Let $M$ be a two dimensional totally real submanifold of class $C^2$ in ${\mathbf{C}}^2$. A continuous map $F:\bar{\Delta }\to {\mathbf{C}}^2$ of the closed unit disk $\bar{\Delta }\subset \mathbf{C}$ into ${\mathbf{C}}^2$ that is holomorphic on the open disk $\Delta$ and maps its boundary $b\Delta$ into $M$ is called an analytic disk with boundary in $M$. Given an initial immersed analytic disk $F^0$ with boundary in $M$, we describe the existence and behavior of analytic disks near $F^0$ with boundaries in small perturbations of $M$ in terms of the homology class of the closed curve $F^0\left(b\Delta \right)$ in $M$. We also prove a regularity...

Let $D\subset \subset {ℂ}^n,n\ge 2$, be a domain with $C^2$-boundary and $K\subset \partial D$ be a compact set such that $\partial D\setminus K$ is connected. We study univalent analytic extension of CR-functions from $\partial D\setminus K$ to parts of $D$. Call $K$ CR-convex if its $A\left(D\right)$-convex hull, $A\left(D\right)-\mathrm{hull}\left(K\right)$, satisfies $K=\partial D\cap A\left(D\right)-\mathrm{hull}\left(K\right)$ ($A\left(D\right)$ denoting the space of functions, which are holomorphic on $D$ and continuous up to $\partial D$). The main theorem of the paper gives analytic extension to $\partial D\setminus A\left(D\right)-\mathrm{hull}\left(K\right)$, if $K$ is CR- convex.

Let $\Omega \subset {ℝ}^2$ be a bounded, convex and open set with real analytic boundary. Let $T_{\Omega }\subset {ℂ}^2$ be the tube with base $\Omega ,$ and let $ℬ$ be the Bergman kernel of $T_{\Omega }$. If $\Omega$ is strongly convex, then $ℬ$ is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation, we relate the off diagonal points where analyticity fails to the Trèves curves. These curves are symplectic invariants which are determined by the CR structure of the boundary of $T_{\Omega }$. Note that Trèves curves...

For a domain $\Omega \subset {ℂ}^n$ let $H\left(\Omega \right)$ be the holomorphic functions on $\Omega$ and for any $k\in \mathbb{N}$ let $A^k\left(\Omega \right)=H\left(\Omega \right)\cap C^k\left(\overline{\Omega }\right)$. Denote by ${𝒜}_D^k\left(\Omega \right)$ the set of functions $f\Omega \to [0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k\left(\Omega \right)$ such that $\left\{\right|f_j\left|\right\}$ is a nonincreasing sequence and such that $f\left(z\right)={lim}_{j\to \infty }\left|f_j\left(z\right)\right|$. By ${𝒜}_I^k\left(\Omega \right)$ denote the set of functions $f\Omega \to (0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k\left(\Omega \right)$ such that $\left\{\right|f_j\left|\right\}$ is a nondecreasing sequence and such that $f\left(z\right)={lim}_{j\to \infty }\left|f_j\left(z\right)\right|$. Let $k\in \mathbb{N}$ and let ${\Omega }_1$ and ${\Omega }_2$ be bounded $A^k$-domains of holomorphy in ${ℂ}^{m_1}$ and ${ℂ}^{m_2}$ respectively. Let $g_1\in {𝒜}_D^k\left({\Omega }_1\right)$, $g_2\in {𝒜}_I^k\left({\Omega }_1\right)$ and $h\in {𝒜}_D^k\left({\Omega }_2\right)\cap {𝒜}_I^k\left({\Omega }_2\right)$. We prove...

Let $\Lambda \subset {ℝ}^n\times {ℝ}^m$ and $k$ be a positive integer. Let $f:{ℝ}^n\to {ℝ}^m$ be a locally bounded map such that for each $(\xi ,\eta )\in \Lambda$, the derivatives $D_{\xi }^{j}f\left(x\right):=\frac{d^j}{d{t}^j}f(x+t\xi ){|}_{t=0}$, $j=1,2,\cdots k$, exist and are continuous. In order to conclude that any such map $f$ is necessarily of class $C^k$ it is necessary and sufficient that $\Lambda$ be not contained in the zero-set of a nonzero homogenous polynomial $\Phi (\xi ,\eta )$ which is linear in $\eta =({\eta }_1,{\eta }_2,\cdots ,{\eta }_m)$ and homogeneous of degree $k$ in $\xi =({\xi }_1,{\xi }_2,\cdots ,{\xi }_n)$. This generalizes a result of J. Boman for the case $k=1$. The statement and the proof of a theorem of Boman for the case $k=\infty$ is...

The $k$-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations $F_k\left[u\right]=0$, where $F_k\left[u\right]$ is the elementary symmetric function of order $k$, $1\le k\le n$, of the eigenvalues of the Hessian matrix $D^{2}u$. For example, $F_1\left[u\right]$ is the Laplacian $\Delta u$ and $F_n\left[u\right]$ is the real Monge-Ampère operator det $D^{2}u$, while $1$-convex functions and $n$-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative $k$-convex functions, and give...