The Bohr inequality for ordinary Dirichlet series

R. Balasubramanian; B. Calado; H. Queffélec

Displaying similar documents to “The Bohr inequality for ordinary Dirichlet series”

The comparison principle and Dirichlet problem in the class $_{p} (f)$ , p > 0

Pham Hoang Hiep (2006)

Annales Polonici Mathematici

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We establish the comparison principle in the class $_{p} (f)$ . The result obtained is applied to the Dirichlet problem in $_{p} (f)$ .

The Dirichlet-Bohr radius

Daniel Carando, Andreas Defant, Domingo A. Garcí, Manuel Maestre, Pablo Sevilla-Peris (2015)

Acta Arithmetica

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Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial $\sum_{n \leq x} a_{n} n^{- s}$ we have $\sum_{n \leq x} | a_{n} | r^{Ω (n)} \leq s u p_{t \in ℝ} | \sum_{n \leq x} a_{n} n^{- i t} |$ . We prove that the asymptotically correct order of L(x) is ${(l o g x)}^{1 / 4} x^{- 1 / 8}$ . Following Bohr’s vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results...

Isometric composition operators on weighted Dirichlet space

Shi-An Han, Ze-Hua Zhou (2016)

Czechoslovak Mathematical Journal

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We investigate isometric composition operators on the weighted Dirichlet space $𝒟_{α}$ with standard weights ${(1 - | z |}^{2})^{α}$ , $α > - 1$ . The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space $𝒟$ . We solve some of these but not in general. We also investigate the situation when $𝒟_{α}$ is equipped with another equivalent norm.

Estimates near the boundary for second order derivatives of solutions of the Dirichlet problem for the biharmonic equation

Vladimir A. Kondratiev, Olga A. Oleinik (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Per ogni soluzione della (1) nel dominio limitato $\Omega$ ,, appartenente a $H_{0}^{2}(\Omega)$ e soddisfacente le condizioni (2), si dimostra la maggiorazione (5), valida nell'intorno di ogni punto $x^{0}$ del contorno; si consente a $\partial\Omega$ di essere singolare in $x^{0}$ .

Weak and strong density results for the Dirichlet energy

Mariano Giaquinta, Domenico Mucci (2004)

Journal of the European Mathematical Society

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Let $𝒴$ be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in $B^{n} \times 𝒴$ with equibounded Dirichlet energies, $B^{n}$ being the unit ball in $ℝ^{n}$ . More precisely, weak limits of graphs of smooth maps $u_{k} : B^{n} \to 𝒴$ with equibounded Dirichlet integral give rise to elements of the space ${cart}^{2, 1} (B^{n} \times 𝒴)$ (cf. [4], [5], [6]). In this paper we prove that every element $T$ in ${cart}^{2, 1} (B^{n} \times 𝒴)$ is the...

Estimates near the boundary for second order derivatives of solutions of the Dirichlet problem for the biharmonic equation

Vladimir A. Kondratiev, Olga A. Oleinik (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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On Dirichlet type spaces on the unit ball of $ℂ^{n}$

Małgorzata Michalska (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we discuss characterizations of Dirichlet type spaces on the unit ball of $ℂ^{n}$ obtained by P. Hu and W. Zhang [2], and S. Li [4].

On some $L^{p}$ -estimates for solutions of elliptic equations in unbounded domains

Sara Monsurrò, Maria Transirico (2015)

Mathematica Bohemica

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In this review article we present an overview on some a priori estimates in $L^{p}$ , $p > 1$ , recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^{p}$ -bounds, $p > 2$ , for...

Continuous pluriharmonic boundary values

Per Åhag, Rafał Czyż (2007)

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 ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

Dirichlet forms on quotients of shift spaces

Manfred Denker, Atsushi Imai, Susanne Koch (2007)

Colloquium Mathematicae

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We define thin equivalence relations ∼ on shift spaces $^{\infty}$ and derive Dirichlet forms on the quotient space $Σ =^{\infty} / \sim$ in terms of the nearest neighbour averaging operator. We identify the associated Laplace operator. The conditions are applied to some non-self-similar extensions of the Sierpiński gasket.

Earnshaw’s Theorem in Electrostatics and a Conditional Converse to Dirichlet’s Theorem

Jeffrey Rauch (2013-2014)

Séminaire Laurent Schwartz — EDP et applications

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For the dynamics $x^{''} = - \nabla_{x} V (x)$ , an equilibrium point $\underset{̲}{x}$ are always unstable when on a neighborhood of $\underset{̲}{x}$ the non constant $V$ satisfies $P (x, \partial) V = 0$ for a real second order elliptic $P$ . The proof uses a result of Kozlov [6].

The equation $- Δ 𝑢 - λ \frac{𝑢}{{| 𝑥 |}^{2}} = {| \nabla 𝑢 |}^{𝑝} + 𝑐 𝑓 (𝑥)$ : The optimal power

Boumediene Abdellaoui, Ireneo Peral (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We will consider the following problem $- Δ u - λ \frac{u}{{| x |}^{2}} = {| \nabla u |}^{p} + c f, u > 0 in Ω,$ where $Ω \subset ℝ^{N}$ is a domain such that $0 \in Ω$ , $N \geq 3$ , $c > 0$ and $λ > 0$ . The main objective of this note is to study the precise threshold $p_{+} = p_{+} (λ)$ for which there is novery weak supersolutionif $p \geq p_{+} (λ)$ . The optimality of $p_{+} (λ)$ is also proved by showing the solvability of the Dirichlet problem when $1 \leq p < p_{+} (λ)$ , for $c > 0$ small enough and $f \geq 0$ under some hypotheses that we will prescribe.

Nonvanishing of a certain Bernoulli number and a related topic

Humio Ichimura (2013)

Acta Arithmetica

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Let $p = 1 + 2^{e + 1} q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e + 1}$ (resp. order $d_{φ}$ dividing q), and let ψₙ be an even character of conductor $p^{n + 1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $K ₙ = ℚ (ζ_{(p - 1) p ⁿ})$ . It is well known that the Bernoulli number $B_{1, χ}$ is not zero, which is shown in an analytic way. In the extreme cases $d_{φ} = 1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $T r_{n / 1} (ξ B_{1, χ}) \neq 0$ for any...

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Sunanda Naik, Karabi Rajbangshi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator $_{a, b}$ by $_{a, b} (f) (z) = Γ (a + 1) / Γ (b + 1) \int_{0}^{1} (f (t) {(1 - t)}^{b}) / ({(1 - t z)}^{a + 1}) d t$ , where a and b are non-negative real numbers. In particular, for a = b = β, $_{a, b}$ becomes the generalized Hilbert operator $_{β}$ , and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that $_{a, b}$ is bounded on Dirichlet-type spaces $S^{p}$ , 0 < p < 2, and on Bergman spaces $A^{p}$ , 2 < p < ∞. Also we find an upper bound for the norm of the operator $_{a, b}$ ....

Results on linear and nonlinear hyperbolic boundary value problems at resonance

Michael W. Smiley (1980)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si considera l’equazione non lineare nell'incognita $u(t,x)$ ((1,1) del testo) soddisfatta in un cilindro $G=(o,T)\times\Omega$ ( $\Omega$ dominio limitato di $\bf{R}^{n}$ ) con condizioni al contorno tipo Dirichlet o Neumann sulla superficie laterale di $G$ e con relazioni omogenee fra $u$ e $u_{t}$ sulle basi. Si stabiliscono per la (1) e nel caso di risonanza alcuni teoremi di perturbazione.

Spectral condition, hitting times and Nash inequality

Eva Löcherbach, Oleg Loukianov, Dasha Loukianova (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Let $X$ be a $μ$ -symmetric Hunt process on a LCCB space $𝙴$ . For an open set $𝙶 \subseteq 𝙴$ , let $τ_{𝙶}$ be the exit time of $X$ from $𝙶$ and $A^{𝙶}$ be the generator of the process killed when it leaves $𝙶$ . Let $r : [0, \infty [\to [0, \infty [$ and $R (t) = \int_{0}^{t} r (s) d s$ . We give necessary and sufficient conditions for $𝔼_{μ} R (τ_{𝙶}) l t; \infty$ in terms of the behavior near the origin of the spectral measure of $- A^{𝙶}$ . When $r (t) = t^{l}$ , $l \geq 0$ , by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l + 1$ for $τ_{𝙶}$ ...