Non standard properties of m-subharmonic functions

Abstract

We survey elements of the nonlinear potential theory associated to m-subharmonic functions and the complex Hessian equation. We focus on properties which distinguish m-subharmonic functions from plurisubharmonic ones. Introduction Plurisubharmonic functions arose as multidimensional generalizations of subharmonic functions in the complex plane (see [LG]). Thus it is not surprising that these two classes of functions share many similarities. There are however many subtler properties which make a plurisubharmonic function in Cn, n > 1 differ from a general subharmonic function. Below we list some of the basic ones: Liouville type properties. it is known ([LG]) that an entire plurisubharmonic function cannot be bounded from above unless it is constant. The function u(z) = −1 ||z||2n−2 in C n, n> 1 is an example that this is not true for subharmonic ones; Integrability. Any plurisubarmonic function belongs to L loc for any 1≤ p <∞. For subharmonic functions this is true only for p < n n−1 as the function u above shows. Symmetries. Any holomorphic mapping preserves plurisubharmonic functions in the sense that a composition of a plurisubharmonic function with a holomporhic mapping is still plurisubharmonic. This does not hold for subharmonic functions in Cn, n> 1. The notion of m-subharmonic function (see [Bl1], [DK2, DK1]) interpolates between subharmonicity and plurisubharmonicity. It is thus expected that the corresponding nonlinear potential theory will share the joint properties of potential and pluripotential theories. Indeed in the works of Li, Blocki, Chinh, Abdullaev and Sadullaev, Dhouib and Elkhadhra, Nguyen and many others the m-subharmonic potential theory was thoroughly developed. In particular S. Y. Li [Li] solved the associated smooth Dirichlet problem under suitable assumptions, proving thus an analogue of the Caffarelli-Nirenberg-Spruck theorem [CNS] who dealt with the real setting. Z. Blocki [Bl1, Bl3] noted that the Bedford-Taylor apparatus from [BT1] and [BT2] can be adapted to m-subharmonic setting. He also described the domain of definition of the complex Hessian operator. L. H. Chinh developed the variational apporach to the complex Hessian equation [Chi1] and studied the associated viscosity theory of weak solutions in [Chi3]. He also developed the theory of m-subharmonic Cegrell classes [Chi1, Chi2]. Abdullaev and Sadullaev in [AS] defined the corresponding m-capacities (this was done also independently by Chinh in [Chi2] and the authors in [DK2]). A. Dhouib and F. Elkhadhra investigated m-subharmonicity with respect to a current [DE] and noticed several interesting phenomena. N. C. Nguyen in [N] investigated existence of solutions to the Hessian equations if a subsolution exists. Arguably the most interesting part of the theory is the one that differs from its pluripotential counterpart. This involves not only new phenomena but also requires new tools. Obviously there are good reasons for such a discrepancy. The very notion of plurisubharmonicity is independent of the Kähler metric in sharp contrast to m-subharmonicity. The fundamental solution for the m-Hessian equation is − 1 |z|2 n m −2 , hence there is stronger than logarithmic singularity at the origin and the function is bounded at infinity. Also it is only Lp integrable for p < nm n−m . The goal of this survey note is to gather such distinctive results for m-subharmonic functions. Our choice is of course subjective and we do not cover many important issues such as m-polar sets or m-subharmonic functions on compact manifolds. First we deal with the symmetries of m-sh functions. We show in particular that the set these symmetries coincides with the set of holomorphic and antiholomorphic orthogonal affine maps for any 1< m< n in sharp contrast to the borderline cases. We also investigate the analogues of upper level sets of Lelong numbers. Following the arguments of Harvey and Lawson ([HL1]) and Chu ([Ch]) we present the proof of the stunning fact that the upper level sets are discrete for m< n. This again is drastically different from the plurisubharmonic case where Siu’s theorem implies the analyticity of such sets when m= n. aJagiellonian University, Kraków, Poland. bJagiellonian University, Kraków, Poland. Dinew · Kołodziej 36 The note is organized as follows: the basic notions and tools are listed in Section 1. In particular we have covered the linear algebraic and potential theoretic properties of m-sh functions. We have also included a fairly brief subsection devoted to weak solutions of general elliptic PDEs. The first part of Section 2 is devoted to the symmetries of m-sh functions. In the second one we construct a particular nonlinear operator Pm. We show that all m-sh functions are subsolutions for Pm and, more importantly, Pm has the same fundamental solution as the m-Hessian operator. We wish to point out that Pm is an example of a much more general construction of an uniformly elliptic operator with the same Riesz characteristic as defined by Harvey and Lawson (see [HL1]). Finally in Section 3 we investigate the upper level sets of analogues of Lelong numbers of m-sh functions. This section depends on the general agruments of Harvey and Lawson ([HL1, HL2]) and Chu ([Ch]). As we deal with the concrete case of m-sh functions our argument is slightly simpler but the main ideas are the same. Dedication. It is our pleasure to dedicate this article to Norm, a great friend and mathematician. Aknowledgements. Both authors were supported by the NCN grant 2013/08/A/ST1/00312. 1 Preliminaries In this section we recall the notions and tools appearing in the potential theory of m-subharmonic functions. 1.1 Linear algebra. Denote by Mn the set of all Hermitian symmetric n× n matrices. Fix a matrix M ∈Mn. By λ(M) = (λ1,λ2, ...,λn) denote its eigenvalues arranged in the decreasing order. Definition 1.1. The m-th symmetric polynomial associated to M is defined by Sm(M) = Sm(λ(M)) = ∑ 0< j1<...< jm≤n λ j1 λ j2 ...λ jm . We recall that S1(M) is the trace of M , whereas Sn(M) is the determinant of M . Next one can define the positive cones Γm as follows Γm = {λ ∈ R| S1(λ)> 0, · · · , Sm(λ)> 0}. (1.1) The following two properties of these cones are classical: 1. (Maclaurin’s inequality) If λ ∈ Γm then ( S j (j) ) 1 j ≥ ( Si (i) ) 1 i for 1≤ j ≤ i ≤ m; 2. (Gårding’s inequality, [Ga]) Γm is a convex cone for any m and the function S 1 m m is concave when restriced to Γm; We refer the Reader to [Bl1] or [W] for further properties of these cones. 1.2 Potential theoretic aspects of m-subharmonic functions. We restirct our considerations to a relatively compact domain Ω ⊂ Cn. We assume n≥ 2 in what follows. Denote by d = ∂ + ∂̄ and d c := i(∂̄ − ∂ ) the standard exterior differentiation operators. By β := dd c |z|2 we denote the canonical Kähler form in Cn. We now define the smooth m-subharmonic functions. Definition 1.2. Given a C2(Ω) function u we call it m-subharmonic in Ω if for any z ∈ Ω the Hessian matrix ∂ 2u ∂ zi∂ z̄ j (z) has eigenvalues forming a vector in the closure of the cone Γm. The geometric properties of the eigenvalue vector can be stated more analytically in the language of differential forms: u is m-subharmonic if and only if the following inequalities hold: (dd u) ∧ β n−k ≥ 0, k = 1, · · · , m. Note that these inequalities depend on the background Kähler form β if n − k ≥ 1. Thus it is meaningful to define msubharomicity with respect to a general Kähler form ω (see [DK1] for details). In this survey however we shall deal only with the standard Kähler form β . In ([Bl1]) Z. Błocki proved, one can relax the smoothness requirement on u and develop a non linear version of potential theory for Hessian operators just as Bedford and Taylor did in the case of plurisubharmonic functions ([BT1], [BT2]). In general m-sh functions are defined as follows: Definition 1.3. Let u be a subharmonic function on a domain Ω ∈ Cn. Then u is called m-subharmonic (m-sh for short) if for any collection of C2-smooth m-sh functions v1, · · · , vm−1 the inequality dd u∧ dd c v1 ∧ · · · ∧ dd c vm−1 ∧ β n−m ≥ 0 holds in the weak sense of currents. The set of all m−ω-sh functions is denoted by SHm(Ω). Remark 1. In the case m = n the m-sh functions are simply plurisubharmonic ones. Also it is enough to test m-subharmonicity of u against a collection of m-sh quadratic polynomials (see [Bl1]). Dolomites Research Notes on Approximation ISSN 2035-6803 Dinew · Kołodziej 37 Using the approximating sequence u j from the definition one can follow the Bedford and Taylor construction from [BT2] of the wedge products of currents given by locally bounded m-sh functions. They are defined inductively by dd u1 ∧ · · · ∧ dd up ∧ β n−m := dd (u1 ∧ · · · ∧ dd up ∧ β n−m). It can be shown (see [Bl1]) that analogously to the pluripotential setting these currents are continuous under monotone or uniform convergence of their potentials. Given an m-sh function u one can always construct locally a dereasing sequence of smooth m-sh approximants through the standard regularizations u ∗ρ" with ρ" being a family of smooth mollifiers. Unlike classical elliptic PDEs one cannot apply the maximum principle for m-sh functions directly as we deal with non-smooth functions in general. Instead one can use the so-called comparison principles which are standard tools in pluripotential theory. Their proofs follow essentially from the same arguments as in the plurisubharmonic case m= n (see [K]): Theorem 1.1. Let u, v be continuous m-sh functions in a domain Ω ⊂ Cn. Suppose that lim infz→∂Ω(u− v)(z)≥ 0 then