3.2. Hölder continuity [02DV]

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3.2. Hölder continuity

Proposition 3.3.

Assume $\omega_{\varphi}^{n}=f\omega^{n}$ , $\omega_{\psi}=g\omega^{n}$ , where $\varphi,\psi\in PSH(X,\omega)$ are continuous and $f,g\in L^{p}(\omega^{n})$ , $p>1$ . Then for all $0<\gamma<2/(2+nq)$ ,

||\varphi-\psi||_{L^{\infty}(X)}\leq C||\varphi-\psi||_{L^{2}(\omega^{n})}^{\gamma},

where $q=p/(p-1)$ denotes the conjugate exponent to $p$ .

Proof.

Fix $\varepsilon>0$ and $\alpha>0$ to be chosen later. It follows from (2) and propositions 2.5, 3.1 that

||\varphi-\psi||_{L^{\infty}(X)}\leq\varepsilon+C_{1}\left[Cap_{\omega}(|\varphi-\psi|>\varepsilon)\right]^{\alpha/n}.

Applying the refined version of lemma 2.2 which involves the uniform bound on $||\varphi||_{L^{\infty}(X)},||\psi||_{L^{\infty}(X)}$ (see inequality (3)), we obtain

Cap_{\omega}(|\varphi-\psi|>\varepsilon)\leq\frac{C_{2}}{\varepsilon^{n+2/q}}\int_{X}|\varphi-\psi|^{2/q}(f+g)\omega^{n}.

It follows thus from Hölder’s inequality that

Cap_{\omega}(|\varphi-\psi|>\varepsilon)\leq\frac{C_{3}||f+g||_{L^{p}}}{\varepsilon^{n+2/q}}\left[||\varphi-\psi||_{L^{2}(\omega^{n})}\right]^{2/q}.

Choose now $\varepsilon:=||\varphi-\psi||_{L^{2}}^{\omega}$ where $0<\gamma<2/(2+nq)$ . Then

Cap_{\omega}(|\varphi-\psi|>\varepsilon)\leq C_{4}\left[||\varphi-\psi||_{L^{2}}\right]^{2/q-\gamma(n+2/q)}.

We infer

||\varphi-\psi||_{L^{\infty}(X)}\leq||\varphi-\psi||_{L^{2}}^{\gamma}+C_{5}||\varphi-\psi||_{L^{2}}^{\gamma^{\prime}},\;\text{ where }\gamma^{\prime}=\frac{\alpha}{n}\left[2/q-\gamma(n+2/q)\right].

We finally choose $\alpha>0$ so large that $\gamma\leq\gamma^{\prime}$ and adjust the value of the constant $C$ : this yields the desired estimate. ∎

Being able to control the $L^{\infty}$ -norm of $\varphi-\psi$ by its $L^{2}$ -norm is a powerful tool. If for instance $\psi=\varphi_{j}$ , $\varphi$ satisfy the assumptions of proposition 3.2 – with $\varphi_{j}$ being uniformly bounded –, and $\varphi_{j}\rightarrow\varphi$ in $L^{1}$ , then $\varphi_{j}\rightarrow\varphi$ in $L^{2}(\omega^{n})$ , hence $(\varphi_{j})$ actually uniformly converges towards $\varphi$ . This yields the continuity of the map

f\in L^{p}(\omega^{n})\mapsto\varphi\in{\mathcal{C}}^{0}(X),

where $\varphi$ is the unique $\omega$ -psh solution to $(\omega+dd^{c}\varphi)^{n}=f\omega^{n}$ , $\sup_{X}\varphi=-1$ . Thus Theorem A is proved.

We now give an application of this estimate, which is new even when the form $\omega$ is Kähler, but requires the manifold $X$ to be homogeneous, i.e. such that its group of holomorphic automorphisms acts transitively on it.⁴⁴ 4 In particular, the cohomology class of $\omega$ is Kähler and $\omega$ itself can be supposed to be Kähler without loss of generality..

Theorem 3.4.

Assume $X$ is a homogeneous manifold. If $\mu=f\omega^{n}$ is a probability measure with density $0\leq f\in L^{p}(\omega^{n})$ , $p>1$ , then the unique solution $\varphi\in PSH(X,\omega)\in{\mathcal{C}}^{0}(X)$ to the normalized Monge-Ampère equation

(\omega+dd^{c}\varphi)^{n}=\mu=f\omega^{n},\,\sup_{X}\varphi=-1,

is Hölder continuous of exponent $\gamma>0$ , for all $\gamma<2/(2+nq)$ , where $q=p/(p-1)$ is the conjugate exponent to $p$ .

Proof.

When $Aut(X)$ , the group of holomorphic automorphisms of $X$ , acts transitively on $X$ , one can regularize $\omega$ -psh functions by averaging over the Haar measure of the connected component of the identity of $Aut(X)$ . This is very similar to the way one regularizes psh functions in $\mathbb{C}^{n}$ by using convolutions with an approximation of the identity for the convolution product. We refer the reader to [Hu] and the Appendix of [G] for more details.

Let $\varphi_{h}$ be the $\omega$ -psh function which is the translate of $\varphi$ by an automorphism which is at distance $h$ from identity. We use the notation $\varphi_{h}$ by analogy with the $\mathbb{C}^{n}$ -situation, where $\varphi_{h}(x)=\varphi(x+h)$ . Since $\varphi$ is bounded, it has gradient in $L^{2}$ , hence

||\varphi_{h}-\varphi||_{L^{2}}\leq C|h|,

by using Cauchy-Schwarz inequality in a local chart. We can thus apply proposition 3.2 to obtain that

||\varphi_{h}-\varphi||_{L^{\infty}}\leq C^{\prime}|h|^{\gamma},

for all $\gamma<2/(2+nq)$ . Since $\varphi_{h}(x)\simeq\varphi(x+h)$ in a local chart, this precisely means that $\varphi$ is Hölder-continuous of exponent $\gamma$ . ∎