We revisit Hardy's inequality in the scope of regular Dirichlet forms following an analytical method. We shall give an alternative necessary and sufficient condition for the occurrence of Hardy's inequality. A special emphasis will be given for the case where the Dirichlet form under consideration is strongly local, extending therefore some known results in the Euclidean case.
A new and simple proof of the embedding of the Hardy--Hilbert space of Dirichlet series into a conformally invariant Hardy space of the half-plane is presented, and the optimal constant of the embedding is computed.
Given a frequency $λ$, we study general Dirichlet series $\sum a_n e^{-λ_n s}$. First, we give a new condition on $λ$ which ensures that a somewhere convergent Dirichlet series defining a bounded holomorphic function in the right half-plane converges uniformly in this half-plane, improving classical results of Bohr and Landau. Then, following recent works of Defant and Schoolmann, we investigate Hardy spaces of these Dirichlet series. We get general results on almost sure convergence which have an harmonic analysis flavour. Nevertheless, we also exhibit examples showing that it seems hard to g…
The aim of this paper is to study when two composition operators on the Hilbert space of Dirichlet series with square summable coefficients belong to the same component or when their difference is compact. As a corollary we show that if a linear combination of composition operators with polynomial symbols of degree at most 2 is compact, then each composition operator is compact.
Given a frequency $λ=(λ_n)$, we study when almost all vertical limits of a $\mathcal{H}_1$-Dirichlet series $\sum a_n e^{-λ_ns}$ are Riesz-summable almost everywhere on the imaginary axis. Equivalently, this means to investigate almost everywhere convergence of Fourier series of $H_1$-functions on so-called $λ$-Dirichlet groups, and as our main technical tool we need to invent a weak-type $(1, \infty)$ Hardy-Littlewood maximal operator for such groups. Applications are given to $H_1$-functions on the infinite dimensional torus $\mathbb{T}^\infty$, ordinary Dirichlet series $\sum a_n n^{-s}$, a…
The main purpose of this article is to survey on some key elements of a recent $\mathcal{H}_p$-theory of general Dirichlet series $\sum a_n e^{-λ_{n}s}$, which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet series $\sum a_n n^{-s}$. In view of an ingenious identification of Bohr, the $\mathcal{H}_p$-theory of ordinary Dirichlet series can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus $\mathbb{T}^\infty$. Extending these ideas, the $\mathcal{H}_p$-theory of $λ$-Dirichlet series is build as a sub-theory of Fourier analysis on what we ca…
By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on $\mathcal{H}^2$, the space of Dirichlet series with square summable coefficients, for the inducing symbol $φ(s)=c_1+c_{q}q^{-s}$ where $q\geq 2$ is a fixed integer. We also give an estimate on the approximation numbers of such an operator.
In this paper, we investigate when weighted composition operators acting on Dirichlet spaces $\mathcal{D}(\mathbb{B}_{N})$ are complex symmetric with respect to some special conjugations, and provide some characterizations of Hermitian weighted composition operators on $\mathcal{D}(\mathbb{B}_{N})$. Furthermore, we give a sufficient and necessary condition for $J$-symmetric weighted composition operators on Hardy spaces $H^2(\mathbb{B}_{N})$ to be unitary or Hermitian, then some new examples of complex symmetric weighted composition operators on $H^2(\mathbb{B}_{N})$ are obtained. We also disc…
We study the Hardy space of translated Dirichlet series $\mathcal{H}_{+}$. It consists on those Dirichlet series $\sum a_n n^{-s}$ such that for some (equivalently, every) $1 \leq p < \infty$, the translation $\sum{a_{n}}n^{-(s+\frac{1}σ)}$ belongs to the Hardy space $\mathcal{H}^{p}$ for every $σ>0$. We prove that this set, endowed with the topology induced by the seminorms $\left\{\Vert\cdot\Vert_{2,k}\right\}_{k\in\mathbb{N}}$ (where $\Vert\sum{a_{n}n^{-s}}\Vert_{2,k}$ is defined as $\big\Vert\sum{a_n n^{-(s+\frac{1}{k})}} \big\Vert_{\mathcal{H}^{2}}$), is a Fréchet space which is Schwartz …
In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces.
We consider continuous semigroups of analytic functions $\{Φ_t\}_{t\geq0}$ in the so-called Gordon-Hedenmalm class $\mathcal{G}$, that is, the family of analytic functions $Φ:\mathbb C_+\to \mathbb C_+$ giving rise to bounded composition operators in the Hardy space of Dirichlet series $\mathcal{H}^2$. We show that there is a one-to-one correspondence between continuous semigroups $\{Φ_{t}\}_{t\geq0}$ in the class $\mathcal G$ and strongly continuous semigroups of composition operators $\{T_t\}_{t\geq0}$, where $T_t(f)=f\circΦ_t$, $f\in\mathcal{H}^2$. We extend these results for the range $p\i…
We characterize the space of multipliers from the Hardy space of Dirichlet series $\mathcal H_p$ into $\mathcal H_q$ for every $1 \leq p,q \leq \infty$. For a fixed Dirichlet series, we also investigate some structural properties of its associated multiplication operator. In particular, we study the norm, the essential norm, and the spectrum for an operator of this kind. We exploit the existing natural identification of spaces of Dirichlet series with spaces of holomorphic functions in infinitely many variables and apply several methods from complex and harmonic analysis to obtain our results.…
Given a Banach space $X$ and $1 \leq p \leq \infty$, it is well known that the two Hardy spaces $H_p(\mathbb{T},X)$ ($\mathbb{T}$ the torus) and $H_p(\mathbb{D},X)$ ($\mathbb{D}$ the disk) have to be distinguished carefully. This motivates us to define and study two different types of Hardy spaces $\mathcal{H}_p(X)$ and $\mathcal{H}^+_p(X)$ of Dirichlet series $\sum_n a_n n^{-s}$ with coefficients in $X$. We characterize them in terms of summing operators as well as holomorphic functions in infinitely many variables, and prove that they coincide whenever $X$ has the analytic Radon-Nikodým Prop…
A result of Helson on general Dirichlet series $\sum a_{n} e^{-λ_{n}s}$ states that, whenever $(a_{n})$ is $2$-summable and $λ=(λ_{n})$ satisfies a certain condition introduced by Bohr, then for almost all homomorphism $ω\colon (\mathbb{R},+) \to \mathbb{T}$ the Dirichlet series $\sum a_{n} ω(λ_{n})e^{-λ_{n}s}$ converges on the open right half plane $[Re>0]$. For ordinary Dirichlet series $\sum a_n n^{-s}$ Hedenmalm and Saksman related this result with the famous Carleson-Hunt theorem on pointwise convergence of Fourier series, and Bayart extended it within his theory of Hardy spaces $\mathcal…
A theorem of Henry Helson shows that for every ordinary Dirichlet series $\sum a_n n^{-s}$ with a square summable sequence $(a_n)$ of coefficients, almost all vertical limits $\sum a_n χ(n) n^{-s}$, where $χ: \mathbb{N} \to \mathbb{T}$ is a completely multiplicative arithmetic function, converge on the right half-plane. We survey on recent improvements and extensions of this result within Hardy spaces of Dirichlet series -- relating it with some classical work of Bohr, Banach, Carleson-Hunt, Cesàro, Hardy-Littlewood, Hardy-Riesz, Menchoff-Rademacher, and Riemann.
We consider composition operators $\mathscr{C}_φ$ on the Hardy space of Dirichlet series $\mathscr{H}^2$, generated by Dirichlet series symbols $φ$. We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of $φ$. The other concerns general symbols, and is based on a geometrical condition on the boundary values of $φ$. Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particula…
We present some open problems and describe briefly some possible research directions in the emerging theory of Hardy spaces of Dirichlet series and their intimate counterparts, Hardy spaces on the infinite-dimensional torus. Links to number theory are emphasized throughout the paper.
For a Dirichlet series symbol $g(s) = \sum_{n \geq 1} b_n n^{-s}$, the associated Volterra operator $\mathbf{T}_g$ acting on a Dirichlet series $f(s)=\sum_{n\ge 1} a_n n^{-s}$ is defined by the integral $f\mapsto -\int_{s}^{+\infty} f(w)g'(w)\,dw$. We show that $\mathbf{T}_g$ is a bounded operator on the Hardy space $\mathcal{H}^p$ of Dirichlet series with $0 < p < \infty$ if and only if the symbol $g$ satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of ${\operatorname{BMOA…
Let $n\ge2$ and $Ω$ be a bounded non-tangentially accessible domain (for short, NTA domain) of $\mathbb{R}^n$. Assume that $L_D$ is a second-order divergence form elliptic operator having real-valued, bounded, measurable coefficients on $L^2(Ω)$ with the Dirichlet boundary condition. The main aim of this article is threefold. First, the authors prove that the heat kernels $\{K_t^{L_D}\}_{t>0}$ generated by $L_D$ are Hölder continuous. Second, for any $p\in(0,1]$, the authors introduce the `geometrical' Hardy space $H^p_r(Ω)$ by restricting any element of the Hardy space $H^p(\mathbb{R}^n)$ to …
Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different properties of dilations systems of a fixed function $φ\in L^2(0,1)$. More precisely, we study when the system $\{φ(nx)\}_n$ is a Bessel sequence, a Riesz sequence, or it satisfies the lower frame bound. We are able to characterize these properties in terms of multipliers of the Hardy space $\mathcal{H}^2$ of Dirichtet series and, also, in terms of Hardy spaces on the infinite polytorus. We also address the multivariate case.
We show that the boundedness of the Hardy-Littlewood maximal operator on a Köthe function space ${\mathbb{X}}$ and on its Köthe dual ${\mathbb{X}}'$ is equivalent to the well-posedness of the $\mathbb{X}$-Dirichlet and $\mathbb{X}'$-Dirichlet problems in $\mathbb{R}^{n}_{+}$ in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versio…
We show that de Branges-Rovnyak spaces include as special cases a number of spaces, such as the Hardy space, the Fock space, the Hardy-Sobolev space and the Dirichlet space. We present a general framework in which all these spaces can be obtained by specializing a sequence that appears in the construction. We show how to exploit this approach to solve interpolation problems in the Fock space.
Hardy-Littlewood type phenomena and the Girela-Peláez conjecture for the Möbius invariant Laplacian operator
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）；Center for Applied Mathematics of Guangxi, Guangxi Normal University（广西师范大学应用数学中心）；Industry-Academia Co-innovation and Research Promotion Headquarters, Kyushu Sangyo University（九州产业大学产学协同创新与研究推进总部）
Critical Norm Profiles for Finite-Prime Composition Operators on the Hardy Space of Dirichlet Series
National Yang Ming Chiao Tung University（国立阳明交通大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Indian Institute of Technology, Kanpur（坎普尔印度理工学院）
For any real $β$ let $H^2_β$ be the Hardy-Sobolev space on the unit disc $\mathbb{D}$. $H^2_β$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $β>1/2$. In this paper, we characterize that for a non-constant analytic function $φ:\mathbb{D}\to\mathbb{D}$, when the composition operator $C_{φ}$ on $H^{2}_{β}$ is Fredholm. For $1/2<β<1$, we also prove that $C_{φ}$ has dense range in $H_{β}^{2}$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $φ(\mathbb{D})$. It follows that if the range of $C_{φ}$ is dense in $H_{β}^{2}$, then $φ$…
We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols $φ$ on a scale of Bergman--type Hilbert spaces $\mathcal{D}_α$. We investigate the optimal $β$ such that the composition operator $\mathcal{C}_φ$ maps $\mathcal{D}_α$ boundedly into $\mathcal{D}_β$. We also prove a new embedding theorem for the non-Hilbertian Hardy space $\mathcal H^p$ into a Bergman space in the half-plane and …
We study composition operators on the Hardy space $\mathcal{H}^2$ of Dirichlet series with square summable coefficients. Our main result is a necessary condition, in terms of a Nevanlinna-type counting function, for a certain class of composition operators to be compact on $\mathcal{H}^2$. To do that we extend our notions to a Hardy space $\mathcal{H}_Λ^2$ of generalized Dirichlet series, induced in a natural way by a sequence of Beurling's primes.
