Regularity of Geodesics in the Spaces of Convex and Plurisubharmonic Functions II

Theorem 0.1. Let the convex function u solve the Dirichlet problem (0.1) as above. Then u is globally Lipschitz and $C_{loc}^{1,1}$ regular. A blow-up of the $C^{1,1}$ norm is only possible at the corners $\partial U \times \{0,1\}$ . Furthermore the solution is $C^{\infty}$ smooth if and only if the gradient images of $u_0$ and $u_1$ agree i.e. $$\partial u_0(U) = \partial u_1(U).$$ In the current note we investigate what happens if we modify somewhat the above setting. First of all in the case of the space S the assumed strict convexity is stronger than the usual one. Typically one works with the local strict convexity i.e. with the space $$\tilde{\mathcal{S}} := \{ u \in C^{\infty}(U) \cap C(\overline{U}) | D^2 u > 0 \text{ on } U, \ ||u||_{C^2} < \infty, u = 0 \text{ on } \partial U \}. \tag{0.2}$$ The reason we have worked with S in [AD21] is that various implicit function arguments were applied and, unless we control the constants involved, all estimates seemed to break close to the boundary. This is also the reason for imposing finite $C^2$ norm in the current note. Thus, roughly speaking, an element in $\tilde{S}$ can have minimal eigenvalue tending to zero as the base point goes to the boundary, but the largest eigenvalue is bounded from above. Note however that the construction described above can also be repeated for $\tilde{\mathcal{S}}$ with the same geodesic equation - the only necessary change is that now the tangent space $T_u\tilde{\mathcal{S}}$ should be the space of test functions $C_0^{\infty}(U,\mathbb{R})$ . Hence a question appears whether the findings from [AD21] remain true in the space $\tilde{\mathcal{S}}$ . Our first observation is that this is the case.

Theorem 0.2. Let the convex function u solve the Dirichlet problem (0.1) as above with $u_0, u_1 \in \tilde{\mathcal{S}}$ . Then u is globally Lipschitz and $C_{loc}^{1,1}$ regular. It is globally $C^{1,1}$ smooth in $U \times (0,1)$ if and only if the gradient images of $u_0$ and $u_1$ agree. A second issue we deal in this note is the optimal regularity for the solution of the geodesic equation when less smoothness is initially assumed. To this end we need some definitions. For any $y \in U$ let $l_y(x)$ denotes an affine support function of u at x. i.e. $u(x) \ge l_y(x)$ throughout U with an equality at y. If u is differentiable at y then obviously $l_y(x)$ is unique and is given by $l_y(x) := u(y) + Du(y) \cdot (x - y)$ .

Theorem 0.5. Let u solve the Dirichlet problem (0.1) as above with $u_0, u_1 \in \tilde{\mathcal{S}}^{1,\alpha}$ for some fixed $\alpha \in (0,1]$ . Then $u \in C^{1,\alpha}_{loc}(U \times (0,1))$ . A blow-up of the $C^{1,\alpha}$ norm is only possible at the corners $\partial U \times \{0,1\}$ . In [Abj19] and [Ras17] the complex analogue of geodesics in domains was considered. There the weak geodesic has to solve a complex Monge-Ampère equation on the product of the domain and a planar annulus A (see the Preliminaries for the details). The analogue of the space $\mathcal S$ is then the space $\mathcal H$ of smooth strictly plurisubharmonic functions vanishing on the boundary. In this situation the problem is much harder and partial results were shown only for special domains (see [Abj19] when the domain is a ball in complex space) or under symmetry assumptions of the boundary data. In [AD21] we considered the case when the boundary data consists of toric plurisub-harmonic functions and the underlying domain is Reinhardt. Thanks to a well known identification (see Lemma 1.4 below) in this setting the problem is reduced to (0.1) on the logarithmic image of the Reinhardt domain. Hence we have shown the following result (see Corollary 0.2 in [AD21]):

Proposition 0.6. Let $\Omega$ be a smoothly bounded strictly pseudoconvex Reinhardt domain in $\mathbb{C}^n$ . Let also $M = \{z \in \mathbb{C}^n | z_1 \cdots z_n = 0\}$ . Suppose that $\phi$ is a weak geodesic solving the problem (1.3). If $\varphi_0, \varphi_1 \in \mathcal{H}$ are toric in the space variables i.e. for all $z, z' \in \Omega$ satisfying |z| = |z'| one has $\varphi_j(z) = \varphi_j(z')$ , j = 0, 1 then $\phi$ is $C^{1,1}$ away from the corner $\partial\Omega \times \partial A$ and $(\Omega \cap M) \times A$ . The problem with $(\Omega \cap M) \times A$ is that the logarithmic map obviously degenerates along the set $\Omega \cap M$ (unless it is empty). Nevertheless in [AD21] (see Remark 2.4 there) we conjectured that $C^{1,1}$ regularity extends past $(\Omega \cap M) \times A$ . Our final result confirms this:

Theorem 0.7. If $\Omega$ , $\varphi_0$ , $\varphi_1$ are as above, then the geodesic $\phi$ is $C^{1,1}$ regular in the $\Omega$ direction past the set $(\Omega \cap M) \times A$ . If $\Omega$ is additionally complete Reinhardt then $\phi$ is $C^{1,1}$ in all directions. The note is organized as follows. In Section 1 we gather some important facts that shall be used later on. Next, in Section 2 we provide some examples. Section 3 is devoted to the proof of Theorem 0.2. In Section 4 we deal with Theorem 0.5. Finally in Section 5 we handle Theorem 0.7. Acknowledgements. The second named author has been partially supported by grant no. 2021/41/B/ST1/01632 from the National Science Center, Poland. The first author was partially funded by the German Research Foundation (DFG) under Germany's Excellence Strategy EXC 2044-390685587 "Mathematics Münster: Dynamics-Geometry-Structure" and by the CRC 1442 "Geometry: Deformations and Rigidity" of the DFG.

Proposition 1.1. Let $U \in \mathbb{R}^n$ be a convex set and $u: U \longmapsto \mathbb{R}$ be a convex function. Then - (1) $\{u^* < \infty\} = \partial u(U);$ - (2) if u is smooth and strictly convex on U then so is $u^*$ on $\partial u(U)$ ; - (3) $u^{**}|_{U} = u$ ; - (4) If u is $C^2$ and strictly convex, then $$D^2u^*(y) = (D^2u(x(y)))^{-1},$$ where $D^2u$ represents the Hessian matrix of u and $(D^2u)^{-1}$ its inverse. x(y) is the unique point where the supremum in the definition of $u^*$ is achieved. A crucial fact linking the Legendre transform and geodesics is the following nice formula whose proof (even in more general situations) can be found in [Ra17].

Proposition 1.2. Let $U \in \mathbb{R}^n$ be a convex set and $u_0$ , $u_1$ be two strictly convex functions vanishing on $\partial U$ . Then the geodesic u joining them is given by the formula <span id="page-3-0"></span> $$u^(y,t) = (1-t)u_0^(y) + tu_1^*(y), \tag{1.2}$$ where $u^*(y,t)$ is the partial Legendre transform done only with respect to the y-variable.

Lemma 1.4. Let $\Omega$ be a Reinhardt domain in $\mathbb{C}^n$ and $\phi$ be a bounded plurisubharmonic function on $\Omega$ , invariant with respect to the toric action. Then: (1) The image U of the mapping $$\Omega \ni z \to (log|z_1|, \cdots, log|z_n|) \in \mathbb{R}^n$$ is a domain in $\mathbb{R}^n$ . $\Omega$ is pseudoconvex if and only if U is convex. - (2) The function $u(x) := \phi(e^{x_1}, \dots, e^{x_n})$ is a convex function in U. Reversely for any bounded convex function u on U the function $\phi$ defined through this formula extends to a toric plurisubharmonic function on $\Omega$ . - (3) $(dd^c\phi)^n = 0$ if and only if $det(D^2u) = 0$ the equivalence continues to hold for bounded singular u and $\phi$ and then the equalities are understood in weak sense of measures- see [Gu02, Kol05]. Finally we recall that a Reinhardt domain $\Omega$ is called complete if for any $z=(z_1,\cdots,z_n)\in\Omega$ and for any $|\lambda_j|\leq 1, \lambda_j\in\mathbb{C}, j=1,\cdots,n$ one has $(\lambda_1z_1,\cdots,\lambda_nz_n)\in\Omega$ .

Proposition 1.5. Let A, B be positive definite matrices of the same dimension. Then for any $t \in [0, 1]$ the inequality $$((1-t)A^{-1} + tB^{-1})^{-1} \le (1-t)A + tB$$ holds in the sense that the difference is a semi-positive definite matrix.

Proposition 1.6. Let $A = (a_{mk})$ and $B = (b_{mk})$ be as above and let $l_j$ , $r_j$ $j = 1, \dots, n$ be any positive numbers. Consider the positive definite matrices $\tilde{A} := (a_{mk}l_ml_k)$ and $\tilde{B} := (b_{mk}r_mr_k)$ . Then we have $$((1-t)(\tilde{A})^{-1} + t(\tilde{B})^{-1})^{-1} \le (1-t)\tilde{A} + tB$$ with $(1-t)A + tB_{mk} = [(1-t)A + tB]_{mk}s_ms_k$ , where $s_i = (\frac{1-t}{l_i} + \frac{t}{r_i})^{-1}$ is the harmonic mean of $l_i$ and $r_i$ . As we were unable to find this exact result in the literature we provide a detailed proof. First we need a lemma that is modeled on Lemma 5.12 from [HP14].

Lemma 1.7. Let A, B and t be as above. Let also <, > denote the scalar product in $\mathbb{R}^n$ . Then for any vector $z \in \mathbb{R}^n$ one has <span id="page-5-0"></span> $$< z, [(1-t)A^{-1} + tB^{-1}]^{-1}z > = inf_{x+y=z} \frac{< x, Ax >}{1-t} + \frac{< y, By >}{t}.$$ (1.4)

Lemma 3.1. Let $(x^0, t^0)$ be any point in $U \times (0, 1)$ . Subtracting a linear function if necessary, we may suppose that $$u \ge 0$$ , $u(x^0, t^0) = 0$ . Then $(x^0, t^0)$ is in the convex hull of (n+1) points (not necessarily distinct) $(x^1, t^1), (x^2, t^2), ...(x^{n+1}, t^{n+1})$ in $\partial(U \times (0, 1))$ with $u(x^i, t^i) = 0$ for all $i \in \{1, 2, ... n + 1\}$ . Next we borrow a crucial observation from [AD21]- see the Claim 1 in the proof of inequality (2.1) there:

Lemma 3.2. Such a convex hull consists of a line segment joining a point from $\overline{U} \times \{0\}$ with a point $\overline{U} \times \{1\}$ which passes through $(x^0, t^0)$ .

Lemma 3.3. Let u(x,t) be a weak geodesic between $\varphi_0$ and $\varphi_1$ in $\tilde{S}$ . We assume furthermore that $\partial \varphi_0(U) = \partial \varphi_1(U)$ . Then the second derivative of u(x,t) in the space x direction is given as follows: $$D_{xx}^{2}u(x,t) = \left((1-t)(D_{xx}^{2}\varphi_{0}(\xi))^{-1} + t(D_{xx}^{2}\varphi_{1}(\eta))^{-1}\right)^{-1}$$ where $D_{xx}^2$ is the Hessian of u with respect the variable x only.

Lemma 3.4. Let $(u_t) = u(x,t)$ be a geodesic path connecting $\varphi_0$ and $\varphi_1$ in $\tilde{\mathcal{S}}$ . If $$\partial \varphi_0(U) = \partial \varphi_1(U)$$ then $$D_{xt}^2 u(x,t) = \left[ (1-t)(D^2 \varphi_0)^{-1}(\xi) + t(D^2 \varphi_1)^{-1}(\eta) \right]^{-1} (\xi - \eta).$$

Lemma 4.1. Let U be a convex open set, C and $\rho$ be positive constants, and $\alpha \in (0,1]$ . Let $u: U \longmapsto \mathbb{R}$ be a globally Lipschitz convex function such that for every $y \in U$ there exists a supporting affine function $l_y$ such that $$u(x) - l_u(x) \le C|x - y|^{1+\alpha}$$ for all $x \in U \cap B_{\rho}(y)$ . Then $u \in C^{1,\alpha}(U)$ . Fix now two neighborhoods $W_1 \subseteq W_2$ of $\partial U \times \{0,1\}$ , such that both $$V_i := U \times (0,1) \setminus \overline{W_i}, \ i = 1, 2$$ are convex. This choice implies that there is $\delta > 0$ such that for any $(y, s) \in W_1$ , $(x, t) \in V_2$ one has $$|(y,s) - (x,t)| \ge \delta.$$ With the aid of Lemmas 4.1 and 3.2, Theorem 0.5 follows from the following bound: there is a $\rho > 0$ dependent on U and $\delta$ , such that for every point $(\overline{x}, \overline{t})$ in $V_2$ and any $(x, t) \in U \times (0, 1)$ such that $|(\overline{x}, \overline{t}) - (x, t)| < \rho$ one has <span id="page-11-1"></span> $$|u(x,t) - u((\overline{x},\overline{t})) - (x - \overline{x}).D_x u((\overline{x},\overline{t})) - (t - \overline{t})D_t u((\overline{x},\overline{t}))|$$ $$\leq C|(x - \overline{x},t - \overline{t})|^{1+\alpha}$$ $$(4.1)$$ for a constant C depending on $V_2$ , $\delta$ , $\rho$ and the $C^{1,\alpha}$ norm of $\varphi_0,\varphi_1$ (but independent on (x,s)). Proof of inequality (4.1). We follow the reasoning from [AD21]. We fix $(\bar{x}, \bar{t}) \in V_2$ . After possibly subtracting an affine support function $l_{\bar{x},\bar{t}}(x,t)$ we can suppose $$u \ge 0$$ , $u(\bar{x}, \bar{t}) = 0$ and $Du(\bar{x}, \bar{t}) = 0$ . The problem (0.1) for this new function u becomes: $$\begin{cases} \det(D_{x,t}^{2}u) = 0 \text{ in } U \times (0,1), \\\nu = \varphi_{0} - l_{\bar{x},\bar{t}} \text{ on } U \times \{0\}, \\\nu = \varphi_{1} - l_{\bar{x},\bar{t}} \text{ on } U \times \{1\}, \\\nu = -l_{\bar{x},\bar{t}} \text{ on } \partial U \times (0,1). \end{cases}$$ (4.2) The inequality we need to prove reads $$u(x,t) \le C|(x-\bar{x},t-\bar{t})|^{1+\alpha} \tag{4.3}$$ for all (x,t) in $U \times (0,1)$ which are at distance less than $\rho$ from $(\bar{x},\bar{t})$ for some constant $\rho$ to be determined later on. By Lemma 3.2 we assume that $(\bar{x}, \bar{t})$ is on the line segment $[(x^0, 0), (x^1, 1)]$ on which u vanishes. Switching the role of the end points, if necessary, that $(\bar{x}, \bar{t})$ is closer to $(x^1, 1)$ than to $(x^0, 0)$ (i.e. we assume $\bar{t} \geq \frac{1}{2}$ ). Claim: there is a constant $\rho > 0$ with the following property: unless both $(x^0, 0)$ , $(x^1, 1)$ belong to $W_1$ then at least one of the rays $[(x^0, 0), (x, t)), [(x^1, 1), (x, t))$ strikes the boundary of $U \times (0, 1)$ at $\overline{U} \times \{0, 1\}$ . Indeed, if $(x^1,1) \in \overline{U} \times \{1\} \setminus \overline{W_1}$ , then choose the ray $[(x^0,0),(x,t))$ and (exploiting the fact that $\hat{t} \geq \frac{1}{2}$ ) the existence of $\rho$ follows from continuity argument. If in turn $(x^1,1) \in W_1$ but $(x^1,1) \in \overline{U} \times \{1\} \setminus \overline{W_1}$ the same argument applies for the ray $[(x^1,1),(x,t))$ except now one uses the fact that $|(\overline{x},\overline{t})-(x,t)| \geq \delta$ instead of $\overline{t} \geq \frac{1}{2}$ . Consider now a ball in $U \times (0,1)$ with center $(\bar{x},\bar{t})$ and radius $\rho$ . For any $(x,t) \in U \times (0,1)$ in the $\rho$ - ball let $(\hat{x},\hat{t})$ be the point where the ray from $(x^0,t^0)$ to (x,t) strikes the boundary of $U \times (0,1)$ . Let also $(\tilde{x},\tilde{t})$ be the point where the ray from $(x^1,t^1)$ to (x,t) strikes the boundary of $U \times (0,1)$ Three cases may occur. Suppose first that $\hat{t} = 1$ i.e. the point $(\hat{x}, \hat{t})$ is on $\overline{U} \times \{1\}$ . In this case the argument from [AD21] applies mutatis mutandis. We recall the details for the sake of completeness. If $(x,t) = t(\hat{x},1) + (1-t)(x^1,0)$ by convexity of u and $u(x^0,0) = 0$ , we have $$u(x,t) \le t\tilde{\varphi}_1(\hat{x}) + (1-t)u(x^0,0) \le \tilde{\varphi}_1(\hat{x}),$$ where $\tilde{\varphi}_1 = \varphi_1 - l_{\bar{x},\bar{t}}$ . Note that $x^1$ is a local minimum point for $\tilde{\varphi}$ . As furthermore $\varphi_1 \in \tilde{\mathcal{S}}^{1,\alpha}$ we have $$\tilde{\varphi}_1(\hat{x}) - \tilde{\varphi}_1(x^1) \le C|\hat{x} - x^1|^{1+\alpha}$$ for a constant C dependent on the $C^{1,\alpha}$ norm of $\varphi_1$ and the global Lipschitz norm of u. It thus suffices to prove that <span id="page-12-0"></span> $$|\hat{x} - x^1| \le C(|(x - \bar{x}, t - \bar{t})|,$$ (4.4) for some C. To this end we consider the plane $\Pi$ spanned by $(x^0,0),(x^1,1)$ and (x,t) (note that if these three points are co-linear then the estimate is trivial). In $\Pi \cap [U \times (0,1)]$ we have $$\frac{|\hat{x} - x^1|}{|(x^0, t^0) - (x^1, t^1)|} = \frac{\sin(\beta)}{\sin(\alpha \pm \beta)}$$ and $$\frac{|(\bar{x},\bar{t}) - (x^0, t^0)|}{|(x,t) - (\bar{x},\bar{t})|} = \frac{\sin(\theta)}{\sin(\beta)},$$ where $\alpha$ is the angle between the line $((x^1,1),(x^0,0))$ and the line $\{t=1\} \cap \Pi$ , $\beta$ is the angle between $((x^0,0),(x^1,1))$ and $((x^0,0),(\hat{x},1))$ and $\theta$ is the angle between $((\bar{x},\bar{t}),(x,t))$ and $((x^0,0),(\hat{x},1))$ . From these two equations we obtain $$\frac{|\hat{x} - x^1|}{|(x,t) - (\bar{x},\bar{t})|} = \frac{\sin(\theta)}{\sin(\alpha \pm \beta)} \frac{|(x^0,0) - (x^1,1)|}{|(\bar{x},\bar{t}) - (x^0,0)|}.$$ Note that the angle $\alpha \pm \beta$ is uniformly bounded from below by a constant dependent on diam(U). Also, trivially, $sin(\theta) \leq 1$ . In order to bound the ratio $$\frac{|(x^0,0) - (x^1,1)|}{|(\bar{x},\bar{t}) - (x^0,0)|}$$ just observe that $|(x^0,0)-(x^1,1)| \leq \sqrt{diam(U)^2+1}$ , while $|(\bar{x},\bar{t})-(x^0,0)| \geq |\bar{t}| \geq \frac{1}{2}$ . Thus the estimate (4.4) is proven in this case. Assume now $\hat{t} \neq 1$ which implies that $(x^1, 1) \in W_1$ . If, in turn, $\tilde{t}=0$ i.e. $(\tilde{x},\tilde{t})\in \overline{U}\times\{0\}$ we apply similar idea. We need an analogue of (4.4) i.e. $$|\tilde{x} - x^0| \le C(|(x - \bar{x}, t - \bar{t})|.$$ (4.5) Arguing as above this boils down to establishing a bound from above on $\frac{|(x^0,0)-(x^1,1)|}{|(\bar{x},\bar{t})-(x^1,1)|}$ . Once again $|(x^0,0)-(x^1,1)| \leq \sqrt{diam(U)^2+1}$ but for the lower bound of $|(\bar{x},\bar{t})-(x^1,1)|$ we exploit that now $(x^1,1) \in W_1$ and hence $|(\bar{x},\bar{t})-(x^1,1)| \geq \delta$ . Finally it remains to analyze the case when both $\tilde{t} \neq 0$ and $\hat{t} \neq 1$ . In this case both points $(x^0,0)$ and $(x^1,1)$ belong to $W_1$ . Once again consider the plane $\Pi$ spanned by $(x^0,0),(x^1,1)$ and (x,t) and let $(\overline{x},1),(\underline{x},0)$ be the vertices of the rectangle $\Pi \cap \overline{U \times (0,1)}$ that are close to $(x^1,1)$ and $(x^0,0)$ , respectively. Note that, by convexity $$u(x,t) \le \frac{t}{\hat{t}}u(\hat{x},\hat{t}) + (1 - \frac{t}{\hat{t}})u(x^0,0)$$ $$= \frac{t}{\hat{t}}(\hat{t}u(\overline{x},1) + (1 - \hat{t})u(\underline{x},0)),$$ where we have used the linearity of u along $\partial U \times [0,1]$ . As $u(\overline{x},1) \leq C|\overline{x}-x^1|^{1+\alpha}$ and $u(\underline{x},0) \leq C|\underline{x}-x^0|^{1+\alpha}$ it suffices to bound the ratios $$\frac{|\overline{x}-x^1|}{|(x,t)-(\overline{x},\overline{t})|}, \ \frac{|\underline{x}-x^0|}{|(x,t)-(\overline{x},\overline{t})|}.$$ Note however that if (p, 1) is the intersection point of the ray $[(x^0, 0), (x, t))$ with $\{t = 1\}$ , while (q, 0) is the intersection point of the ray $[(x^1, 1), (x, t))$ with $\{t = 0\}$ one obviously has $$|\overline{x} - x^1| \le |p - x^1|, |\underline{x} - x^0| \le |q - 1|,$$ whereas the ratios $$\frac{|p-x^1|}{|(x,t)-(\bar{x},\bar{t})|}, \ \frac{|q-x^0|}{|(x,t)-(\bar{x},\bar{t})|}$$ can be bounded in exactly the same way as in the previous two cases (note that it does not matter that p and q stick out of $\Pi \cap \overline{U \times (0,1)}$ ). This finishes the proof. Remark 4.2. Just as in [AD21] the obtained bound is stronger than purely interior one. It implies that the blow-up of the $C^{1,\alpha}$ norm can occur only at the wedge $\partial U \times \{0,1\}$ .

Lemma 5.1. Let $\varphi$ be a toric plurisubharmonic function from $\tilde{\mathcal{H}}$ and u be as above. Then $$\varphi(z_1, z_2, ..., z_n) = u(\log |z_1|, \log |z_2|, , , \log |z_n|) =: u(x),$$ and hence $$\frac{\partial^2 \varphi}{\partial z_i \partial \overline{z}_j}(z) = \frac{\partial}{\partial z_i} \left( \frac{\partial u}{\partial x_j} \frac{1}{2\overline{z}_j} \right) = \frac{\partial^2 u}{\partial x_i \partial x_j}(x) \frac{1}{4z_i \overline{z}_j}$$ As a corollary of the above lemma note that $\frac{\partial^2 u}{\partial x_i \partial x_j}(x) \frac{1}{4z_i \overline{z}_j}$ extends smoothly past $\{z_i z_j = 0\} \cap \Omega$ . The lemma can also be restated as <span id="page-14-0"></span> $$\frac{\partial^2 u}{\partial x_i \partial x_j}(x) = \frac{\partial^2 \varphi}{\partial z_i \partial \bar{z}_j}(e^x) 4e^{x_i} e^{x_j}, \tag{5.2}$$ where we used the suggestive notation $e^x = (e^{x_1}, \dots, e^{x_n})$ . Next proposition is essentially contained in [Ras17]. We refer to [AD21] for more details on the topic.

Proposition 5.2. Let $\Omega$ be a bounded Reinhardt domain in $\mathbb{C}^n$ with U - its logarithmic image. Let also $\varphi$ be a weak geodesic joining $\varphi_0$ and $\varphi_1$ from $\tilde{\mathcal{H}}$ . If $u_0(x) := \varphi_0(e^x)$ , $u_1(x) := \varphi_1(e^x)$ , then $u(x,t) := \varphi(e^x, e^t)$ is a weak geodesic joining $u_0, u_1 \in \tilde{\mathcal{S}}$ . This identification justifies the proof of Proposition 0.6 (see also Corollary 0.2 in [AD21]) through the already proven regularity for geodesics in $\tilde{\mathcal{S}}$ and the logarithmic map. The major problem that occurs in this approach is the obvious degeneracy of the logarithmic map along the set $M = \{z_1 \cdots z_n = 0\}$ . Henceforth we shall assume that $\Omega \cap M$ is not empty for otherwise the regularity analysis is finished. Note that $\Omega \cap M \neq \emptyset$ implies that U is unbounded, which as we shall see below additionally complicates computations. We are ready to state the main result of this section:

Theorem 5.3. Let $\Omega$ be a bounded Reinhardt domain in $\mathbb{C}^n$ . Let also $\phi$ be a weak geodesic joining $\varphi_0$ and $\varphi_1$ from $\tilde{\mathcal{H}}$ . Then $\varphi$ is $C^{1,1}_{loc}$ regular in the spatial directions and the blow-up of the $C^{1,1}$ norm can occur only at the corner $\partial\Omega \times \partial A$ . If $\Omega$ is complete Reinhardt then the $C^{1,1}_{loc}$ bound holds in all directions. Furthermore $\varphi$ is globally $C^{1,1}$ smooth (and strictly plurisubharmonic) in the spatial direction if and only if $\partial u_0(U) = \partial u_1(U)$ . Remark 5.4. It is very likely that under the assumption $\partial u_0(U) = \partial u_1(U) \phi$ is in fact $C^{\infty}$ smooth and hence a classical geodesic in any complete smoothly bounded strictly pseudoconvex Reinhardt domain. This however requires a thorough higher order analysis across the set $\Omega \cap M$ . We hope to address this problem in the future.

Definition 0.3. Let U be a convex domain in $\mathbb{R}^n$ and u be a convex function on U. u is said to be strictly convex at $y \in U$ if for some affine support function $l_y$ we have $${u(x) = l_u(x)} = {y}.$$ u is said to be strictly convex if it is strictly convex at every point $y \in U$ . Below we describe the space of strictly convex $C^{1,\alpha}$ regular functions with Dirichlet boundary data. <span id="page-1-0"></span>^&lt;sup>1</sup>Both authors thank Eleonora Di Nezza for discussions on the topic.

Definition 0.4. Let U be a convex domain in $\mathbb{R}^n$ and $\alpha \in (0,1]$ be a constant. Then the space $\tilde{\mathcal{S}}^{1,\alpha}$ is given by $$\tilde{\mathcal{S}}^{1,\alpha}:=\{u\in C^{1,\alpha}(U)\cap C(\overline{U})|u\text{ strictly convex on }U,\ ||u||_{C^1}<\infty, u=0\text{ on }\partial U\}.$$ The $C^{1,\alpha}$ regularity for the degenerate Monge-Ampère equation was also intensively studied - we refer to [DPF15] and especially the recent paper [CTW22] for an up-to-date results. Once again the geodesic equation is posed in a non-smooth domain and results from these papers cannot be used directly. Our second result is that, thanks to the specific boundary data, the geodesic equation has the same $C^{1,\alpha}$ regularity theory as in the case of space $\tilde{S}$ .

Definition 1.3. Let $\Omega$ be a domain in $\mathbb{C}^n$ . An upper semicontinuous function $\phi$ is called plurisubharmonic, if its restriction to any complex line L is subharmonic or constantly $-\infty$ on each component of $L \cap \Omega$ . The space of plurisubharmonic functions will be denoted by $PSH(\Omega)$ . These are the complex analogues of convex functions. As we shall deal only with rotationally invariant plurisubharmonic functions in this note it is worth recalling that $\phi$ is plurisubharmonic and rotationally invariant on its domain of definition iff the function $u(x) := \phi(e^{x_1}, \dots, e^{x_n})$ is convex. The geodesic equation in this setting reads (see [AD21]) <span id="page-4-1"></span> $$\begin{cases} \phi \in PSH(\Omega \times A) \cap C(\overline{\Omega \times A}); \\ (dd_{z,\zeta}^{c}\phi)^{n+1} = 0 & \text{in } \Omega \times A; \\ \phi = \varphi_{0} & \text{in } \Omega \times \{|z| = 1\}; \\ \phi = \varphi_{1} & \text{in } \Omega \times \{|z| = e\}; \\ \phi = 0 & \text{in } \partial\Omega \times A, \end{cases}$$ (1.3) where $A = \{z \in \mathbb{C} | 1 < |z| < e\}$ denotes an annulus in $\mathbb{C}$ and $\phi(z,\zeta) = \varphi_t(z)$ with $t = \log |\zeta|$ . Finally $(dd_{z,\zeta}^c)^n$ is the complex Monge-Ampère operator in joint variables $(z,\zeta)$ which, for smooth input $\phi$ , is simply the determinant of the complex Hessian of $\phi$ . Suppose now that $\Omega$ is a Reinhardt domain i.e. if $z=(z_1,\cdots,z_n)\in\Omega$ , then $(e^{i\theta_1}z_1,\cdots,e^{i\theta_n}z_n)\in\Omega$ for any $\theta_j\in[0,2\pi),\ j=1,\cdots,n$ . Recall the following classical fact (see also [AD21]):