Non-homogeneous directional equations: Slice solutions belonging to functions of bounded $L$-index in the unit ball

Andriy Bandura; Tetyana Salo; Oleh Skaskiv

Displaying similar documents to “Non-homogeneous directional equations: Slice solutions belonging to functions of bounded $L$ -index in the unit ball”

On spaces of holomorphic functions in ℂⁿ

Diana D. Jiménez S., Lino F. Reséndis O., Luis M. Tovar S. (2014)

Banach Center Publications

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Following the line of Ouyang et al. (1998) to study the $_{p}$ spaces of holomorphic functions in the unit ball of ℂⁿ, we present in this paper several results and relations among $_{p} (ₙ)$ , the α-Bloch, the Dirichlet $_{p}$ and the little $_{p, 0}$ spaces.

On highly nonintegrable functions and homogeneous polynomials

P. Wojtaszczyk (1997)

Annales Polonici Mathematici

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We construct a sequence of homogeneous polynomials on the unit ball $_{d}$ in $ℂ^{d}$ which are big at each point of the unit sphere . As an application we construct a holomorphic function on $_{d}$ which is not integrable with any power on the intersection of $_{d}$ with any complex subspace.

On some extremal problems in classes of holomorphic functions $S_{p}^{} (ρ)$ , $S_{p}^{)} (ρ)$ , $Σ_{p}^{} (ρ)$ , $M C (ρ)$ at additional condition

B. Platynowicz (1980)

Matematički Vesnik

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On L₁-subspaces of holomorphic functions

Anahit Harutyunyan, Wolfgang Lusky (2010)

Studia Mathematica

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We study the spaces $H_{μ} (Ω) = f : Ω \to ℂ h o l o m o r p h i c : \int_{0}^{R} \int_{0}^{2 π} | f (r e^{i φ}) | d φ d μ (r) < \infty$ where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, $H_{μ} (Ω)$ is either isomorphic to l₁ or to ${(\sum \oplus A ₙ)}_{(1)}$ . Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.

Shilov boundary for holomorphic functions on some classical Banach spaces

María D. Acosta, Mary Lilian Lourenço (2007)

Studia Mathematica

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Let $_{\infty} (B_{X})$ be the Banach space of all bounded and continuous functions on the closed unit ball $B_{X}$ of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let $_{u} (B_{X})$ be the subspace of $_{\infty} (B_{X})$ of those functions which are uniformly continuous on $B_{X}$ . A subset $B \subset B_{X}$ is a boundary for $_{\infty} (B_{X})$ if $∥ f ∥ = s u p_{x \in B} | f (x) |$ for every $f \in_{\infty} (B_{X})$ . We prove that for X = d(w,1) (the Lorentz sequence space) and X = C₁(H), the trace class operators, there is a minimal closed boundary for $_{\infty} (B_{X})$ . On the other hand, for X = , the Schreier...

Spaces of geometrically generic configurations

Yoel Feler (2008)

Journal of the European Mathematical Society

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Let $X$ denote either ${ℂℙ}^{m}$ or $ℂ^{m}$ . We study certain analytic properties of the space $ℰ^{n} (X, g p)$ of ordered geometrically generic $n$ -point configurations in $X$ . This space consists of all $q = (q_{1}, \dots, q_{n}) \in X^{n}$ such that no $m + 1$ of the points $q_{1}, \dots, q_{n}$ belong to a hyperplane in $X$ . In particular, we show that for a big enough $n$ any holomorphic map $f : ℰ^{n} ({ℂℙ}^{m}, g p) \to ℰ^{n} ({ℂℙ}^{m}, g p)$ commuting with the natural action of the symmetric group $𝐒 (n)$ in $ℰ^{n} ({ℂℙ}^{m}, g p)$ is of the form $f (q) = τ (q) q = (τ (q) q_{1}, \dots, τ (q) q_{n})$ , $q \in ℰ^{n} ({ℂℙ}^{m}, g p)$ , where $τ : ℰ^{n} ({ℂℙ}^{m}, g p) \to 𝐏𝐒𝐋 (m + 1, ℂ)$ is an $𝐒 (n)$ -invariant holomorphic map. A similar result holds true for mappings of the configuration...

$J$ -holomorphic discs and real analytic hypersurfaces

William Alexandre, Emmanuel Mazzilli (2014)

Annales de l’institut Fourier

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We give in $ℝ^{6}$ a real analytic almost complex structure $J$ , a real analytic hypersurface $M$ and a vector $v$ in the Levi null set at $0$ of $M$ , such that there is no germ of $J$ -holomorphic disc $γ$ included in $M$ with $γ (0) = 0$ and $\frac{\partial γ}{\partial x} (0) = v$ , although the Levi form of $M$ has constant rank. Then for any hypersurface $M$ and any complex structure $J$ , we give sufficient conditions under which there exists such a germ of disc.

Equidistribution towards the Green current for holomorphic maps

Tien-Cuong Dinh, Nessim Sibony (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $f$ be a non-invertible holomorphic endomorphism of a projective space and $f^{n}$ its iterate of order $n$ . We prove that the pull-back by $f^{n}$ of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to $f$ when $n$ tends to infinity. We also give an analogous result for the pull-back of positive closed $(1, 1)$ -currents and a similar result for regular polynomial automorphisms of $ℂ^{k}$ .

New characterizations for weighted composition operator from Zygmund type spaces to Bloch type spaces

Xin-Cui Guo, Ze-Hua Zhou (2015)

Czechoslovak Mathematical Journal

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Let $u$ be a holomorphic function and $ϕ$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u C_{ϕ}$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$ , $ϕ$ , their derivatives, and $ϕ^{n}$ , the $n$ -th power of $ϕ$ . Moreover, we obtain some similar estimates for the essential norms of the operators $u C_{ϕ}$ , from which sufficient and necessary conditions of compactness of $u C_{ϕ}$ follows immediately. ...

Pluriharmonic extension in proper image domains

Rafał Czyż (2009)

Annales Polonici Mathematici

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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D ₁ \times \dots \times D_{s}$ , j=1,...,s, s ≥ 3. Also let $Ω_{π}$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f : \partial Ω_{π} \to ℝ$ that can be extended to a real-valued pluriharmonic function in $Ω_{π}$ .

Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric

Ngaiming Mok (2012)

Journal of the European Mathematical Society

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We study the extension problem for germs of holomorphic isometries $f : (D; x_{0}) \to (Ω; f (x_{0}))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d s_{D}^{2}$ on $D$ and $d s_{Ω}^{2}$ on $Ω$ . Our main focus is on boundary extension for pairs of bounded domains $(D, Ω)$ such that the Bergman kernel $K_{D} (z, w)$ extends meromorphically in $(z, \bar{w})$ to a neighborhood of $\bar{D} \times D$ , and such that the analogous statement holds true for the Bergman kernel $K_{Ω} (ς, ξ)$ on $Ω$ . Assuming that $(D; d s_{D}^{2})$ and $(Ω; d s_{Ω}^{2})$ are complete Kähler manifolds, we prove that...

Equidistribution towards the Green current

Vincent Guedj (2003)

Bulletin de la Société Mathématique de France

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Let $f : ℙ^{k} \to ℙ^{k}$ be a dominating rational mapping of first algebraic degree $λ \geq 2$ . If $S$ is a positive closed current of bidegree $(1, 1)$ on $ℙ^{k}$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks $λ^{- n} {(f^{n})}^{*} S$ converge to the Green current $T_{f}$ . For some families of mappings, we get finer convergence results which allow us to characterize all $f^{*}$ -invariant currents.

Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ

Kuzman Adzievski (2006)

Annales Polonici Mathematici

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We study questions related to exceptional sets of pluri-Green potentials $V_{μ}$ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials $V_{μ}$ are defined by $V_{μ} (z) = \int_{B} l o g (1 / | ϕ_{z} (w) |) d μ (w)$ , where for a fixed z ∈ B, $ϕ_{z}$ denotes the holomorphic automorphism of B satisfying $ϕ_{z} (0) = z$ , $ϕ_{z} (z) = 0$ and $(ϕ_{z} \circ ϕ_{z}) (w) = w$ for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...

Linearization of germs: regular dependence on the multiplier

Stefano Marmi, Carlo Carminati (2008)

Bulletin de la Société Mathématique de France

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We prove that the linearization of a germ of holomorphic map of the type $F_{λ} (z) = λ (z + O (z^{2}))$ has a $𝒞^{1}$ -holomorphic dependence on the multiplier $λ$ . $𝒞^{1}$ -holomorphic functions are $𝒞^{1}$ -Whitney smooth functions, defined on compact subsets and which belong to the kernel of the $\bar{\partial}$ operator. The linearization is analytic for $| λ | \neq 1$ and the unit circle $𝕊^{1}$ appears as a natural boundary (because of resonances,roots of unity). However the linearization is still defined at most points of $𝕊^{1}$ , namely those points which lie “far enough...

Moser's Inequality for a class of integral operators

Finbarr Holland, David Walsh (1995)

Studia Mathematica

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Let 1 < p < ∞, q = p/(p-1) and for $f \in L^{p} (0, \infty)$ define $F (x) = (1 / x) ʃ_{0}^{x} f (t) d t$ , x > 0. Moser’s Inequality states that there is a constant $C_{p}$ such that $s u p_{a \leq 1} s u p_{f \in B_{p}} ʃ_{0}^{\infty} e x p [a x^{q} {| F (x) |}^{q} - x] d x = C_{p}$ where $B_{p}$ is the unit ball of $L^{p}$ . Moreover, the value a = 1 is sharp. We observe that $F = K_{1}$ f where the integral operator $K_{1}$ has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for...

Holomorphic line bundles and divisors on a domain of a Stein manifold

Makoto Abe (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $D$ be an open set of a Stein manifold $X$ of dimension $n$ such that $H^{k} (D, 𝒪) = 0$ for $2 \leq k \leq n - 1$ . We prove that $D$ is Stein if and only if every topologically trivial holomorphic line bundle $L$ on $D$ is associated to some Cartier divisor $𝔡$ on $D$ .

Area differences under analytic maps and operators

Mehmet Çelik, Luke Duane-Tessier, Ashley Marcial Rodriguez, Daniel Rodriguez, Aden Shaw (2024)

Czechoslovak Mathematical Journal

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Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping $h$ and that of $z h$ , we study various $L^{2}$ norms for $T_{ϕ} (h)$ , where $T_{ϕ}$ is the Toeplitz operator with symbol $ϕ$ . In Theorem , given polynomials $p$ and $q$ we find a symbol $ϕ$ such that $T_{ϕ} (p) = q$ . We extend some of our results to the polydisc.

Displaying similar documents to “Non-homogeneous directional equations: Slice solutions belonging to functions of bounded L -index in the unit ball”

Displaying similar documents to “Non-homogeneous directional equations: Slice solutions belonging to functions of bounded $L$ -index in the unit ball”