Convex Domains Of Finite Type Research Articles

Sobolev and Hölder estimates for homotopy operators of the [formula omitted]-equation on convex domains of finite multitype

We construct homotopy formulas for the ∂‾-equation on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain Ω⊂Cn that has q-type mq for 1≤q≤n, our ∂‾ solution operator Hq on (0,q)-forms has (fractional) Sobolev boundedness Hq:Hs,p→Hs+1/mq,p and Hölder–Zygmund boundedness Hq:Cs→Cs+1/mq for all s∈R and 1<p<∞. We also demonstrate the Lp-boundedness Hq:Hs,p→Hs,prq/(rq−p) for all s∈R and 1<p<rq, where rq:=(n−q+1)mq+2q.

Dyadic decomposition of convex domains of finite type and applications

In this paper, we introduce a dyadic structure on convex domains of finite type via the so-called dyadic flow tents. This dyadic structure allows us to establish weighted norm estimates for the Bergman projection P on such domains with respect to Muckenhoupt weights. In particular, this result gives an alternative proof of the $$L^p$$ boundedness of P. Moreover, using extrapolation, we are also able to derive weighted vector-valued estimates and weighted modular inequalities for the Bergman projection.

Formula omitted]-corona problem and convex domains of finite type

We prove that the Hp-corona problem has a solution for two generators in convex domains of finite type in Cn, n≥2.

Bekollé-Bonami estimates on some pseudoconvex domains

We establish a weighted Lp norm estimate for the Bergman projection for a class of pseudoconvex domains. We obtain an upper bound for the weighted Lp norm when the domain is, for example, a bounded smooth strictly pseudoconvex domain, a pseudoconvex domain of finite type in C2, a convex domain of finite type in Cn, or a decoupled domain of finite type in Cn. The upper bound is related to the Bekollé-Bonami constant and is sharp. When the domain is smooth, bounded, and strictly pseudoconvex, we also obtain a lower bound for the weighted norm. As an additional application of the method of proof, we obtain the result that the Bergman projection is weak-type (1,1) on these domains.

A remark on boundary Schwarz lemma for the convex domain of finite type

Let Ω be a bounded convex domain of finite type with smooth boundary. If f is a holomorphic self-mapping of Ω which fixes a boundary point P, we obtain some estimates of the eigenvalues of the complex Jacobian matrix of f at P. These estimates can be thought of as a generalization of the boundary Schwarz lemma on strongly pseudoconvex domains.

Lorentz boundedness and compactness characterization of integral commutators on spaces of homogeneous type

We study the Lorentz boundedness properties of Calderón-type commutators and relate those properties to the spaces of bounded mean oscillation and vanishing mean oscillation. In application, we obtain Lorentz boundedness and compactness characterization of integral commutators with singular kernel of Calderón–Zygmund type on strictly pseudoconvex domains in ℂn, or convex domain of finite type in ℂn.

Boundary behavior of the Carathéodory and Kobayashi–Eisenman volume elements

We study the boundary limits of the Carath\'eodory and Kobayashi-Eisenman volume elements on smoothly bounded convex finite type domains and also on Levi corank one domains.

Composition Operators Between Hardy Spaces on Linearly Convex Domains in $$\mathbb {C}^2$$ C 2

We study composition operators acting between Hardy spaces $$H^p(\Omega )$$ , where $$\Omega \subset \mathbb {C}^2$$ is a smoothly bounded, $$\mathbb {C}$$ -linearly convex domain admitting the so-called F-type at all boundary points. This F-type domains contain certain convex domains of finite type and many cases of infinite type in the sense of Range. Criteria for boundedness and compactness of such composition operators are established. Our approach is based on the Cauchy–Leray kernel.

Zeros Sets of Hp Functions in Lineally Convex Domains of Finite Type in ℂ n $ \mathbb {C}^{n}$

In this note, we extend N. Th. Varopoulos result on zero sets of Hp functions of strictly pseudo-convex domains in $\mathbb {C}^{n}$ to lineally convex domains of finite type.

Weighted and boundary $$L^{p}$$ L p estimates for solutions of the $$\overline{\partial }$$ ∂ ¯ -equation on lineally convex domains of finite type and applications

We obtain sharp weighted estimates for solutions of the equation ∂ u = f in a lineally convex domain of finite type. Precisely we obtain estimates in the spaces L p (Ω,δ γ), δ being the distance to the boundary, with two different types of hypothesis on the form f : first, if the data f belongs to L p Ω,δ γ Ω , γ > −1, we have a mixed gain on the index p and the exponent γ; secondly we obtain a similar estimate when the data f satisfies an apropriate anisotropic L p estimate with weight δ γ+1 Ω. Moreover we extend those results to γ = −1 and obtain L p (∂ Ω) and BMO(∂ Ω) estimates. These results allow us to extend the L p (Ω,δ γ)-regularity results for weighted Bergman projection obtained in [CDM14b] for convex domains to more general weights.

Zero sets of $${\mathcal {H}}^p$$ H p functions in convex domains of finite type

We give a condition under which a divisor \(\hat{X}\) in a bounded convex domain of finite type D in \(\mathbb {C}^n\) is the zero set of a function in a Hardy space \({\mathcal {H}}^p(D)\) for some \(p>0\). This generalizes Varopoulos’ result (Pac J Math 88:189–246, 1980) on zero sets of \({\mathcal {H}}^p\)-functions in strictly convex domains of \(\mathbb {C}^n\).

Almost strictly pseudo-convex domains. Examples and applications

In this work we introduce a class of smoothly bounded domains Ω in Cn with few non strictly pseudo-convex points in @ Ω with respect to a certain Minkowski dimension. We call them almost strictly pseudo-convex, aspc. For these domains we prove that a canonical measure associated to a separated se- quence of points in Ω which projects on the set of weakly pseudo-convex points is automatically a geometric Carleson measure. This class of aspc domains con- tains of course strictly pseudo-convex domains but also pseudo-convex domains of finite type in C2 , domains locally diagonalizable, convex domains of finite type 2 in Cn , domains with real analytic boundary and domains like |z1 | |z2 | 2 } < 1, which are not of finite type. + exp{1 As an application we study interpolating sequences for convex domains of finite type in Cn . After proving a Carleson-type embedding theorem, we get that if Ω is a convex domain of finite type in Cn and if S ⇢ Ω is a dual bounded sequence of points in H p (Ω), if p = 1 then for any q < 1, S is H q (Ω) interpolating with the linear extension property and if p < 1 then S is H q () interpolating with the linear extension property, provided that q < min( p, 2).

EXTENSION AND RESTRICTION FOR BERGMAN SCALE OF SPACES AND ONE-DIMENSIONAL SUBVARIETIES ON CONVEX FINITE TYPE DOMAINS

We prove that the extension problem from one-dimensional subvarieties with values in Bergman space $H^{1}(D)$ on convex finite type domains can be solved by means of appropriate measures. We obtain also almost optimal results concerning the extension problem for other Bergman spaces and one-dimensional varieties.

Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type

In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with $$C^\infty $$ boundary being of finite type in the sense of D’Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also show that bounded domains which are locally convexifiable and have finite type in the sense of D’Angelo have Gromov hyperbolic Kobayashi metric. The proofs use ideas from the theory of the Hilbert metric.

On estimates for weighted Bergman projections

In this note we show that the weighted $L^{2}$-Sobolev estimates obtained by P. Charpentier, Y. Dupain \& M. Mounkaila for the weighted Bergman projection of the Hilbert space $L^{2}\left(\Omega,d\mu_{0}\right)$ where $\Omega$ is a smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^{n}$ and $\mu_{0}=\left(-\rho_{0}\right)^{r}d\lambda$, $\lambda$ being the Lebesgue measure, $r\in\mathbb{Q}_{+}$ and $\rho_{0}$ a special defining function of $\Omega$, are still valid for the Bergman projection of $L^{2}\left(\Omega,d\mu\right)$ where $\mu=\left(-\rho\right)^{r}d\lambda$, $\rho$ being any defining function of $\Omega$. In fact a stronger directional Sobolev estimate is established. Moreover similar generalizations are obtained for weighted $L^{p}$-Sobolev and lipschitz estimates in the case of pseudoconvex domain of finite type in $\mathbb{C}^{2}$ and for some convex domains of finite type.

BOUNDARY BEHAVIOR OF HOLOMORPHIC DISCS IN CONVEX FINITE TYPE DOMAINS

In this paper, we study holomorphic discs in a domain with a plurisubharmonic peak function at a boundary point. The aim is to describe boundary behavior of holomorphic discs in convex finite type domains in the complex Euclidean space in term of a special local neigh-borhood system at a boundary point.

Gain of Regularity in Extension Problem on Convex Domains

We investigate the extension problem from higher codimensional linear subvarieties on convex domains of finite type. We prove that there exists a constantdsuch that on Bergman spacesHp(D)with1≤p&lt;dthere appears the so-called “gain regularity.” The constantddepends on the minimum of the dimension and the codimension of the subvariety. This means that the space of functions which admit an extension to a function in the Bergman spaceHp(D)is strictly larger thanHp(D∩A), whereAis a subvariety.

Estimates for Solutions of the $\bar{\partial}$ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type

In the last ten years, the resolution of the equation $\bar{\partial}u=f$ with sharp estimates has been intensively studied for convex domains of finite type in $\mathbb{C}^{n}$ by many authors. Generally, they used kernels constructed with holomorphic support function satisfying “good” global estimates. In this paper, we consider the case of lineally convex domains. Unfortunately, the method used to obtain global estimates for the support function cannot be carried out in that case. Then we use a kernel that does not directly give a solution of the $\bar{\partial}$ -equation, but only a representation formula which allows us to end the resolution of the equation using Kohn’s L 2 theory. As an application, we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.

Area integral, Littlewood-Paley g-function and BMOA on convex domains of finite type

Let Ω be a smoothly bounded convex domain of finite type in ℂ n , ∂Ω be the boundary of Ω, S α (f) and g(f) be the area integral and Littlewood-Paley g-function on ∂Ω, respectively. If f e BMOA, then S α (f) < ∞ a.e. on ∂Ω, and there exists a constant C such that ‖S α (f)‖* ੽ C‖f‖*. The same result also holds for g(f).

Some Aspects of the Kobayashi and Carathéodory Metrics on Pseudoconvex Domains

The purpose of this article is to consider two themes, both of which emanate from and involve the Kobayashi and the Carathéodory metric. First, we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in C 2, and on convex finite type domains in C n using the scaling method. Applications include an alternate proof of the Wong–Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point, and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in C 2 and convex finite type domains in C n in terms of Euclidean parameters. Second, a version of Vitushkin’s theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for C 1-isometries of the Kobayashi and Carathéodory metrics on a smoothly bounded strongly pseudoconvex domain.