Search results for "analytic function"

Łojasiewicz exponents, the integral closure of ideals and Newton polyhedra

2003

We give an upper estimate for the Łojasiewicz exponent $\ell(J,I)$ of an ideal $J\subseteq A(K^{n})$ with respect to another ideal I in the ring $A(K^{n})$ of germs analytic functions $f$ : $(K^{n},\mathrm{O})\rightarrow K$ , where $K=C$ or $R$ , using Newton polyhedrons. In particular, we give a method to estimate the Łojasiewicz exponent $\alpha_{0}(f)$ of a germ $f\in A(K^{n})$ that can be applied when $f$ is Newton degenerate with respect to its Newton polyhedron.

Lipschitz-type conditions on homogeneous Banach spaces of analytic functions

2017

Abstract In this paper we deal with Banach spaces of analytic functions X defined on the unit disk satisfying that R t f ∈ X for any t > 0 and f ∈ X , where R t f ( z ) = f ( e i t z ) . We study the space of functions in X such that ‖ P r ( D f ) ‖ X = O ( ω ( 1 − r ) 1 − r ) , r → 1 − where D f ( z ) = ∑ n = 0 ∞ ( n + 1 ) a n z n and ω is a continuous and non-decreasing weight satisfying certain mild assumptions. The space under consideration is shown to coincide with the subspace of functions in X satisfying any of the following conditions: (a) ‖ R t f − f ‖ X = O ( ω ( t ) ) , (b) ‖ P r f − f ‖ X = O ( ω ( 1 − r ) ) , (c) ‖ Δ n f ‖ X = O ( ω ( 2 − n ) ) , or (d) ‖ f − s n f ‖ X = O ( ω …

A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

2016

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain $\Omega_{\boldsymbol\varepsilon}$ obtained by making a small hole of size $\varepsilon_1 \varepsilon_2 $ in an open regular subset $\Omega$ of $\mathbb{R}^n$ at distance $\varepsilon_1$ from the boundary $\partial\Omega$. As $\varepsilon_1 \to 0$, the perforation shrinks to a point and, at the same time, approaches the boundary. When $\boldsymbol\varepsilon \to (0,0)$, the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by $u_{\bolds…

Well-posedness and singularity formation for the Camassa-Holm equation

2006

We prove the well-posedness of Camassa--Holm equation in analytic function spaces both locally and globally in time, and we investigate numerically the phenomenon of singularity formation for particular initial data.

Weakly compact composition operators between algebras of bounded analytic functions

1999

A Unifying Approach to Weyl Type Theorems for Banach Space Operators

2013

Weyl type theorems have been proved for a considerably large number of classes of operators. In this paper, by introducing the class of quasi totally hereditarily normaloid operators, we obtain a theoretical and general framework from which Weyl type theorems may be promptly established for many of these classes of operators. This framework also entails Weyl type theorems for perturbations f(T + K), where K is algebraic and commutes with T, and f is an analytic function, defined on an open neighborhood of the spectrum of T + K, such that f is non constant on each of the components of its domain.

Cluster sets and quasiconformal mappings

2010

Certain classical results on cluster sets and boundary cluster sets of analytic functions, due to Iversen, Lindelof, Noshiro, Tsuji, Ohtsuka, Pommerenke, Carmona, Cufi and others, are extended to n-dimensional quasiconformal mappings. Unlike what is usually the case in the context of analytic functions, our considerations are not restricted to mappings of a disk or ball only. It is shown, for instance, that quasiconformal cluster sets and boundary cluster sets, taken at a non-isolated boundary point of an arbitrary domain, coincide. More refined versions are established in the special case where the domain is the open unit ball. These include cluster set considerations of the induced radial…

Norm, essential norm and weak compactness of weighted composition operators between dual Banach spaces of analytic functions

2017

Abstract In this paper we estimate the norm and the essential norm of weighted composition operators from a large class of – non-necessarily reflexive – Banach spaces of analytic functions on the open unit disk into weighted type Banach spaces of analytic functions and Bloch type spaces. We also show the equivalence of compactness and weak compactness of weighted composition operators from these weighted type spaces into a class of Banach spaces of analytic functions, that includes a large family of conformally invariant spaces like BMOA and analytic Besov spaces.

Remarks on mapping properties for the Bargmann transform on modulation spaces

2011

We investigate the mapping properties for the Bargmann transform and prove that this transform is isometricand bijective from modulation spaces to convenient Banach spaces of analytic functions.

Fredholm composition operators on algebras of analytic functions on Banach spaces

2010

AbstractWe prove that Fredholm composition operators acting on the uniform algebra H∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.