domain $U\subset {\mathbb {C}}^n$ is $\ell$-strictly pseudoconvex for some $\ell \ge 1$, then there exist a positive constant L, a neighborhood V of bU and a defining function $\rho \in C^{\ell }(V)$ of bU such that for all $a\in U\cap V$ and $b\in {\mathbb {C}}^n$ with $D'\rho (a)(b)=0$ and $\Vert b\Vert <d_{U\cap V}(a)$, we have

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi } \rho (a+e^{i\theta }b)d\theta \ge \rho (a)+L\Vert b\Vert ^2. \end{aligned}$$

Proof

Since U is $\ell$-strictly pseudoconvex, there exist a positive constant $L_0$, a neighborhood V of bU, and a defining function $\rho \in C^{\ell }(V)$ of bU such that for all $a\in U\cap V$ and $b\in {\mathbb {C}}^n$ of small norm (with size lower semicontinuously depending on a),

$$\begin{aligned} & \frac{1}{2\pi }\int _0^{2\pi }-\log |\rho |(a+e^{i\theta }b) d\theta \\ & \quad \ge -\log |\rho |(a)+\frac{L_0}{|\rho (a)|}\Vert b\Vert ^2. \end{aligned}$$

For each $n\in {\mathbb {N}}$, define the function $\rho _{1/n}$ as in Proposition 2.1, and take $\sigma _n=(-\log (-\rho ))*\rho _{1/n}$, which is defined and smooth on $(U\cap V)_{1/n}$. Then, as in the proof of Proposition 2.1, there is $n_1\ge n$ such that for all $m\ge n_1$, $\sigma _m$ is strictly plurisubharmonic on average on $(U\cap V)_{1/n}$ with respect to the function $\frac{L_0}{2|\rho |}$, and because of the proof of Proposition 2.2, we have that if $L=L_0/2$, for all $a\in (U\cap V)_{1/n}$, $b\in {\mathbb {C}}^n$ and $m\ge n_1$,

$$\begin{aligned} D'D''\sigma _m(a)(b,b)\ge \frac{L}{|\rho |(a)}\Vert b\Vert ^2. \end{aligned}$$

Because of distribution theory, it is clear that $\sigma _m$ converges uniformly on each $\overline{(U\cap V)_{\frac{2}{n}}}$ to $-\log (-\rho )$, and likewise, the vector-valued function $D'\sigma _m$ converges uniformly on each $\overline{(U\cap V)_{\frac{2}{n}}}$ to $D'(-\log (-\rho ))=\frac{-1}{\rho }D'\rho$. Let $r_m=-e^{-\sigma _m}$ on $(U\cap V)_{1/m}$, which again converges uniformly on each $\overline{(U\cap V)_{\frac{2}{n}}}$ to $\rho$.

Pick $n\in {\mathbb {N}}$. Now choose $\epsilon _n>0$ such that $\epsilon _n<\min \Big (\frac{L}{M_n+1}, 1\Big )$, where $M_n=\sup _{a\in \overline{(U\cap V)_{2/n}}}(L/|\rho (a)|+|\rho (a)|)$. Without loss of generality, for all $m\ge n_1$, we have $|e^{-\sigma _m}-|\rho | \; |\le \epsilon _n$ on $\overline{(U\cap V)_{\frac{2}{n}}}$, and that $|\; |D'\sigma _m(a)(b)|^2-|\frac{1}{|\rho (a)|}D'\rho (a)(b)|^2\; |\le \epsilon _n\Vert b\Vert ^2$ when $a\in \overline{(U\cap V)_{\frac{2}{n}}}$ and $b\in {\mathbb {C}}^n$.

Then, when $a\in \overline{(U\cap V)_{2/n}}$ and $D'\rho (a)(b)=0$, we have that for $m\ge n_1$,

$$\begin{aligned}&D'D''r_m(a)(b,b)\\&\quad =e^{-\sigma _m(a)}D'D''\sigma _m(a)(b,b)-e^{-\sigma _m(a)}|D'\sigma _m(a)(b)|^2\\&\quad \ge e^{-\sigma _m(a)} \frac{L}{|\rho (a)|}\Vert b\Vert ^2-e^{-\sigma _m(a)}|D'\sigma _m(a)(b)|^2\\&\quad \ge \Big (\frac{|\rho (a)|-\epsilon _n}{|\rho (a)|}\Big )L\Vert b\Vert ^2-(|\rho (a)|+\epsilon _n)\\&\qquad \times \Big (\Big |\frac{1}{\rho (a)}D'\rho (a)(b)\Big |^2+\epsilon _n\Vert b\Vert ^2\Big )\\&\quad =(L-\epsilon _n\Big (\frac{L}{|\rho (a)|}+|\rho (a)|\Big )-\epsilon _n^2)\Vert b\Vert ^2 \end{aligned}$$

where $L-\epsilon _n\Big (\frac{L}{|\rho (a)|}+|\rho (a)|\Big )-\epsilon _n^2\ge L-\epsilon _n(M_n+1)>0$. Then, following the proof of Proposition 2.2, each $r_m$ with $m\ge n_1$ satisfies for $a\in (U\cap V)_{2/n}$ as well as b with $D'\rho (a)(b)=0$ and $\Vert b\Vert <d_{(U\cap V)_{2/n}}(a)$,

$$\begin{aligned} & \frac{1}{2\pi }\int _0^{2\pi } (r_m(a+e^{i\theta }b)-r_m(a))d\theta \\ & \quad \ge (L-\epsilon _n(M_n+1))\Vert b\Vert ^2. \end{aligned}$$

so taking the limit as $m\rightarrow \infty$,

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\! (\rho (a+e^{i\theta }b)-\rho (a))d\theta \ge (L-\epsilon _n(M_n+1))\Vert b\Vert ^2. \end{aligned}$$

As $\epsilon _n\rightarrow 0$ and then $n\rightarrow \infty$, we conclude what we desire. $\square$

Proposition 3.2 is significant since it confirms a expected behavior of a defining function of an $\ell$-strictly pseudoconvex domain. However, if $\ell =1$, it does not recover a condition at the boundary resembling equation (3.1), so we consider it separately.

Definition 10

Given a domain $U\subset X$ with a $C^{1}$ 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, $\rho \in C^{1}(V)$ a defining function of bU, and $\varphi \in C(bU)$ a positive function, such that for all $w\in bU$ and $b\in {\mathbb {C}}^n$ with $D'\rho (w)(b)=0$ and $\Vert b\Vert$ small (lower semicontinuously depending on w), we have

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\rho (w+e^{i\theta }b)d\theta \ge \rho (w)+\varphi (w)\Vert b\Vert ^2. \end{aligned}$$

(3.4)

We will soon explore examples of domains that are 1-strictly pseudoconvex at the boundary, both in finite and infinite dimensions. However, first, let us see that the following is a sufficient condition for $\ell$-strict pseudoconvexity when $\ell \ge 1$.

Proposition 3.3

Let $\ell \ge 1$. If U in a Banach space X is a bounded open domain that admits a $C^{\ell }$ defining function of bU, $r:V\supset bU\rightarrow {\mathbb {R}}$, which is uniformly plurisubharmonic on average on $U\cap V$, then U is $\ell$-strictly pseudoconvex.

Proof

Let $r:V\supset bU \rightarrow {\mathbb {R}}$ be a $C^{\ell }$ defining function of bU, which is uniformly plurisubharmonic on average on $U\cap V$. Then, we can find $L>0$ such that, for all $a\in U\cap V$ and $b\in {\mathbb {C}}^n$ of small norm (with size depending lower semicontinuously on a), we have that $a+\overline{{\mathbb {D}}}b$ is contained in $U\cap V$ and

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi } r(a+e^{i\theta }b)d\theta \ge r(a)+L\Vert b\Vert ^2. \end{aligned}$$

(3.5)

Fix $a\in U\cap V$ and $b\in X$ of small norm as before. Since $a+\overline{{\mathbb {D}}}\cdot b\subset U\cap V$, we have that $r(a+e^{i\theta }b)<0$ for each $\theta \in [0,2\pi ]$.

Since $x\mapsto -\log (-x)$ is convex on $(-\infty ,0)$, we obtain by Jensen’s inequality that

$$\begin{aligned} & \frac{1}{2\pi }\int _0^{2\pi }-\log (-r)(a+e^{i\theta }b)d\theta \nonumber \\ & \quad \ge -\log (-\frac{1}{2\pi }\int _0^{2\pi }r(a+e^{i\theta }b)d\theta ). \end{aligned}$$

(3.6)

Moreover, according to equation (3.5), we have

$$\begin{aligned} & 0<\frac{-1}{2\pi }\int _{0}^{2\pi } r(a+e^{i\theta }b)d\theta \le -(r(a)+L\Vert b\Vert ^2), \end{aligned}$$

so using that $-\log$ is decreasing on $(0,\infty )$, we obtain that

$$\begin{aligned} -\log (\frac{-1}{2\pi }\int _{0}^{2\pi } r(a+e^{i\theta }b)d\theta )\ge -\log (-(r(a)+L\Vert b\Vert ^2)). \end{aligned}$$

(3.7)

According to equations (3.6) and (3.7), we obtain

$$\begin{aligned}&\frac{1}{2\pi }\int _0^{2\pi }-\log (-r)(a+e^{i\theta }b)d\theta \\&\quad \ge -\log (-(r(a)+L\Vert b\Vert ^2)) \\&\quad =-\log (-r(a)(1-\frac{L\Vert b\Vert ^2}{[-r(a)]}))\\&\quad =-\log (-r(a)) -\log (1-\frac{L\Vert b\Vert ^2}{[-r(a)]})\\&\quad \ge -\log (-r(a))+L\Vert b\Vert ^2/[-r(a)] \end{aligned}$$

as desired. $\square$

Note that if U bounded admits a $C^1$ defining function $\rho :V\supset bU\rightarrow {\mathbb {R}}$, which is uniformly plurisubharmonic on average on $U\cap V$, then there exists $\delta >0$ such that for all $c\in (-\delta ,0)$, $U_{\rho ,c}:=(U\setminus V)\cup \{z\in V: \rho (z)< c\}$ is 1-strictly pseudoconvex and, moreover, 1-strictly pseudoconvex at the boundary.

We present examples of domains that are 1-strictly pseudoconvex at the boundary. The following concept of strong ${\mathbb {C}}$-linear convexity was succinctly studied for the ambient space ${\mathbb {C}}^n$ in [9].

Definition 11

A domain $U\subset X$ is called strongly ${\mathbb {C}}$-linearly convex if bU is of class $C^1$ and there is a $C^1$ defining function $\rho :V\supset bU\rightarrow {\mathbb {R}}$ that satisfies that for a certain $C> 0$,

$$\begin{aligned} |D'\rho (w)(w-z)|\ge C\Vert w-z\Vert ^2 \text { for all }w\in bU, \; z\in {\overline{U}} \end{aligned}$$

(in ${\mathbb {C}}^n$, if this condition holds for a defining function, then it passes to every defining function).

Definition 12

A domain U in X is called weakly linearly convex if for every $w\in bU$, there exists an affine complex hyperplane $\pi$ such that $w\in \pi \subset X\setminus U$.

Example 3

Domains U in a Banach space X that are strongly ${\mathbb {C}}$-linearly convex are weakly linearly convex since, given $w\in bU$, we have that $w\in w+T_w^{{\mathbb {C}}}(bU)$, and for all $z\in U$, there exists $C>0$ such that

$$\begin{aligned} & \inf _{b\in T_w^{{\mathbb {C}}}(bU)}\Vert w+b-z\Vert \ge \inf _{b\in T_w^{{\mathbb {C}}}(bU)} \frac{1}{\Vert D'\rho (w)\Vert } \\ & \quad \cdot |D'\rho (w)(w+b-z)| \ge \frac{C}{\Vert D'\rho (w)\Vert } \Vert w-z\Vert ^2>0. \end{aligned}$$

Domains that are weakly linearly convex satisfy that their intersection with each finite-dimensional subspace are again weakly linearly convex, and thus holomorphically convex in each intersection with a finite-dimensional space (see [6, Ch. IV, §4.6]); hence, the weakly linearly convex domains are pseudoconvex [15, Ch. VIII, §37].

Now, we show that domains strongly ${\mathbb {C}}$-linearly convex $U\subset {\mathbb {C}}^n$ with a $C^{1,1}$ boundary admit a $C^1$ defining function $\rho : V\supset bU\rightarrow {\mathbb {R}}$ satisfying equation (3.4) for all $w\in bU$ and $b\in {\mathbb {C}}^n$ with $D'\rho (w)(b)=0$ and $\Vert b\Vert <\min (d_{V}(w),1)$.

Consider the function

$$\begin{aligned} \rho (z)={\left\{ \begin{array}{ll} -d_U(z), & \text { if }z\in U\\ d_U(z), & \text { if }z\in U^c \end{array}\right. }. \end{aligned}$$

Since bU is $C^{1,1}$, $\rho$ is a $C^1$ defining function in a neighborhood V of bU.

We fix $w\in bU$ and $b\in T_w^{{\mathbb {C}}}(bU)$ with $\Vert b\Vert <\min (d_{V}(w),1)$. For every $\theta \in [0,2\pi ]$, we know that $d_U(w+e^{i\theta }b)=\inf _{w'\in bU}\Vert w+e^{i\theta }b-w'\Vert$, and for each $w'\in bU$, let us consider two cases:

• If $\Vert w-w'\Vert \le \Vert b\Vert /2$, then $\Vert w+e^{i\theta }b-w'\Vert \ge \Vert b\Vert -\Vert w-w'\Vert \ge \Vert b\Vert /2\ge \Vert b\Vert ^2/2$.

• If $\Vert w-w'\Vert \ge \Vert b\Vert /2$, then as we did before, we obtain that

  $$\begin{aligned} & \Vert w+e^{i\theta }b-w'\Vert \ge \inf _{b'\in T_w^{{\mathbb {C}}}(bU)}\Vert w+b'-w'\Vert \\ & \quad \ge \frac{C}{\Vert D'\rho (w)\Vert } \Vert w-w'\Vert ^2\ge \frac{C}{\Vert D'\rho (w)\Vert } \Vert b\Vert ^2/4. \end{aligned}$$

Then, taking $\varphi (w)=\min \Big (\frac{1}{2}, \frac{C}{4\Vert D'\rho (w)\Vert }\Big )$, we obtain that $d_U(w+e^{i\theta }b)\ge \varphi (w) \Vert b\Vert ^2$. Since $w+e^{i\theta }b\notin U$ for every $\theta \in [0,2\pi ]$, $\rho (w+e^{i\theta }b)-\rho (w)=d_U(w+ e^{i\theta }b)\ge \varphi (w)\Vert b\Vert ^2$; hence,

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi } (\rho (w+e^{i\theta }b)-\rho (w))d\theta \ge \varphi (w)\Vert b\Vert ^2. \end{aligned}$$

We will return to domains 1-strictly pseudoconvex at the boundary, but for now, let us discuss domains U that are 1-strictly pseudoconvex at the boundary as well as $\ell$-strictly pseudoconvex. Obviously, this will occur when a defining function of bU is uniformly plurisubharmonic on average on its entire domain. Thus, we will be interested in the following notion.

Definition 13

If $\ell \ge 1$ and U in a Banach space X is a bounded open domain that admits a $C^{\ell }$ defining function of bU that is uniformly plurisubharmonic on average, then we will say that U is $\ell$-uniformly pseudoconvex.

The reader may recall that a bounded domain in ${\mathbb {C}}^n$ with a $C^2$ boundary is strictly pseudoconvex if and only if it is 2-uniformly pseudoconvex.

We present nontrivial examples of 1-uniformly pseudoconvex domains, of which at least $B_{\ell _2}$ is 2-uniformly pseudoconvex.

Proposition 3.4

If $n\in {\mathbb {N}}$ and $p \in (1,2)$, $B_{\ell _p^n}$ is 1-uniformly pseudoconvex.

Proof

Let $n\in {\mathbb {N}}$ and $p \in (1,2)$. We have seen that the norm $\Vert \cdot \Vert _p$ of $\ell _p^n$ is uniformly plurisubharmonic on average on $2 B_{\ell _p^n}$; consequently, so is the function $r=\Vert \cdot \Vert _p-1$. Moreover, as shown in [3], the norm $\Vert \cdot \Vert _p$ is of class $C^1$ except at 0, and hence, so is r. To complete the proof that $bB_{\ell _p^n}$ has 1 degree of differentiability, we observe that r satisfies that

$$\begin{aligned} \{z\in (2B_{\ell _p^n})\setminus \{0\}: r(z)<0\}=B_{\ell _p^n}\setminus \{0\} \end{aligned}$$

and the gradient of r is not null at every element of $bB_{\ell _p^n}$, since for each point z in $bB_{\ell _p^n}$ there is $i\in \{1,\cdots , n\}$ such that $z_i\ne 0$, and for such i, we have

$$\begin{aligned} & \frac{\partial |z_i|^p}{\partial z_i}\Big |_z=\frac{\partial (z_i\overline{z_i})^{p/2}}{\partial z_i}\Big |_z=p/2(z_i\overline{z_i})^{p/2-1}\overline{z_i}\\ & \quad =p/2|z_i|^{p-2}\overline{z_i}\ne 0 \end{aligned}$$

so $\nabla (\Vert \cdot \Vert _p-1)|_z\ne 0$. $\square$

Proposition 3.5

If $p\in (1,2]$, then $B_{\ell _p}$ is 1-uniformly pseudoconvex.

Proof

Let $p\in (1,2]$. As in the previous proposition, we know that the norm $\Vert \cdot \Vert _p$ of $\ell _p$ is uniformly plurisubharmonic on average on $2 B_{\ell _p}$, and hence, so is $r=\Vert \cdot \Vert _p-1$. Again, the norm $\Vert \cdot \Vert _p$ is of class $C^1$ except at 0, and thus, r is $C^1$ except at 0 and r satisfies

$$\begin{aligned} \{z\in (2B_{\ell _p})\setminus \{0\}: r(z)<0\}=B_{\ell _p}\setminus \{0\} \end{aligned}$$

where the derivative of r is not null at points z in $bB_{\ell _p}$, because for each z in $bB_{\ell _p}$ there exists $n\in {\mathbb {N}}$ such that $z_n\ne 0$, therefore, if $e_n$ denotes the n-th element of the canonical basis of $\ell _p$ and $\theta =\arg (z_n) \in [0, 2\pi )$, we have that

$$\begin{aligned} Dr(z)(e^{i\theta }e_n)&=\lim _{h\rightarrow 0^+}\frac{\Vert z+he^{i\theta }e_n\Vert -\Vert z\Vert }{h}\\&=\lim _{h\rightarrow 0^+}\frac{(1-|z_n|^p+(|z_n|+h)^p)^{1/p}-1}{h}\\&=\lim _{h\rightarrow 0^+}\frac{|z_n|^{p-1}h+o(|h|)}{h}, \end{aligned}$$

where the last equality has been obtained with a Taylor series, leading us to obtain that $\Vert Dr(z)\Vert \ge |Dr(z)(e^{i\theta }e_n)|=|z_n|^{p-1}> 0$. We conclude that $B_{\ell _p}$ is 1-uniformly pseudoconvex.$\square$

Suitable modifications to the defining functions $\Vert \cdot \Vert _p-1$ by plurisubharmonic functions can lead to examples of domains that are at least 1-strictly pseudoconvex at the boundary, as in the next examples.

Proposition 3.6

If $p\in (1,2]$, then $\{z\in {\mathbb {C}}^n: -\text {Re}(z_1)+\Vert z\Vert _p<1 \}$ is 1-strictly pseudoconvex at the boundary.

Proof

From Proposition 2.3, the function $r(z)=-\text {Re}(z_1)+\Vert z\Vert _p-1$ is strictly plurisubharmonic on average. Additionally, since the function r is in particular a continuous plurisubharmonic function, $\{z\in {\mathbb {C}}^n: r(z)<0\}$ is pseudoconvex [15, Ch. VIII, §38].

Finally, the gradient of r is not null at each $z\in b\{z'\in {\mathbb {C}}^n: -\text {Re}(z'_1)+\Vert z'\Vert _p<1 \}$ because, if $z_j\ne 0$ for some $j>1$, then

$$\begin{aligned} \frac{\partial r}{\partial z_j}(z)=\frac{1}{2}\Vert z\Vert _p^{1-p}|z_j|^{p-2}\overline{z_j}\ne 0, \end{aligned}$$

or if $\text {Im}(z_1)\ne 0$, then by finding a Taylor series, one obtains that

$$\begin{aligned} \frac{\partial r}{\partial y_1}(z)=\Vert z\Vert _p^{1-p}|z_1|^{p-2}\text {Im}(z_1)\ne 0; \end{aligned}$$

otherwise, $\text {Im}(z_1)=z_2=\cdots =z_n=0$, and since $-\text {Re}(z_1)+\Vert z\Vert _p=1$, we have that $\text {Re}(z_1)=-1/2$, where

$$\begin{aligned} & \frac{\partial r}{\partial x_1}(-\frac{1}{2} e_1)=-1+\Vert z\Vert _p^{1-p}|z_1|^{p-2}\text {Re}(z_1)\big |_{(-\frac{1}{2}e_1)}\\ & \quad =-1+(\frac{1}{2})^{1-p}(\frac{1}{2})^{p-2}(-\frac{1}{2})=-2. \end{aligned}$$

$\square$

Proposition 3.7

If $p\in (1,2]$, then $\{z\in \ell _p: -\text {Re}(z_1)+\Vert z\Vert _p<1 \}$ is 1-strictly pseudoconvex at the boundary.

The proof is analogous to that of finite dimension.

According to distribution theory, given $\ell \ge 1$, a bounded open domain $U\subset {\mathbb {C}}^n$ is $\ell$-uniformly pseudoconvex if and only if there exist open domains $V_1$ and $V_2$ such that $V_2\supset \overline{V_1} \supset bU$, there exist positive constants L, M, and $M'$ and a sequence of pointwise bounded $C^{\ell +1}$ functions on $V_2$, $\{r_m\}$, satisfying

• (i) each $r_m$ is L-uniformly plurisubharmonic on average on $V_2$,

• (ii) $M'\ge \Vert \nabla r_m(z)\Vert \ge M$ for all $z\in \overline{V_1}$ and

• (iii) there exists a $C^{\ell }$ function, $r:V_2\rightarrow {\mathbb {R}}$, satisfying $U\cap V_2=\{z\in V_2:r(z)<0\}$ and whenever $\alpha$ is a multi-index with $0\le |\alpha |\le \ell$, we have that the following holds uniformly on $\overline{V_1}\subset {\mathbb {R}}^{2n}$:

  $$\begin{aligned} \frac{\partial ^{\alpha }r}{\partial x^{\alpha }}=\lim _{m\rightarrow \infty } \frac{\partial ^{\alpha }r_m}{\partial x^{\alpha }}. \end{aligned}$$

Equivalently, $\{r_m\}$ above can be replaced by a family of $C^{\infty }$ functions.

Based on the previous remark and the Arzelà-Ascoli theorem, we define what comes next.

Definition 14

Let us call $U\Subset {\mathbb {C}}^n$ a 0-uniformly pseudoconvex domain when there exist positive constants L, M and $M'$, bounded neighborhoods $V_1$ and $V_2$ of bU with $V_2\supset \overline{V_1}$ and a function $r:V_2\rightarrow {\mathbb {R}}$ such that $U\cap V_2=\{z\in V_2: r(z)<0\}$ and r is the limit of a sequence of pointwise bounded $C^1$ functions given on $V_2$, $\{r_m\}$, such that each $r_m$ is L-uniformly plurisubharmonic on average on $V_2$ and $M'\ge \Vert \nabla r_m(z)\Vert \ge M$ for all $z\in \overline{V_1}$.

As before, 0-uniformly pseudoconvex domains are clearly pseudoconvex, and $\ell$-uniformly pseudoconvex domains are 0-uniformly pseudoconvex.

Although strong and strict pseudoconvexity are the same notion in finite dimensions, for infinite dimensions we now define strong pseudoconvexity and leave open the general concept of strict pseudoconvexity.

Definition 15

We will say that a bounded open set U in a Banach space X is strongly pseudoconvex if $U\cap M$ is 0-uniformly pseudoconvex for each finite-dimensional subspace M of X.

Theorem 3.8

$B_{\ell _1}$ is strongly pseudoconvex.

Proof

We have that $r(z)=\Vert z\Vert _1-1$ on $2B_{\ell _1}{\setminus }\{0\}$ satisfies $B_{\ell _1}{\setminus }\{0\}=\{z \in 2B_{\ell _1}{\setminus }\{0\}: r(z)<0\}$. Additionally, r is the pointwise limit of $\Vert \cdot \Vert _{p_n}-1$ for $2> p_n\rightarrow 1^+$, where each $\Vert \cdot \Vert _{p_n}-1$ is $(\sqrt{\frac{1}{2 e}+4}-2)$-uniformly plurisubharmonic on average because of the main result in [13] and because we can argue as in the proofs of [2, Thm. 2.4 and Thm. 4.1] and Proposition 2.3, indeed, for $p\in [1,2)$, $z\in 2B_{\ell _1}{\setminus }\{0\}$ and $b\in B_{\ell _1}$,

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi } \Vert z+e^{i\theta }b\Vert _p d\theta&\ge (\frac{1}{2\pi }\int _0^{2\pi } \Vert z+\frac{1}{\sqrt{e}}e^{i\theta }b\Vert _p^2d\theta )^{1/2}\\&\ge (\frac{1}{2\pi }\int _0^{2\pi } \Vert z+\frac{1}{\sqrt{e}}e^{i\theta }b\Vert _p^pd\theta )^{1/p}\\&\ge (\Vert z\Vert _p^2+\frac{1}{2e}\Vert b\Vert _p^2)^{1/2}\\&\ge \Vert z\Vert _p+(\sqrt{\frac{1}{2e}+4}-2)\Vert b\Vert _p^2. \end{aligned}$$

The $C^1$ functions $\{|\Vert \cdot \Vert _p-1|\}_{p\in (1,2)}$ on $2B_{\ell _1}\setminus \{0\}$ are bounded by 1 because $\Vert \cdot \Vert _p\le \Vert \cdot \Vert _1$. Clearly, the same holds when we restrict those functions to finite-dimensional subspaces.

Now, let M be a finite-dimensional subspace of $\ell _1$. Then, since all norms are equivalent in M, we will just bound the operator norm of $D(\Vert \cdot \Vert _p-1)(z)$ for $p\in (1,2)$ and $z\in \overline{2B_M}{\setminus }\{0\}$. As $\Vert z\Vert _1\le C_M \Vert z\Vert _2$ for all $z\in M$, for a constant $C_M\ge 1$ only depending on M, we have

$$\begin{aligned} & \Vert D (\Vert \cdot \Vert _p-1)(z)\Vert \ge \frac{D (\Vert \cdot \Vert _p-1)( z)(z)}{\Vert z\Vert _1}\\ & \quad =\lim _{h\rightarrow 0^+}\frac{\Vert z+hz/\Vert z\Vert _1\Vert _p-\Vert z\Vert _p}{h}=\Vert z\Vert _p/\Vert z\Vert _1 \end{aligned}$$

which is greater than $\Vert z\Vert _2/\Vert z\Vert _1\ge 1/C_M$. Now, to bound the gradient from above, recall that

$$\begin{aligned} \Vert D (\Vert \cdot \Vert _p-1)\Vert =\sup _{w\ne 0} \frac{|D (\Vert \cdot \Vert _p-1)(z)(w)|}{\Vert w\Vert _1} \end{aligned}$$

Note that for $e_n$ the n-th element of the canonical basis of $\ell _1$ and $\theta _n=\text {arg}(z_n)$, we have, as in the proof of Proposition 3.5, that

$$\begin{aligned} D(\Vert \cdot \Vert _p-1)(z)(e^{i\theta _n}e_n)&=\Vert z\Vert _p^{1-p}|z_n|^{p-1},\\ D(\Vert \cdot \Vert _p-1)(z)(ie^{i\theta _n}e_n)&=0. \end{aligned}$$

Then, if $<\cdot ,\cdot>$ represents the usual inner product in ${\mathbb {R}}^2$, for any $w\ne 0$ in M,

$$\begin{aligned}&|D (\Vert \cdot \Vert _p-1)(z)(w)|\\&\quad =\Big |\sum _n <e^{i\theta _n}, w_n> D(\Vert \cdot \Vert _p-1)(z)(e^{i\theta _n}e_n)\Big |\\&\quad \le \sum _n |w_n| \Vert z\Vert _p^{1-p}|z_n|^{p-1}\\&\quad \le \Vert z\Vert _p^{1-p} \Vert z\Vert _{\infty }^{p-1} \Vert w\Vert _1\\&\quad \le \Vert z\Vert _2^{1-p} \Vert z\Vert _1^{p-1}\Vert w\Vert _1. \end{aligned}$$

Hence

$$\begin{aligned} \Vert D (\Vert \cdot \Vert _p-1)(z)\Vert _2\le \Vert z\Vert _2^{1-p} \Vert z\Vert _1^{p-1}\le (C_M)^{p-1}\le C_M. \end{aligned}$$

$\square$

Corollary 3.9

$B_{\ell _1^n}=\{z\in {\mathbb {C}}^n:\sum _{j=1}^n |z_j|<1\}$ is 0-uniformly pseudoconvex.

We continue our discussion of $\ell$-uniform pseudoconvexity with a characterization of 0-uniformly pseudoconvex domains as a certain limit of 1-uniformly pseudoconvex domains.

Proposition 3.10

In ${\mathbb {C}}^n$, a domain U is 0-uniformly pseudoconvex if and only if it is exhausted by an increasing sequence $\{U_m\}$ of 1-uniformly pseudoconvex domains given by a respective family of $C^1$ defining functions $r_m: V'\supset {\overline{V}}\supset bU_m \cup bU \rightarrow {\mathbb {R}}$ such that $\infty>\limsup |r_m(z)|\ge \liminf |r_m(z)|>0$ for each $z\in {\overline{V}}{\setminus } bU$, and such that there exist common positive bounds L, M, and $M'$ satisfying that each $r_m$ is L-uniformly plurisubharmonic on average on V and $M' \ge \Vert Dr_m(z)\Vert \ge M$ for all $z \in {\overline{V}}$ compact and $w\in {\mathbb {C}}^n$.

Proof

Suppose that we can exhaust U with an increasing sequence $\{U_m\}$ of 1-uniformly pseudoconvex domains, where each $bU_m$ is given by the $C^1$ defining function $r_m:V'\supset {\overline{V}}\supset bU_m\cup bU\rightarrow {\mathbb {R}}$ so that the family $\{r_m\}$ satisfies $\infty>\limsup |r_m(z)|\ge \liminf |r_m(z)|>0$ for each $z\in {\overline{V}}{\setminus } bU$, each $r_m$ is L-uniformly plurisubharmonic on average on V, and for all z in the compact set ${\overline{V}}$ as well as any $w\in {\mathbb {C}}^n$, it follows that $M' \ge \Vert Dr_m(z)\Vert \ge M$. Then, we can use the Arzelà–Ascoli theorem to find a subsequence $\{r_{m_k}\}$ of $\{r_m\}$ as restricted to ${\overline{V}}$ that converges uniformly to a function $r:{\overline{V}}\rightarrow {\mathbb {R}}$, and it follows that $U\cap V=\{z\in V: r(z)<0\}$.

Conversely, suppose that r is the uniform limit on ${\overline{V}}$ compact of a sequence of $C^1$ functions on $V'\supset {\overline{V}}$, $\{r_m\}$, which are L-uniformly plurisubharmonic on average and such that $M'\ge \Vert Dr_m(z)\Vert \ge M$ for all $z\in {\overline{V}}$, and where $r: V'\supset {\overline{V}}\supset bU \rightarrow {\mathbb {R}}$ satisfies $U\cap V'=\{z\in V':r(z)<0\}$. Let $V_0$ be a neighborhood of bU with $\overline{V_0}\subset V$. For each $m\in {\mathbb {N}}$, take $c_m=\min _{{\overline{V}}\setminus U} r_m$, and let $U_m=(U{\setminus } \overline{V_0})\cup \{z\in V: r_m(z)-c_m<0\}$. It is clear that $c_m \rightarrow 0$ and that $\cup U_m=U$. By subtracting small positive numbers $d_m$ to each $c_m$ with $d_m\rightarrow 0$ and then passing to a subsequence, we may assume that $\{U_m\}$ is an increasing sequence of open sets whose union is U, where $bU_m$ has $C^1$ defining function $r_m-(c_m-d_m)$ defined on $V_0$. We conclude that U satisfies the desired conditions. $\square$

For $\ell \ge 1$, the reader can similarly show a characterization of $\ell$-uniformly pseudoconvex domains in ${\mathbb {C}}^n$ as a certain limit of $(\ell +1)$-uniformly pseudoconvex domains. In particular, the following holds.

Proposition 3.11

A domain U in ${\mathbb {C}}^n$ is 1-uniformly pseudoconvex if and only if it is exhausted by an increasing sequence $\{U_m\}$ of strongly pseudoconvex domains given by a respective family of $C^2$ defining functions $r_m: V'\supset {\overline{V}}\supset bU_m \cup bU \rightarrow {\mathbb {R}}$ such that

• (i) $\infty>\limsup |r_m(z)|\ge \liminf |r_m(z)|>0$ for each $z\in {\overline{V}}{\setminus } bU$,

• (ii) each family $\{\frac{\partial r_m}{\partial x_i}\}$ is equicontinuous on ${\overline{V}}$ for $i=1, \cdots , 2n$ and

• (iii) there exist common positive bounds L, M, and $M'$ such that $D'D''r_m(z)(w,w)\ge L \Vert w\Vert ^2$ and $M' \ge \Vert Dr_m(z)\Vert \ge M$ for all $z \in {\overline{V}}$ compact and $w\in {\mathbb {C}}^n$. there exists a positive bound K such that $K\ge \Vert D^2 r(z)\Vert$ for all $z\in {\overline{V}}$.

Equivalently, $\{r_m\}$ above can be replaced by a family of $C^{\infty }$ functions.

Let us use the previous approximation result to show an invariance property of 1-uniformly pseudoconvex domains in ${\mathbb {C}}^n$, which in particular yields that the image of a 1-uniformly pseudoconvex domain in ${\mathbb {C}}^n$ under an affine isomorphism 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 ${\mathbb {C}}^n$ is still 1-uniformly pseudoconvex, when the domain of the biholomorphism contains ${\overline{U}}$.

Proof

Suppose that U is a 1-uniformly pseudoconvex domain that is exhausted by the increasing sequence $\{U_m\}$ of strongly pseudoconvex domains given by the respective family of $C^2$ defining functions $r_m: V'\supset {\overline{V}}\supset bU_m \cup bU \rightarrow {\mathbb {R}}$ such that $\infty>\limsup |r_m(z)|\ge \liminf |r_m(z)|>0$ for each $z\in {\overline{V}}{\setminus } bU$, for which there exist common positive bounds L, M, and $M'$ such that $D'D''r_m(z)(w,w)\ge L \Vert w\Vert ^2$ and $M' \ge \Vert Dr_m(z)\Vert \ge M$ for all z in the compact set ${\overline{V}}$ and $w\in {\mathbb {C}}^n$, and such that each family $\{\frac{\partial r_m}{\partial x_i}\}$ is equicontinuous, $i=1, \cdots , 2n$.

Then, as in [17, Ch. II, §2.6], if $F:W'\supset {\overline{W}}\supset U \rightarrow {\mathbb {C}}^n$ is a biholomorphic map, then F(U) is exhausted by the increasing sequence of domains $\{F(U_m)\}$ given by the respective family of defining functions $\rho _m=r_m\circ F^{-1}: F(W'\cap V')\supset F({\overline{W}}\cap {\overline{V}})\supset bF(U_m)\cup bF(U)\rightarrow {\mathbb {R}}$, which satisfy for $z\in {\overline{W}}\cap {\overline{V}}$ and $w\in {\mathbb {C}}^n$,

• (i) $\infty>\limsup |\rho _m\circ F(z)|\ge \liminf |\rho _m\circ F(z)|>0\text { when }z\notin bU$,

• (ii) $D'D''r_m(z)(w,w)=D'D''\rho _m(F(z))(F'(z)w, F'(z)w)$, and

• (iii) $Dr_m(z)(w)=D\rho _m(F(z))(F'(z)w)$,

where each $F'(z)$ is a nonsingular ${\mathbb {C}}$-linear map, so it defines an isomorphism on ${\mathbb {C}}^n$.

Hence, each domain $F(U_m)$ is still strongly pseudoconvex with

• (i) $D'D''\rho _m(z')(w', w'){\ge } \frac{L}{\max _{z_1\in F({\overline{W}}\cap {\overline{V}})}\Vert F'(F^{-1}(z_1))\Vert ^2}\Vert w'\Vert ^2$,

• (ii) $M'\cdot \max _{z_1\in F({\overline{W}}\cap {\overline{V}})}\Vert (F^{-1})'(z_1)\Vert \ge \Vert D\rho _m(z')\Vert \ge \frac{M}{\max _{z_1\in F({\overline{W}}\cap {\overline{V}})}\Vert F'(F^{-1}(z_1))\Vert }$

for all $z' \in F({\overline{W}}\cap {\overline{V}})$ and $w'\in {\mathbb {C}}^n$. It is also easy to check that each family $\{\frac{\partial \rho _m}{\partial x_i}\}$ ($i=1,\cdots , 2n$) is equicontinuous on $F({\overline{W}}\cap {\overline{V}})$.

Thus, F(U) is strongly pseudoconvex with boundary F(bU). $\square$

Let us conclude this section by stating the invariance property of 0-uniformly pseudoconvex domains U under biholomorphisms whose domain contain ${\overline{U}}$; its proof is an easy consequence of the corresponding result for 1-uniformly pseudoconvex domains. As well as with 1-uniformly pseudoconvex domains, we have extended our family of examples of 0-uniformly pseudoconvex domains. Moreover, we found that strongly pseudoconvex domains are invariant under affine isomorphisms between Banach spaces.

Proposition 3.13

The biholomorphic image of a 0-uniformly pseudoconvex domain U in ${\mathbb {C}}^n$ is still 0-uniformly pseudoconvex, when the domain of the biholomorphism contains ${\overline{U}}$.

4 Cauchy–Riemann equations

In this section, we will present some solutions to the (inhomogeneous) Cauchy–Riemann equations in infinite dimensions. The importance of this lies in that strongly pseudoconvex domains with $C^4$ boundary in finite dimensions admit solutions that extend continuously to the boundary to the inhomogeneous Cauchy–Riemann equations for (0, 1)-forms with bounded conditions [8], while the ball of $\ell _1$ admits solutions to the inhomogeneous Cauchy–Riemann equations but with Lipschitz conditions [12]. Having discovered that the ball of $\ell _1$ is strongly pseudoconvex, at the very least here we solve the inhomogeneous Cauchy–Riemann equations for (0, 1)-forms with Lipschitz conditions on any domain affinely equivalent to the ball of $\ell _1$, a type of domain that we have also proved is strongly pseudoconvex.

The following necessary notions can be found in [15, Ch. V].

Definition 16

If X and Y denote complex Banach spaces, let $X_{{\mathbb {R}}}$ and $Y_{{\mathbb {R}}}$ denote the respective previous spaces seen as real Banach spaces. For every $m \in {\mathbb {N}}$, $L(^m X_{{\mathbb {R}}}, Y_{{\mathbb {R}}})$ denotes the continuous m-linear mappings $A:X_{{\mathbb {R}}}^m \rightarrow Y_{{\mathbb {R}}}$, while $L^a(^m X_{{\mathbb {R}}}, Y_{{\mathbb {R}}})$ denotes the continuous m-linear mappings $A:X_{{\mathbb {R}}}^m \rightarrow Y_{{\mathbb {R}}}$ that are alternating, i.e.,

$$\begin{aligned} & A(x_{\sigma (1)}, \cdots , x_{\sigma (m)})=(-1)^{\sigma } A(x_1, \cdots , x_m), \; \\ & \quad \text { for all } \sigma \in S_m \text { and } x_1, \cdots , x_m \in X. \end{aligned}$$

Additionally, given $m \in {\mathbb {N}}$ and $p,q \in {\mathbb {N}}_0$ such that $p+q=m$, $L^a(^{p,q} X_{{\mathbb {R}}}, Y_{{\mathbb {R}}})$ is the subspace of $A \in L^a(^m X_{{\mathbb {R}}}, Y_{{\mathbb {R}}})$ such that

$$\begin{aligned} & A(\lambda x_1, \cdots , \lambda x_m)=\lambda ^p {\bar{\lambda }}^q A(x_1, \cdots , x_m), \;\\ & \quad \text { for all } \lambda \in {\mathbb {C}} \text { and } x_1, \cdots , x_m \in X; \end{aligned}$$

while $L^{apq}(^m X_{{\mathbb {R}}}, Y_{{\mathbb {R}}})$ denotes the subspace of all $A \in L(^m X_{{\mathbb {R}}}, Y_{{\mathbb {R}}})$ which are alternating in the first p variables and are alternating in the last q variables.

Given $A \in L(^m X_{{\mathbb {R}}}, Y_{{\mathbb {R}}})$, define its alternating part $A^a \in L^a(^m X_{{\mathbb {R}}}, Y_{{\mathbb {R}}})$ by

$$\begin{aligned} & A^a(x_1, \cdots , x_m)=\frac{1}{m!}\sum _{\sigma \in S_m} (-1)^{\sigma } A(x_{\sigma (1)}, \cdots , x_{\sigma (m)}), \;\\ & \quad \text { for all }x_1,\cdots , x_m \in X. \end{aligned}$$

Given U an open subset of X and $p,q \in {\mathbb {N}}_0$, let $C_{p,q}^{\infty }(U,Y):= C^{\infty }(U, L^a(^{p,q} X_{{\mathbb {R}}}, Y_{{\mathbb {R}}}))$. Then, for each $f \in C_{p,q}^{\infty }(U,Y)$, ${\bar{\partial }}f \in C_{p,q+1}^{\infty }(U,Y)$ is given by

$$\begin{aligned} {\bar{\partial }}f(x)=(m+1)[D''f(x)]^a, \; \text { for all }x \in U. \end{aligned}$$

Remark 1

Since $D''f(x) \in L(X_{{\mathbb {R}}}, L^a(^m X_{{\mathbb {R}}}, Y_{{\mathbb {R}}}) )=L^{a1m}(^{m+1} X_{{\mathbb {R}}}, Y_{{\mathbb {R}}})$, Proposition 18.6 in [15] implies that $\text { for all }t_1, \cdots , t_{m+1} \in X$,

$$\begin{aligned}&{\bar{\partial }}f(x)(t_1, \cdots , t_{m+1})\\&\quad =(m+1)[D''f(x)]^{a1m}(t_1, \cdots , t_{m+1})\\&\quad =\frac{m+1}{m+1} \sum _{\sigma \in S_{1m}} (-1)^{\sigma } D''f(x) (t_{\sigma (1)}, \cdots , t_{\sigma (m+1)}) \end{aligned}$$

where $S_{1m}$ denotes the set of all permutations $\sigma \in S_{m+1}$ such that $\sigma (2)< \cdots < \sigma (m+1)$, so

$$\begin{aligned} & {\bar{\partial }}f(x)(t_1, \cdots , t_{m+1})=\sum _{j=1}^{m+1}(-1)^{j-1} D''f(x)(t_j) \\ & \quad (t_1, \cdots , t_{j-1}, t_{j+1}, \cdots , t_{m+1}). \end{aligned}$$

The ${\bar{\partial }}$ problem for $g \in C_{p,q+1}^{\infty }(U,Y)$ asks whether the equation ${\bar{\partial }}g=0$ implies the existence of $f \in C_{p,q}^{\infty }(U,Y)$ such that ${\bar{\partial }}f=g$.

We are interested in the case when $p=q=0$, since in this case we have the following known result [12]:

Theorem 4.1

Suppose that g is a complex-valued (0, 1)-form on $B_{\ell _1}$ that is Lipschitz continuous on all balls $r B_{\ell _1}$, $0<r<1$. If g is a ${\overline{\partial }}$-closed form, i. e. ${\overline{\partial }}g=0$, then there is a continuously differentiable function f on $B_{\ell _1}$ that solves ${\overline{\partial }}f=g$. If, in addition, g is m times continuously differentiable, $m=2,3,\dots$, then so is f.

We can draw some conclusions from the last result. To apply the following proposition, keep in mind that examples of spaces isomorphic to $\ell _1$ include the infinite-dimensional complemented subspaces of $\ell _1$ [16].

Proposition 4.2

Suppose that $U=T(B_{\ell _1})$ for $T=a+L$ an affine isomorphism of Banach spaces $T: \ell _1\rightarrow X$. If $g\in C^{\infty }_{0,1}(U, {\mathbb {C}})$, ${\overline{\partial }}g=0$, and g is Lipschitz continuous on all $a+r(U-a)$, $0<r<1$; then, there exists $f\in C^{\infty }(U,{\mathbb {C}})$ such that ${\overline{\partial }}f=g$.

Proof

Let $M_r$ be the Lipschitz constant of g on $a+r(U-a)$, for $0<r<1$. Define ${\tilde{g}}\in C^{\infty }_{0,1}(B_{\ell _1}, {\mathbb {C}})$ by ${\tilde{g}}(x)(t)=g(Tx)(Lt)$. Note that ${\tilde{g}}$ is Lipschitz continuous on all balls $rB_{\ell _1}$, $0<r<1$, because given $r\in (0,1)$ as well as $x_1, x_2\in r B_{\ell _1}$ and $t\in \ell _1$ we have that $Tx_1, Tx_2\in a+r(U-a)$, hence,

$$\begin{aligned} |{\tilde{g}}(x_1)(t)-{\tilde{g}}(x_2)(t)|&= |g(Tx_1)(Lt)-g(Tx_2)(Lt)|\\&\le M_r \Vert Tx_1-Tx_2\Vert \Vert Lt\Vert \\&= M_r\Vert Lx_1-Lx_2\Vert \Vert Lt\Vert \\&\le M_r \Vert L\Vert ^2 \Vert x_1-x_2\Vert \Vert t\Vert \end{aligned}$$

i.e., $\Vert {\tilde{g}}(x_1)-{\tilde{g}}(x_2)\Vert \le M_r\Vert L\Vert ^2 \Vert x_1-x_2\Vert$.

Moreover, we also know that ${\overline{\partial }}{\tilde{g}}=0$ because given $x\in B_{\ell _1}$ as well as $t_1, t_2 \in \ell _1$ we have that, since L is linear,

$$\begin{aligned} {\overline{\partial }}({\tilde{g}})(x)(t_1,t_2)&=D''({\tilde{g}})(x)(t_1)(t_2)-D''({\tilde{g}})(x)(t_2)(t_1)\\&=D''g(Tx)(Lt_1)(Lt_2)-D''g(Tx)(Lt_2)(Lt_1)\\&={\overline{\partial }}g(Tx)(Lt_1,Lt_2)\\&=0. \end{aligned}$$

Then, there exists ${\tilde{f}}\in C^{\infty }(B_{\ell _1}, {\mathbb {C}})$ such that ${\overline{\partial }}{\tilde{f}}={\tilde{g}}$. Consequently, $f={\tilde{f}}\circ T^{-1}\in C^{\infty }(U, {\mathbb {C}})$ satisfies that ${\overline{\partial }}f=g$ since for all $x\in U$ and $t\in X$,

$$\begin{aligned} {\overline{\partial }}f(x)(t)&=D''({\tilde{f}}\circ T^{-1})(x)(t)\\&=D''{\tilde{f}}(T^{-1}x)(L^{-1}t)\\&={\overline{\partial }}{\tilde{f}}(T^{-1}x)(L^{-1}t)\\&={\tilde{g}}(T^{-1}x)(L^{-1}t)\\&=g(TT^{-1}x)(LL^{-1}t)\\&=g(x)(t). \end{aligned}$$

$\square$

We can extend Proposition 4.2 to images over bounded biholomorphisms at the price of global Lipschitz continuity. Note, however, that we do not know which domains are biholomorphic to the unit ball of $\ell _1$ and that when we deal with automorphisms of the ball of $\ell _1$, these are simply affine isomorphisms [1].

Proposition 4.3

Suppose that $U=\phi (B_{\ell _1})$ for a bounded biholomorphism $\phi :B_{\ell _1}\rightarrow U\subset X$. If $g\in C^{\infty }_{0,1}(U, {\mathbb {C}})$, ${\overline{\partial }}g=0$, and g is Lipschitz continuous on U, then there exists $f\in C^{\infty }(U,{\mathbb {C}})$ such that ${\overline{\partial }}f=g$.

Proof

Define ${\tilde{g}}\in C^{\infty }_{0,1}(B_{\ell _1}, {\mathbb {C}})$ by ${\tilde{g}}(x)(t)=g(\phi x)(D'\phi (x) t)$. Then, ${\tilde{g}}$ is Lipschitz continuous on all balls $rB_{\ell _1}$ because so is $\phi$ due to Schwarz’ lemma [15, §7] and the convexity of each $rB_{\ell _1}$. Additionally, because $D'\phi$ is bounded on each $rB_{\ell _1}$ due to Cauchy Inequality [15, §7] and because of the Cauchy integral formula [15, §7] we further have that $D'\phi$ is also Lipschitz on each ball $rB_{\ell _1}$. Indeed, if M is the Lipschitz constant of g on U, we have that for all $r\in (0,1)$ as well as $x_1, x_2 \in r B_{\ell _1}$ and $t\in \ell _1$,

$$\begin{aligned}&|{\tilde{g}}(x_1)(t)-{\tilde{g}}(x_2)(t)|\\&\quad = |g(\phi x_1)(D'\phi (x_1)t)-g(\phi x_2)(D'\phi (x_2)t)|\\&\quad \le |g(\phi x_1)(D'\phi (x_1)t)-g(\phi x_2)(D'\phi (x_1)t)|\\ &+|g(\phi x_2)(D'\phi (x_1)t)-g(\phi x_2)(D'\phi (x_2)t)|\\&\quad \le M \Vert \phi x_1-\phi x_2\Vert \Vert D'\phi (x_1) t\Vert \\&\qquad +\Vert g(\phi x_2)\Vert \Vert D'\phi (x_1)(t)-D'\phi (x_2)(t)\Vert \\&\quad \le M\cdot \frac{2\Vert \phi \Vert }{1-r} \Vert x_1-x_2\Vert \cdot \Vert D'\phi (x_1)\Vert \Vert t\Vert \\&\qquad +(M\cdot 2\Vert \phi \Vert +\Vert g(\phi (0))\Vert )\frac{\Vert t\Vert }{(1-r)/2}\cdot 2\Vert \phi \Vert \Vert x_1-x_2\Vert \\&\quad \le M\cdot \frac{2\Vert \phi \Vert }{1-r} \cdot \frac{\Vert \phi \Vert }{1-r}\Vert x_1-x_2\Vert \Vert t\Vert \\&\qquad +(M\cdot 2\Vert \phi \Vert +\Vert g(\phi (0))\Vert )\frac{4\Vert \phi \Vert }{1-r} \Vert x_1-x_2\Vert \Vert t\Vert . \end{aligned}$$

Moreover, ${\overline{\partial }}{\tilde{g}}=0$ because given $x\in B_{\ell _1}$ and $t_1, t_2 \in \ell _1$, since $D'\phi (x)$ is linear,

$$\begin{aligned} {\overline{\partial }}({\tilde{g}})(x)(t_1,t_2)&=D''({\tilde{g}})(x)(t_1)(t_2)-D''({\tilde{g}})(x)(t_2)(t_1)\\&=D''g(\phi x)(D'\phi (x)t_1)(D'\phi (x)t_2)\\&\quad -D''g(\phi x)(D'\phi (x)t_2)(D'\phi (x)t_1)\\&={\overline{\partial }}g(\phi x)(D'\phi (x)t_1,D'\phi (x)t_2)\\&=0. \end{aligned}$$

Consequently, there exists ${\tilde{f}}\in C^{\infty }(B_{\ell _1}, {\mathbb {C}})$ such that ${\overline{\partial }}{\tilde{f}}={\tilde{g}}$. Then, $f={\tilde{f}}\circ \phi ^{-1}\in C^{\infty }_{0,1}(U, {\mathbb {C}})$ satisfies the following for $x\in U$ and $t\in X$:

$$\begin{aligned}&{\overline{\partial }}f(x)(t)\\&\quad =D''({\tilde{f}}\circ \phi ^{-1})(x)(t)\\&\quad =D''{\tilde{f}}(\phi ^{-1}x)(D'\phi ^{-1}(x)t)\\&\quad ={\overline{\partial }}{\tilde{f}}(\phi ^{-1}x)(D'\phi ^{-1}(x)t)\\&\quad ={\tilde{g}}(\phi ^{-1}x)(D'\phi ^{-1}(x)t)\\&\quad =g(\phi \phi ^{-1}x)( D'\phi (\phi ^{-1}x)D'\phi ^{-1}(x)t)\\&\quad =g(x)(t). \end{aligned}$$

i.e., ${\overline{\partial }}f=g$, as we wanted. $\square$

We also prove a generalization of the following result in [11].

Theorem 4.4

If $\Omega \subset \ell _1$ is pseudoconvex and $g\in C_{0,1}(\Omega , {\mathbb {C}})$ is Lipschitz continuous and ${\overline{\partial }}$-closed, then the equation ${\overline{\partial }}f=g$ has a solution $f\in C^1(\Omega , {\mathbb {C}})$.

As already mentioned in [11], Lipschitz continuous (0, 1)-forms are not holomorphically invariant. The best result we achieved for biholomorphic mappings is the following.

Proposition 4.5

Suppose that $U=\phi (\Omega )$ for a bounded biholomorphism $\phi :V\subset \ell _1 \rightarrow W\subset X$ such that $\Omega \subset V_{\delta }:=\{z\in V: d(z, bV)>\delta \}$ and that $\Omega$ is convex and open. If $g\in C^{\infty }_{0,1}(U, {\mathbb {C}})$, ${\overline{\partial }}g=0$, and g is Lipschitz continuous on U, then there exists $f\in C^{\infty }(U,{\mathbb {C}})$ such that ${\overline{\partial }}f=g$.

Proof

Proceed as in the proof of Proposition 4.3, using Theorem 4.4 instead of Theorem 4.1 and restricting $\phi$ as well as $D'\phi$ to $\Omega$ instead of $rB_{\ell _1}$. $\square$

5 Conclusion

A suitable definition of strong pseudoconvexity has been provided along with relevant examples such as any domain biholomorphic to the ball of $\ell _p$ with $1\le p \le 2$ (with the domain of the biholomorphism containing the closure of such a ball). However, the notion of strict pseudoconvexity remains open. Since we know that the inhomogeneous Cauchy–Riemann equations with bounded conditions have solutions that extend continuously to the boundary on strongly pseudoconvex domains with four degrees of smoothness in finite dimensions [8], such as the Euclidean ball of ${\mathbb {R}}^n$, and the same equations but with Lipschitz conditions are solvable on the ball of $\ell _1$ [12], we wondered whether we could reach similar conclusions for strongly pseudoconvex domains. At the very least, we show that domains affinely isomorphic to the ball of $\ell _1$ or $\ell _1^n$ admit solutions to the inhomogeneous Cauchy–Riemann equations with Lipschitz conditions, yet it remains unclear whether the same is true for the ball of $\ell _p$ or $\ell _p^n$, $1<p< 2$. Moreover, we show that domains boundedly biholomorphic to the ball of $\ell _1$ also admit solutions to the inhomogeneous Cauchy–Riemann equations with bounded conditions, yet nontrivial examples of those are unknown to us.