Existence and Hölder continuity to solutions of the complex Monge–Ampère-type equations with measures vanishing on pluripolar subsets

In this paper, we establish existence of Hölder continuous solutions to the complex Monge–Ampère-type equation with measures vanishing on pluripolar subsets of a bounded strictly pseudoconvex domain [Formula: see text] in [Formula: see text].

Atomic decompositions in weighted Bergman spaces of analytic functions on strictly pseudoconvex domains

We construct an atomic decomposition of the weighted Bergman spaces Ap?(D) (0 < p ? 1, ? > -1) of analytic functions on a bounded strictly pseudoconvex domain D in Cn with smooth boundary. The atoms used are atoms in the real-variable sense.

Weak solution of parabolic complex Monge–Ampère equation II

We study the equation [Formula: see text] in [Formula: see text], where [Formula: see text] and [Formula: see text] is a bounded strictly pseudoconvex domain in [Formula: see text], with the boundary condition [Formula: see text] and the initial condition [Formula: see text]. In this paper, we consider the case, where [Formula: see text] is smooth and [Formula: see text] is an arbitrary plurisubharmonic function in a neighborhood of [Formula: see text] satisfying [Formula: see text].

Derivations on Toeplitz Algebras

Let H2(Ω) be the Hardy space on a strictly pseudoconvex domain Ω ⊂ ℂn, and let A ⊂ L∞(∂Ω) denote the subalgebra of all L∞-functions ƒ with compact Hankel operator Hƒ. Given any closed subalgebra B ⊂ A containing C(Ω), we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra 𝒯(B) ⊂ B(H2(Ω). In particular, we show that every derivation on 𝒯(A) is inner. These results are new even for n = 1, where it follows that every derivation on T(H∞ +C) is inner, while there are non-inner derivations on T(H∞ + C(∂ℝn)) over the unit ball Bn in dimension n > 1.

Variétés Singulières et Extension des Fonctions Holomorphes

In this paper, we study a problem of extension of holomorphic functions given on a complex hypersurface with singularities on the boundary of a strictly pseudoconvex domain.

Lp extension of holomorphic functions from submanifolds to strictly pseudoconvex domains with non-smooth boundary

Let D be a bounded strictly pseudoconvex domain in ℂn (with not necessarily smooth boundary) and let X be a submanifold in a neighborhood of . Then any Lp (1 ≥ p < ∞) holomorphic function in X ∩ D can be extended to an Lp holomorphic function in D.

Approximate formulas for canonical homotopy operators for the $\bar \partial$ complex in strictly pseudoconvex domains

Let $D=\{ \rho <0 \}$ be a smoothly bounded strictly pseudoconvex domain in $\boldsymbol C^n$ and $\rho$ a strictly plurisubharmonic smooth defining function. We construct explicit homotopy operators for the $\bar \partial$ complex, which are approximately equal to the homotopy operators that are canonical with respect to the metric $\Omega = i\varphi(-\rho)\partial \bar \partial \log(1/-\rho)$ and weights $(-\rho)^\alpha$, where $\varphi$ is a strictly positive smooth function. We also obtain an explicit operator which is approximately equal to the canonical homotopy operator for $\bar \partial_b$ on $\partial D$. From the explicit operators we obtain regularity results for these canonical operators, including $C^\infty$ regularity and $L^p$-boundedness for the orthogonal projections onto Ker $\bar \partial$ and Ker $\bar \partial_b$. Previously it has been proved, in the ball case and $\varphi \equiv 1$, that the boundary values of the canonical operators coincide with the values of well-known explicit operators due to Henkin and Skoda et al. Previously Lieb and Range have constructed an explicit homotopy operator which is approximately equal to the canonical operator with respect to the metric $i\varphi \partial \bar \partial_\rho$.