Continuous solutions to Monge-Amp\`ere equations on Hermitian manifolds for measures dominated by capacity

We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge-Amp\`ere equation on a compact Hermitian manifold for a very general measre on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh-Nguyen-Sibony. As a consequence, we give a characterization of measures admitting H\"older continuous quasi-plurisubharmonic potential, inspired by the work of Dinh-Nguyen.


introduction
Let (X, ω) be a compact Hermitian manifold of dimension n. The study of the complex Monge-Ampère equation in this setting was initiated by Cherrier [Ch87], and the counterpart of the Calabi-Yau theorem [Yau76] on compact Hermitian manifolds was proven by Tosatti and Weinkove [TW10b]. Later Dinew and the authors, in a series of papers [DK12], [KN15,KN17,KN19a], obtained weak continuous solutions for more general densities on the right hand side of the equation, by extending the pluripotential methods employed before on the Kähler manifolds. In this paper we deal with yet more general measures on the right hand side.
If ω is Kähler, then the first named author obtained in [Ko98,Ko03] the unique continuous ω-plurisubharmonic (ω-psh for short) solution to the complex Monge-Ampère equation with the right hand side being a measure in one of the classes F (X, h) satisfying a bound in terms of the Bedford-Taylor capacity and a weight function h (the precise definition is given in the next section). We prove here the generalization of this result to Hermitian manifolds.
Theorem 1.1. Let µ ∈ F (X, h) be such that µ(X) > 0. Then, there exists a continuous ω-psh function u and a constant c > 0 solving the equation (ω + dd c u) n = c µ.
If we assume further that the right hand side is strictly positive and absolutely continuous with respect to the Lebesgue measure, then we prove a stability of solutions and their uniqueness extending the main theorem of [KN19a], (Theorem 4.1). Our method is adaptable to the Monge-Ampère type equations [Ng16]. As a consequence, we get the existence and uniqueness of continuous ω-psh solutions of these equations (Corollary 3.3).
The families of measures which belong to F (X, h), for some h, include those having densities in L p , p > 1, or even broader Orlicz spaces, but also measures singular with respect to ω n , for instance smooth forms on totally real submanifolds (see e.g. [Ko98], [BJZ], [Vu16]). We shall distinguish classes H(τ ) which are unions (over C > 0) of F (X, h 1 ) with h 1 (x) = Cx nτ and fixed τ > 0; and F (X, h 2 ) with h 2 (x) = Ce αx for some C, α > 0. The latter was introduced by Dinh, Nguyen and Sibony [DNS10], who called the measures in this class (the union over C > 0, α > 0) moderate. They proved that any measure locally dominated by the Monge-Ampère measure of a Hölder continuous psh function is moderate.
Later, Dinh and Nguyen [DN14] characterized the measures locally dominated by the Monge-Ampère measure of a Hölder continuous psh function via the associated functionals acting on P SH(ω) (the set of all ω-psh functions) when ω is Kähler. In the last section we give a similar description in the Hermitian setting. Let us define Let µ be a positive Radon measure on X andμ : P SH(ω) → R the associated functional given byμ Theorem 1.2. The measure µ belongs to H(τ ) andμ is Hölder continuous with respect to L 1 -distance on S if and only if there exists a Hölder continuous ω-psh function u and a constant c > 0 solving (ω + dd c u) n = c µ.
Notice that the Hölder continuity ofμ on the larger subset {v ∈ P SH(ω) : sup X v = 0} implies the H(τ ) property and the Hölder continuity on S (Propostion 5.5). The latter properties are independent. The examples [DDGHKZ, Example 5.5] or [DN14, Example 2.5] belong to H(τ ) for every τ > 0, but they do not admit Hölder continuous potentials. On the other hand, the well-known conjecture of Dinh, Nguyen, Sibony [DN14, Problem 1.5] predicted that the moderate property implies the Hölder continuity of the Monge-Ampère potential or equivalently the Hölder continuity on S of the functional associated to this measure.
As it was shown in [KN17] (inspired by [DDGHKZ]) the existence of Hölder continuous solution is a local problem. We apply Theorem 1.2 to get main results of [Ph10] and [Vu16] in the Hermitian setting. For example, this gives a Hölder continuous ω-psh potential for a smooth volume form of a compact smooth real hypersuface in X.
Let us indicate some motivations behind the study of the Monge-Ampère on Hermitian manifolds with measures on the right hand side. Unlike in the Kähler case, one solves the equation not only for a function but also for a constant on the right hand side. The range of those constants for a given manifold seems to have a geometrical meaning. It comes up in constructions of ω -psh functions with logarithmic poles like in [TW12], [Ng16] (where one solves the equation for approximants of Dirac measures); in connection to problems involving holomorphic Morse inequalities (see [KT19]), and others. The parabolic Monge-Ampère on a Hermitian manifold, the Chern-Ricci flow, is recently intensively studied (see [Gil11,Gil13], [FTWZ16], [Ni17], [TW13,TW15] [TWY15], [Zh17]). The flow is expected to play an important role in the classification of complex surfaces. In the context of parabolic equations the pluripotential estimates are also useful. For example, To [To18] (independently, Nie [Ni17] in particular cases) used results in [DK12] and [Ng16] to prove a conjecture by Tosatti and Weinkove [TW13]. The geometric applications of pluripotential theory on Hermitian manifolds are discussed at length in surveys by Dinew [Di16,Di19].
Another new topic is the complex dynamics on compact Hermitian manifolds. There the measures having interesting properties are often singular with respect to the volume form. In a recent paper Vu [Vu19] showed that for any holomorphic dominant endomorphism f of X there exists an equilibrium measure µ f associated to f . The understanding of this measure is a central problem in complex dynamics (as in the Kähler setting). By [Vu19, Theorem 1.1] and Theorem 1.2 one gets that µ f admits a Hölder continuous ω-psh potential. We refer to [DF18] for results on the push-forwards of measures by dominant meromorphic maps between complex manifolds.
Acknowledgement. The authors are partially supported by NCN grant 2017/ 27/B/ST1/ 01145. The second author is also partially supported by the start-up grant G04190056 of KAIST.

preliminaries
In this section we recall and extend some results from [KN15,KN19a,KN16b]. Their statements are often more technical than the counterparts in the Kähler setting [Ko05].
Let h : R + → (0, ∞) be an increasing function such that In particular, lim x→∞ h(x) = +∞. Such a function h is called admissible. In what follows we often omit to stress that h is admissible. If h is admissible, then so is .
Recall that the analogue of Bedford-Taylor capacity on compact complex manifolds is where P SH(ω) is the set of ω-psh functions on X and ω n w := (ω + dd c w) n . This capacity is equivalent to the Bedford-Taylor capacity [BT82] defined locally (see [Ko05,).
Let µ be a positive Radon measure satisfying for any Borel set E ⊂ X and some F h . Let us denote by F (X, h) the set of all measures that are dominated by the capacity cap ω in the sense of (2.3) for some admissible h.
Some particular families of measures which satisfy (2.3) were mentioned in Introduction. Another fairly general family is given in the following example. Note that these measures are often singular with respect to the Lebesgue measure and their potentials may not be Hölder continuous.  Then, µ ∈ F (X, h) for some admissible function h.
Let us fix a finite covering of X: is the coordinate ball centered at x j of radius s > 0. Take s so small that B(x j , 3s), j ∈ J, are still coordinate balls. Let χ j be the partition of unity subordinate to {B j (s)} j∈J }. The first observation is that if µ satisfies (2.3) on X, then in each chart B j (3s) the same property holds for subsets of the smaller ball.
for an admissible function h 0 depending only on h, ω, X and Ω, where cap(K, Ω) is the relative capacity of Bedford and Taylor [BT82].
Proof. The proof follows by the monotonicity of h and the fact that where C 1 is a uniform bound for plurisubharmonic functions on B(x j , 3s) such that v j = 0 on ∂Ω and dd c v j ≥ ω in Ω (see [Ko05,page 53]). Thus, we can take The proof is completed.
The second observation is the following.
Let ρ ε be the standard smoothing kernel on B(0, 3s). Then, is the sequence of smooth measures which converge weakly to µ as ε tends to 0. Moreover, µ ε ∈ F (X, h 0 ) for an admissible function h 0 when ε is small enough.
Proof. Since the cover is finite, it is enough to show that each smooth measure of the right hand side belongs to F (Ω, h 0 ) for an admissible function h 0 . By Lemma 2.2 it follows that µ Uj ∈ F (Ω, h 0 ). Thanks to [Ko98, Eq.(3.5.1)] the convolutions with smoothing kernels preserve the inequality (2.6) when ε is small enough.
where C is a dimensional constant.
Proof of Lemma 2.6. By [KN15, Lemma 5.4] The lemma follows from the assumption on the measure ω n ϕ = µ.
Notice that we used the fact that (s) is also increasing. Hence |S| ≤ Cε with C = C(ω). Thus, This is the desired stability estimate.
There is always a uniform lower bound for the volume of Monge-Ampère measures dominated by capacity. This is essentially [KN19a, Proposition 2.4].
Proposition 2.7. Consider µ ∈ F (X, h) such that µ(X) > 0. Suppose w ∈ P SH(ω) ∩ C(X) and c > 0 solve Then there exists a constant V min > 0 depending only on X, ω, h such that whenever Proof. Suppose c ≤ 2 n . We shall see that this leads to a contradiction for some positive V min . Firstly, we have ω n w ≤ 2 n µ. Therefore, the Monge-Ampère measure ω n w satisfies the inequality (2.3) for the admissible function h(x)/2 n . The inequality (2.11) for 0 < t ≤ 1 3 min{ 1 2 n , 1 2 7 B } then gives: where S = inf X w and C > 0 depends only on n, B. It implies that The formula (2.9) for the function κ 0 (x) corresponding to ω n w is It is defined on (0, cap ω (X)). Since κ 0 (x) is an increasing function it has the inverse 0 (x). It follows from (2.8) that for 0 < t ≤ 1 3 min{ 1 2 n , 1 2 7 B } we have 0 (t) ≤ cap ω ({w < S + t}). Coupling this with (2.15) we obtain Then, (2.16) and the above choices lead to a contradiction Thus the proposition is proven.

Existence of continuous solutions
In this section we generalize the results of [KN15,KN17] on the existence of continuous solutions of the Monge-Ampère equation. This is also the extension of [Ko98,Ko05] from Kähler to Hermitian setting. We prove the first theorem in the introduction.
Theorem 3.1. Let µ ∈ F (X, h) be such that µ(X) > 0. Then, there exists a continuous ω-psh function u and a constant c > 0 solving the equation Proof. The proof follows the scheme of the one in [KN17, Theorem 1.3]. We only clarify the differences. Let µ ε be the approximating sequence from Lemma 2.3. By [KN15, Theorem 0.1] there exist u ε ∈ P SH(ω) ∩ C 0 (X) and a constant c ε > 0 solving The main difficulty lies in proving the uniform upper bound for constants {c ǫ } which requires a bit different approach compared to [KN15,KN17]. Since lim ε→0 µ ε (X) = µ(X), there exist a ball U = B(a, s) ⊂ U ′ = B(a, 2s) in the finite open cover (2.5) and a positive constant C 1 > 0 such that for every small ε > 0, where we recall µ U is the restriction of χ a µ to U , and χ a is the smooth function in the partition of unity subordinate to {B(x j , s)}. Let us denote Ω = B(a, 3s). Thanks to [CKNS85] there is v ε ∈ P SH(Ω) ∩ C ∞ (Ω) such that (dd c v ε ) n = µ U * ̺ ε + εω n and v ε = 0 on ∂Ω. By Lemma 2.3 and [Ko96] it follows that It is clear that µ ε ≥ µ U * ̺ ε on Ω. Let us write µ U * ̺ ε = R ε ω n for a smooth function R ε in Ω. Using the mixed forms type inequality [KN19a, Lemma 2.2] we have Fix a strictly plurisubharmonic function ρ Ω in Ω such that ω ≤ dd c ρ Ω . Then the Demailly's version of the Chern-Levine-Nirenberg inequality [De85] gives Notice that X |u ε |ω n is uniformly bounded (see e.g. [DK12, Proposition 2.5]). These combined with (3.2) and (3.3) give the uniform upper bound for {c ε } ε>0 . The uniform lower bound of this sequence follows from [KN15, Lemma 5.9] as µ ε (X) is uniformly bounded. By the proof of [KN15, Corollary 5.6] it follows u ε L ∞ (X) < C. Now we continue as in the proof of [KN17, Theorem 1.3]. Since the sequence {u ε } ε>0 normalized by sup X u ε = 0 is a compact subset of L 1 (X), passing to a subsequence, we may assume that (3.4) u ε −→ u in L 1 (X); moreover, u ∈ P SH(ω) ∩ L ∞ (X) and also lim ε→0 c ε = c > 0. We wish to apply Proposition 2.5 to conclude that the convergence (3.4) is in C 0 (X). This amounts to showing that Finally, we get that u ε converges to u in C 0 (X), which is a solution to ω n u = cµ.
Corollary 3.2. Suppose that µ j ∈ F (X, h) and it is smooth for every j ≥ 1. Let µ j converge weakly to µ ∈ F (X, h) as j → +∞. For j ≥ 1 let us solve (ω + dd c v j ) n = e vj µ j .
Then v j converges uniformly to a continuous ω-psh function v as j → +∞. Consequently, v is the unique continuous ω-psh solution to ω n v = e v µ.  (ω + dd c v) n = e λv µ.
Proof. It is a simple application of Corollary 3.2 for the approximating sequence µ ε from Lemma 2.3.

stability of solutions
We prove a stability estimate for measures belonging to F (X, h) which are strictly positive, absolutely continuous with respect to the Lebesgue measure. We use the following notation: the L p -norms for 0 < p < ∞ are Theorem 4.1. Assume f, g ∈ L 1 (X) and f ω n , gω n ∈ F (X, h). Consider two bounded ω-psh solutions u, v of ω n u = f ω n , ω n v = gω n with sup X u = sup X u = 0. Suppose that f ≥ c 0 > 0. Fix γ > 2 + n(n + 1). Then, We will adapt the proof of [KN19a, Theorem 3.1] with necessary changes. First, it is enough to assume that f, g are smooth.
Lemma 4.2. Let f j , g j ∈ F (X, h) be smooth sequences of functions converging in L 1 (X) to f, g respectively. Let u j , v j ∈ P SH(ω) ∩ C ∞ (X) be such that u j ց u and v j ց u as j → +∞. Assume ϕ j , ψ j solve (ω + dd c ϕ j ) n = e ϕj−uj f j ω n , (ω + dd c ψ j ) n = e ψj−vj g j ω n .
Proof. We use the argument of [KN19a, Remark 3.11] pointed out by a referee of that paper. By Corollary 3.2 the sequence {ϕ j } converges uniformly to the solution u 0 of (ω + dd c u 0 ) n = e u0 e −u f ω n . It follows from the uniqueness of u that u 0 = u. Similarly, {ψ j } converges uniformly to v. The conclusion follows.
Proof of Theorem 4.1. We fix the notation as in the proof of [KN19a, Theorem 3.1]. For t ∈ R define We need to replace [KN19a, Lemma 3.4] by the following statement. The proof is similar up to some technicalities. For the reader's convenience we give all details here.

It follows that
Next, we construct a barrier function by putting for z ∈ X \ Ω(t 1 ).
If 2ε α ≤ s ≤ 1, then Let us use the notation m s : Then m s (τ ) ≤ m s . By the above definitions we have We are going to show that for s = 2ε α and τ = ε α /2. Suppose it was false. By (4.6) we have for 0 < t < t1−t0 2 . To go further we need to estimate the integrals: f ω n for 0 < t << s, τ . By the modified comparison principle [KN15, Theorem 0.2] for every 0 < t < min{ τ 3 16B , t1−t0 2 }. Hence, a simple estimate from below gives Using (4.5) for s = 2ε α we get f ω n .

The Dinh-Nguyen theorem on Hermitian manifolds
In this section we give a characterization of measures leading to Hölder continuous solutions of the Monge-Ampère equation on compact Hermitian manifolds, which is an analogue of the Dinh-Nguyen theorem [DN14]. If ω is Kähler, [DN14] says that a positive Radon measure admits a Hölder continuous ω-psh potential if and only if the associated functional is Hölder continuous on {w ∈ P SH(ω) : sup X v = 0} with respect to the L 1 -distance. Let us denote S = S(ω) := u ∈ P SH(ω) : −1 ≤ u ≤ 0, sup X u = 0 .
The L 1 -distance, with respect to the Lebesgue measure, between u, v ∈ P SH(ω) is given by A measure µ gives the natural functionalμ : P SH(ω) → R defined bŷ Following Dinh-Nguyen [DN14] we say that Definition 5.1.μ is Hölder continuous on S if it is Hölder continuous with respect to the L 1 distance.
In other words there exist a uniform exponent α > 0 and a uniform constant C > 0 such that for every u, v ∈ S, Since max{u, v} ∈ S for every u, v ∈ S, this inequality is equivalent to We are going to show that the Hölder continuity property on S is local. Let Ω be a strictly pseudoconvex domain in C n and define The L 1 distance (with respect to the Lebesgue measure) between ϕ, ψ ∈ S 0 is defined similarly: Let ν be a positive Borel measure on Ω. It also gives a natural functionalν on P SH(Ω) defined byν (ϕ) = Ω ϕdν.
Definition 5.2.ν is locally Hölder continuous on S 0 (Ω) if for a fixed Ω ′ ⊂⊂ Ω, there exists a constant C = C(Ω ′ , Ω) > 0 and an exponent α > 0 such that for every ϕ, ψ ∈ S 0 (Ω) Proof. Suppose thatμ is locally Hölder continuous on each local coordinate chart. Let u, v ∈ S. We wish to show that there exist C, α > 0 such that Let B(a, r) be a local coordinate ball in the finite covering (2.5). Let ρ be a strictly plurisubharmonic function on U := B(a, 2r) such that dd c ρ ≥ ω. Define ϕ := u + ρ, ψ := v + ρ. By local Hölder continuity ofμ we have Summing up over all j ∈ J of the cover, we get thatμ is Hölder continuous on S.
For the reverse direction, assume now thatμ is Hölder continuous S. Let B(a, r), U be the coordinate balls above. Take ϕ, ψ ∈ S 0 (U ). Let χ be a ωpsh function on X such that χ = 0 outside U and χ ≤ −3δ on B(a, r) for some andψ analogously. Then, using the assumption Note that on U we have |φ −ψ| ≤ δ|ϕ − ψ|. It follows that This is the local Hölder continuous property ofμ on U .
There are plenty of examples of measures which are locally Hölder continuous on S 0 (Ω) (see [Ng17,Ng18]). We give below a sufficient condition. Let us consider the class Then, the Hölder continuity of a functional on E ′ 0 (Ω) is considered with respect to L 1 -distance [Ng17, Definition 2.3].
Let us consider the following classes of measures: H(τ ) = {µ ∈ F (X, h 1 ) : h 1 (x) = C 1 x nτ for some C 1 , τ > 0} , and the moderate measures, which by definition, are in F (X, h 2 ) with h 2 (x) = C 2 e αx for some C 2 , α > 0. For the latter the stability estimate of its potential has a nicer form, i.e., the function defined in (2.10) is Γ(s) = Cs α with α > 0.
We observe that the proof of [DN14, Proposition 4.4] holds true for a general Hermitian metric ω. This gives a sufficient condition for moderate measures.
Another sufficient condition for a measure to be moderate, due to Dinh, Nguyen and Sibony [DNS10], is as follows.
Lemma 5.6. If there exists a Hölder continuous ω-psh function ϕ and a constant C > 0 such that µ ≤ Cω n ϕ , then µ is moderate andμ is Hölder continuous on S. Proof. These properties are local by [KN17, Lemma 1.2] and Lemma 5.3. Therefore, we only prove them in a local coordinate chart. Let U := B(x, r) ⊂ Ω := B(x, 2r). Then, we can assume µ is compactly supported in U and µ ≤ (dd c ϕ) n for some Hölder continuous plurisubharmonic function ϕ in Ω. By [DNS10, Corollary 1.2] (see also [Ng17, Lemma 2.7, Proposition 2.9]) we get that µ is moderate andμ is Hölder continuous on E ′ 0 (Ω). Thus, it is also Hölder continuous on S. Remark 5.7. If ω is Kähler, then under the assumption of the lemma µ is indeed Hölder continuous on {v ∈ P SH(ω) : sup X v = 0}. However, due to the torsion terms dd c ω and dω ∧ d c ω in the general Hermitian case, it seems the Hölder continuity only holds on the smaller set S.
We are ready to prove the Dinh-Nguyen type characterization on Hermitian manifolds.
Theorem 5.8. A positive Radon measure µ belongs to H(τ ) andμ is Hölder continuous on S if and only if there exists a Hölder continuous ω-psh function u and a constant c > 0 such that (5.6) (ω + dd c u) n = c µ.
Proof. The second condition implies the first by Lemma 5.6. It remains to show the reverse direction. The last theorem allows to extend results of Pham [Ph10] and Vu [Vu16] from the Kähler to the Hermitian setting.
Proof. By Theorem 5.8 it is sufficient to show that f dµ belongs to H(τ ) for some τ > 0 and that the corresponding functional is Hölder continuous on S. These properties are local. We may assume that supp µ ⊂ U := B(a, r) ⊂ Ω := B(a, 2r) in C n . By [Ng17, Lemma 2.15, Corollary 2.14] it follows thatμ is Hölder continuous on E ′ 0 (Ω), then so is the functional of f dµ. Finally, by [Ng17, Propositon 2.9] we have that f dµ is moderate.
One example of measures satisfying the assumption of the proposition above is given by the smooth volume form of a smooth hypersurface as in Pham [Ph10].
Corollary 5.10. Let S be a compact smooth real hypersurface in X and dV S is its smooth volume form. Then, for every 0 ≤ f ∈ L p (S, dV S ) with p > 1 and S f dV S > 0, there exist a constant c > 0 and a Hölder continuous ω−psh function u solving (ω + dd c u) n = cf dV S .
Later on, Vu [Vu16] proved the result for a generic CR immersed C 3 −submanifold of X. The Kähler assumption in his paper is needed only to use the characterization of [DN14]. Given our results above we get immediately the statement of his result in the Hermitian setting. Actually, we can also simplify a bit his arguments by using the local Hölder continuity criterion (Lemma 5.3).