Poincar\'e- and Sobolev- type inequalities for complex $m$-Hessian equations

By using quasi-Banach techniques as key ingredient we prove Poincar\'e- and Sobolev- type inequalities for $m$-subharmonic functions with finite $(p,m)$-energy. A consequence of the Sobolev type inequality is a partial confirmation of B\l ocki's integrability conjecture for $m$-subharmonic functions.

By these definitions we get that the 1-Hessian operator is the classical Laplace operator defined on 1-admissible functions that are just the subharmonic functions. Furthermore, the n-Hessian operator is the real Monge-Ampère operator defined on n-admissible functions that are the same as the convex functions. Therefore, for k = 2, . . . , n − 1, the k-Hessian operator can be regarded as a sequence of nonlinear partial differential operators linking the classical Laplace operator to the real Monge-Ampère operator. The natural progression is then to extend the set of k-admissible functions together with the real k-Hessian operator. This was done in the famous trilogy written by Trudinger and Wang [41,43,44] (especially [43]). For k = 1, .., n, the k-Hessian integral is formally defined as When k = 1 we see that I 1 (u) = Ω |Du| 2 is the Dirichlet energy integral from potential theory that goes back to the work of Gauß, Dirichlet, Riemann, among many others, while for k = n I n (u) = Ω (−u) det D 2 u is the fundamental integral in the variational theory for the real Monge-Ampère equations (see e.g. [8,9,10,11,22]). The k-Hessian integral was introduced by Chou [21]. For further information about the k-Hessian integral see e.g. [49]. Now let 0 ≤ l < k ≤ n, and let Ω ⊂ R n , n ≥ 2, be a smoothly bounded (k − 1)convex domain, and let u be an k-admissible function that vanishes on ∂Ω. Then there exists a constant C(l, k, n, Ω) depending only on n, l, k and Ω such that , and this can be interpreted as a type of the classical Poincaré inequality and therefore motivates calling (1.1) a Poincaré type inequality for k-Hessian operators. Inequality (1.1) was first proved by Trudinger and Wang [42] (for an alternative proof see [29]). Under the same requirements on Ω and u the Sobolev type inequality that is of our interest states then that there exists a constant C(k, n, Ω) depending only on k, n and Ω such that: (2) if k = n 2 , then u L q ≤ C(k, n, Ω) I k (u) 1 k+1 , for q < ∞; If k = 1, then we have and for k = n, The Sobolev type inequalities (1)-(3) for n-admissible functions was first proved by Chou [21], while for the general case they were proved by Wang [48] (see also [40]). Now to the complex setting. Let n ≥ 2 and 1 ≤ m ≤ n. Mimicking the real case above we say that a C 2 -function u defined in a bounded domain in C n is msubharmonic or m-admissible if the elementary symmetric functions are positive σ l (λ(u)) ≥ 0 for l = 1, . . . , m, where this time λ(u) = (λ 1 , . . . , λ n ) are eigenvalues of the complex Hessian matrix D 2 C u = [ ∂ 2 u ∂zj ∂z k ]. The complex m-Hessian operator on a C 2 -function u is then defined by . In the complex case we get that the complex 1-Hessian operator is the classical Laplace operator defined on 1-subharmonic functions that are just the subharmonic functions, while the complex n-Hessian operator is the complex Monge-Ampère operator defined on n-subharmonic functions that is the plurisubharmonic functions. An early encounter of the complex m-Hessian operator is the work of Vinacua [45] from 1986. That work was later published in article form in [46]. The extension of m-subharmonic functions and the complex m-Hessian operator to non-smooth admissible functions was done by Błocki in 2005 ([16]). There he also introduced pluripotential methods to the theory of complex Hessian operators. Standard notations and terminology in the real and complex case differ in part, and so instead of I k above, we shall use the following notation in the complex case: For p > 0, p ∈ R, and m = 1, .., n, let and we call e p,m (u) for the (p, m)-energy of u. Thus, for k = 1 we have that e 1,1 (u) = I 1 (u), but notice the difference in the definition of e 0,1 (u) compared to I 0 (u). For the early work on the theory of variation for the complex n-Hessian operator see e.g. [12,13,20,26,27,31].
To be able to prove the Poincaré-and Sobolev-type inequalities for m-subharmonic functions we need classes of m-subharmonic functions that, in a general sense, vanishes on the boundary and additionally they should have finite (p, m)-energy. Denote these classes with E p,m (Ω) (see Section 2 for details).
Our Poincaré type inequality for the complex m-Hessian operator is: Let n ≥ 2, 1 ≤ l < k ≤ n, and p ≥ 0. Assume that Ω is a bounded B k -regular domain in C n . Then there exits a constant C(p, l, k, n, Ω) > 0, depending only on p, l, k, n, and Ω, such that for any u ∈ E p,k (Ω) we have e p,l (u) where (dd c u) n is the standard notation for the complex Monge-Ampère operator in pluripotential theory. Furthermore, if p = 1, l = 1, and k = n, then we have that When Ω is assumed to have the stronger convexity property known as strongly k-pseudoconvexity, and p = 1, then inequality (1.2) was proved by Hou [29]. In Theorem 4.3 we find the optimal constant in (1.2) for the cases p = 0, and p = 1, in the case of the unit ball Ω = B.
Our Sobolev type inequality for complex m-Hessian equations is: Theorem 5.4. Let n ≥ 2, 1 ≤ m ≤ n, and p ≥ 0. Assume that Ω be a bounded mhyperconvex domain in C n . There exists a constant C(p, q, m, n, Ω) > 0, depending only on p, q, m, n, and Ω such that for any function u ∈ E p,m (Ω), and for 0 < q < (m+p)n n−m , we have u L q ≤ C(p, q, m, n, Ω)e p,m (u) For p = 0, we have for m = 1 and m = n, respectively, Furthermore, for p = 1, we have for m = 1 and m = n, respectively, For the complex n-Hessian operator with p = 1, inequality (1.3) was proved by Berman and Berndtsson in [15] (see also [28]). Later their result was generalized by the authors to the case when p is any positive number, and when Ω is a nhyperconvex domain in C n , or a compact Kähler manifold ( [3]). The case when Ω is assumed to have the stronger convexity assumption of strongly k-pseudoconvexity, and p = 1, then inequality (1.3) was proved by Zhou [50].
After proving Theorem 5. 4 we give examples that shows that the following inequalities are not in general possible: It is well known that all n-subharmonic functions are locally in L p for any p > 0. In general, this fact is no longer valid for m-subharmonic functions. Błocki proved that if u is m-subharmonic function, then u ∈ L p loc for p < n n−m . Motivated by the real case he then conjectured that any m-subharmonic function is in L p loc for p < nm n−m ( [16]). Later, Dinew and Kołodziej partially confirmed this conjecture under the extra assumption that the m-subharmonic functions ( [24]). For the relation of this conjecture with the so called integrability exponent, and the Lelong number, of m-subharmonic functions see [14]. As an immediate consequence of our Theorem 5.4 is that we get that Błocki's conjecture is true for functions in the Cegrell class E m (Ω) (Corollary 5.8). The inequalities under investigation are very helpful in solving the Dirichlet problem for the complex Hessian type equation, and the solution of those equations can be used for the construction of certain metrics on compact Kähler and Hermitian manifolds (see e.g. [15,28]). Furthermore, the optimal constant in these inequalities are connected to the isoperimetric inequality and therefore classically to symmetrization of functions (see e.g. [39]).
Both our proofs of Theorem 4.2, and Theorem 5.4, uses the theory of quasi-Banach spaces (Theorem 3.2).

Preliminaries
Here we give some necessary background. We start with the definition of msubharmonic functions and the m-Hessian operator. Let Ω ⊂ C n , n ≥ 2, be a bounded domain, 1 ≤ m ≤ n, and define C (1,1) to be the set of (1, 1)-forms with constant coefficients. With this set Definition 2.1. Let n ≥ 2, and 1 ≤ m ≤ n. Assume that Ω ⊂ C n is a bounded domain, and let u be a subharmonic function defined on Ω. Then we say that u is m-subharmonic if the following inequality holds Let σ k be k-elementary symmetric polynomial of n-variable, i.e., It can be proved that a function u ∈ C 2 (Ω) is m-subharmonic if, and only if, for all k = 1, . . . , m, and all z ∈ Ω. Here, λ 1 (z), . . . , λ n (z) are the eigenvalues of the complex Hessian matrix ∂ 2 u ∂zj ∂z k (z) . For C 2 smooth m-subharmonic function u, the complex m-Hessian operator is defined by where dV 2n is the Lebesgue measure in C n .
To be able to have sufficiently many m-subharmonic functions that vanishes in some sense on the boundary we need some suitable convexity condition on our underlying domain. In this paper we need m-hyperconvexity (Definition 2.2), and B m -regularity (Definition 2.3).
(2) B n -regular domains are B-regular domains from pluripotential theory, while B 1 -regular domains are regular domains in potential theory.
(3) Every B m -regular domain is m-hyperconvex. On the other hand, the bidisc D × D in C 2 is 2-hyperconvex, but not B 2 -regular while it is both 1hyperconvex and B 1 -regular. For proofs, and further information about these convexity notions see [5].
Next, we shall recall the function classes that are of our interest. As said in the introduction we shall use the following notations: and Ω H m (ϕ) < ∞ .
, that converges pointwise to u on Ω, as j tends to ∞, and sup j e p,m (ϕ j ) < ∞.
In [32,33], it was proved that for u ∈ E p,m (Ω) the complex Hessian operator, H m (u), is well-defined, and Theorem 2.5. Let n ≥ 2, 1 ≤ m ≤ n, and p > 0. Assume that Ω be a bounded m-hyperconvex domain in C n . For u 0 , u 1 , . . . , u m ∈ E p,m (Ω) we have where C ≥ 1 depends only on p, m, n and Ω.

quasi-Banach spaces
In this section we introduce the necessary background of the theory of quasi-Banach spaces to be able to prove Theorem 3.2 which subsequently will be used in both the proof of Theorem 4.2 and Theorem 5.4. Let X be a real vector space. We say that K is a cone in the vector space X if it is a non-empty subset of X that satisfies: (1) K + K ⊆ K , (2) αK ⊆ K for all α ≥ 0 , and It should be noted that in some texts the name proper convex cone is used instead. Furthermore, δK = K − K is vector subspace of X. Let us recall the definition of a quasi-norm and a quasi-Banach space.
Definition 3.1. A quasi-norm · 0 on a cone K is a mapping · 0 : K → [0, ∞) with the following properties: (1) x 0 = 0 if, and only if, x = 0; (2) tx 0 = t x 0 for all x ∈ K and t ≥ 0; (3) there exists a constant C ≥ 1 such that for all x, y ∈ K we have that The constant C in (3.1) is often refereed to the modulus of concavity of the quasinorm · . Now one can extend · 0 to the vector space δK by The classical Aoki-Rolowicz theorem for quasi-Banach spaces ( [7,37]) states that every quasi-normed space X is q-normable for some 0 < q ≤ 1. In other words, X can be endowed with an equivalent quasi-norm |||·||| that is q-subadditive, and therefore we can define the following metric d(x, y) = |||x − y||| q on X. The vector space X is called a quasi-Banach space if it is complete with respect to the metric d induced by the quasi-norm · . Note that it follows from the definition of quasi-norm that for any x 1 , . . . , x k ∈ δK holds (3. 2) The cone K in a vector space X generates a vector ordering defined on δK by letting x y whenever x − y ∈ K.
Theorem 3.2. Let X be a real vector space, K ⊂ X a cone, and let · 0 be a quasi-norm on K, such that (δK, · ) is a quasi-Banach space, and K is closed in δK. Assume that Ψ : X → [0, ∞] is a function that satisfies:

The following conditions are then equivalent:
(1) there exists a constant B > 0 such that for all x ∈ K holds (2) Ψ is finite on K.
Proof. The implication (1)⇒(2) is clear. To prove the opposite implication (2)⇒(1) we shall argue by contradiction. Assume that there does not exists any constant B as above. Therefore, by using homogeneity of Ψ we can assume that there exists a sequence x j ∈ K such that where C is the modulus of concavity of the quasi-norm · 0 . Let us define We shall prove that {y k } is a Cauchy sequence. By (3.2) we have that for k > l Therefore, there exists y ∈ δK such that y k → y, as k → ∞. But since the cone K is closed we get that y ∈ K.
On the other hand, by the same argument as above we get that for any m ∈ N we have and therefore by (3.3) and monotonicity of Ψ

This is impossible by our assumption.
Remark. Note that condition b) in Theorem 3.2 can be replaced by upper semicontinuity of Ψ.  Then for any v ∈ δE p,k (Ω) define It was proved in [1,34] that (δE p,k , · ) is a quasi-Banach space for p = 1, and a Banach space for p = 1. Furthermore, the cone E p,k (Ω) is closed in δE p,k (Ω).
Let µ be a positive Radon measure µ, and p > 0. Then we define The functional Ψ 1 satisfies conditions a) and b) in Theorem 3.2. This example will be used in our proof of the Sobolev type inequality (Theorem 5.4). In this special case Theorem 3.2 was proved by Cegrell, see [18], and Lu [32,33]. Inspired by Ψ 1 , we define for 1 ≤ l ≤ n the following: This functional, Ψ 2 , shall be used in the proof of the Poincaré type inequality (Theorem 4.2).
In this space one can consider the following functional: For p > 0, and a msubharmonic function u define The functional, Ψ 4 , satisfies conditions a) and b) in Theorem 3.2. In this special case Theorem 3.2 was proved in [4] in order to characterize E p,k (Ω)

A Poincaré type inequality in B k -regular domains
The aim of this section is to prove the Poincaré type inequality in B k -regular domains for k-subharmonic functions. First we need the following lemma.
Then we have is a finite measure, and therefore µ is also finite and u ∈ E 0 l (Ω). Hence, E 0 k (Ω) ⊂ E 0 l (Ω). Case p > 0: Assume that u ∈ E p,k (Ω). Then by definition there exists a decreasing sequence u j ∈ E 0 k (Ω) such that lim j→∞ u j = u and sup j e p,k (u j ) < ∞.
Case p = 0: Assume that u ∈ E 0,k (Ω). By definition there exists a decreasing sequence u j ∈ E 0 k (Ω) such that lim j→∞ u j = u and sup j e 0,k (u j ) < ∞.
Now to the proof of the Poincaré type inequality.
Theorem 4.2. Let n ≥ 2, 1 ≤ l < k ≤ n, and p ≥ 0. Assume that Ω is a bounded B k -regular domain in C n . Then there exits a constant C(p, l, k, n, Ω) > 0, depending only on p, l, k, n, and Ω, such that for any u ∈ E p,k (Ω) we have e p,l (u) Proof. Using the functionals Ψ 2 and Ψ 3 (from Example 3.3 and Example 3.4) the proof follows from Theorem 3.2 and Lemma 4.1.
Next, we shall determine the optimal constant in Theorem 4.2 for the unit ball in C n in the cases p = 0 and p = 1. Proof. Case p = 0: We shall start proving that there exists a constant C > 0 such that for any u ∈ E 0,m (B) it holds Set β = dd c (|z| 2 − 1), and note that |z| 2 − 1 is an exhaustion function for B. Then for any u ∈ E 0,m (B). We get by [30] Thus, This constant is optimal since we have equality in the Poincrè type inequality for the function |z| 2 − 1.
Case p = 1: As in the case above we shall set β = dd c (|z| 2 − 1), and note that |z| 2 − 1 is an exhaustion function for the unit ball B. Let u ∈ E p,k (B), p > 0. Then by using Hölder's inequality, Theorem 2.5 and integration by parts we get Remark. In [42], Trudinger and Wang used the real Hessian quotient operator S k S l to establish the optimal constant in the Poincaré inequality for the real Hessian operator. More precisely, they prove that the optimal constant is attained by the solution u 0 of the equation S k (u0) S l (u0) = 1. We suspect that this is also the case in the complex setting. With Theorem 4.3 in mind we suspect that the optimal constant is C(p, k, l, n, Ω) = (e p,k (u 0 )) 1 p+l − 1 p+k , where p > 0 and u 0 ∈ E p,k (Ω) is the unique negative k-subharmonic function such that H k (u 0 ) = H l (u 0 ). We refer to [25,38], and reference therein for results concerning such functions in Euclidean spaces as well as on compact manifolds.

A Sobolev type inequality in m-hyperconvex domains
Let us first recall the notion of m-capacity. Let n ≥ 2, 1 ≤ m ≤ n. For an arbitrary bounded domain Ω ⊂ C n , and for any K ⋐ Ω define The following lemma was proved by Dinew and Kołodziej [24]. Lemma 5.1. Let n ≥ 2, 1 ≤ m ≤ n, and let Ω ⊂ C n be a m-hyperconvex domain. Then for 1 < α < n n−m there exists a constant C(α) > 0 such that for any K ⋐ Ω, V 2n (K) ≤ C(α) cap α m (K). We will also need the following two lemmas. Proof. By [35,36] we have for any s, t > 0 Taking t = s we get Lemma 5.3. Let n ≥ 2, 1 ≤ m ≤ n, p ≥ 0, and assume that Ω ⊂ C n is a mhyperconvex domain. Then we have that E p,m (Ω) ⊂ L q (Ω), for any 0 < q < n(m+p) n−m . Proof. Assume first that u ∈ E 0 m (Ω), and let p ≥ 0. Let us define λ(s) = V 2n ({u < −s}).
Then by Lemma 5.1, and Lemma 5.2, we get that for 0 < α < n n−m λ(s) ≤ C 1 cap α m ({u < −s}) ≤ C 2 s −(m+p)α e p,m (u) α , where C 1 and C 2 are constants not depending on u. For q > 0 we then have where C 3 is a constant not depending on u. From (5.1) we have Ω (−u) q dV 2n < ∞ if, and only if, Next, if we take a function u ∈ E p,m (Ω), then there exists a decreasing sequence u j ∈ E 0 m (Ω) such that u j ց u and sup j e p,m (u) < ∞. By the first part of the proof there are constants A, B do not depending on u j such that u j L q ≤ A + Be p,m (u j ) α , and by passing to the limit we get Now we can state and prove the Sobolev type inequality in arbitrary m-hyperconvex domains.
Theorem 5.4. Let n ≥ 2, 1 ≤ m ≤ n, and p ≥ 0. Assume that Ω be a bounded mhyperconvex domain in C n . There exists a constant C(p, q, m, n, Ω) > 0, depending only on p, q, m, n, and Ω such that for any function u ∈ E p,m (Ω), and for 0 < q < Proof. This follows from Lemma 5.3 and Theorem 3.2.
We now give examples that shows that the following inequalities are not in general possible:  Then we have Hence, if β > α p+m p , then e p,m (u) = c(n, m) 1 j α(m+p) (j β − 1) p → ∞, as j → ∞, and c(n, m) = 2π n ( n m − 1) m m!(n − m)! (see [47] for details).
On the other hand, one can check that if 0 < q < Hence, a contradiction is obtained. Thus, we can not in general have u L ∞ ≤ Ce p,m (u) 1 m+p .
Example 5.7. Similarly as before we consider the following functions defined on the unit ball B in C n u j (z) = j max 1 − |z| 2− 2n m , − 1 j .
Then we have that u j L ∞ = −u j (0) = 1, but at the same time This shows that we can not in general have e p,m (u) 1 n+p ≤ C u L ∞ .
As an immediate consequence of Theorem 5.4 is that we get that Błocki's integrability conjecture is true for functions in the Cegrell class E m (Ω) (Corollary 5.8). Before stating this result let us recalling the definition of E m (Ω). Let Ω be a bounded m-hyperconvex domain in C n . We say that u ∈ E m (Ω) if for any ω ⋐ Ω there exists u ω ∈ E 0,m (Ω) such that u = u ω on ω.