In this article we examine Dirichlet type spaces in the unit polydisc, and multipliers between these spaces. These results extend the corresponding work of G. D. Taylor in the unit disc.
In addition, we consider functions on the polydisc whose restrictions to lower dimensional polydiscs lie in the corresponding Dirichet type spaces. We see that such functions need not be in the Dirichlet type space of the whole polydisc. Similar observations are made regarding multipliers.
We obtain new results on Fourier multipliers for Dirichlet-Hardy spaces. As a consequence, we establish a Littlewood-Paley type inequality which yields a simple proof that the Dirichlet monomials form a Schauder basis for p>1.
The Hadamard product of two power series is obtained by multiplying them coefficientwise. In this paper we characterize those power series that act as Hadamard multipliers on all weighted Dirichlet spaces on the disk with superharmonic weights, and we obtain sharp estimates on the corresponding multiplier norms. Applications include an analogue of Fejér's theorem in these spaces, and a new estimate for the weighted Dirichlet integrals of dilates.
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.…
We consider the space $\mathcal{H}(ces_p)$ of all Dirichlet series whose coefficients belong to the Cesàro sequence space $ces_p$, consisting of all complex sequences whose absolute Cesàro means are in $\ell^p$, for $1<p<\infty$. It is a Banach space of analytic functions, for which we study the maximal domain of analyticity and the boundedness of point evaluations. We identify the algebra of analytic multipliers on $\mathcal{H}(ces_p)$ as the Wiener algebra of Dirichlet series shifted to the vertical half-plane $\mathbb{C}_{1/q}:=\{s\in\mathbb{C}:\Re s>1/q\}$, where $1/p+1/q=1$.
Using Carleson measure theorem of weighted Bergman spaces, we provide a complete characterization of embedding theorem for Dirichlet type spaces. As an application, we study the Volterra integral operator and multipliers for Dirichlet type spaces.
Boundedness of weighted composition operators $W_{u,φ}$ acting on the classical Dirichlet space $\mathcal{D}$ as $W_{u,φ}f= u\, (f\circ φ)$ is studied in terms of the multiplier space associated to the symbol $φ$, i.e., ${\mathcal{M}(ϕ)}=\{ u \in {\mathcal D}: W_{u,ϕ} \hbox{ is bounded on } {\mathcal D} \}$. A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of $W_{u,φ}$ in $\mathcal{D}$ whenever $φ$ is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen, Lindström, Nieminen and Saukko to the context of the Dirichl…
Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix…
In this paper, certain classes of Hilbert spaces of Dirichlet series with weighted norms and their corresponding multiplier algebras will be explored. For a sequence $\{w_n\}_{n=n_0}^\infty $ of positive numbers, define \[\mathcal H^\textbf{w}=\left\{\sum_{n=n_0}^\infty a_nn^{-s}:\sum_{n=n_0}^\infty |a_n|^2 w_n<\infty\right\}.\] Hedenmalm, Lindqvist and Seip considered the case in which $w_n\equiv 1$ and classified the multiplier algebra of $\mathcal H^\textbf{w}$ for this space in \cite {HLS}. In \cite{M}, McCarthy classified the multipliers on $\mathcal H^\textbf{w}$ when the weights are giv…
We study a concept of inner function suited to Dirichlet-type spaces. We characterize Dirichlet-inner functions as those for which both the space and multiplier norms are equal to 1.
The classical Dirichlet space is a complete Pick space, hence by a theorem of Agler and McCarthy, there exists an embedding $b$ of the unit disc into a $d$-dimensional ball such that composition with $b$ realizes the Dirichlet space as a quotient of the Drury-Arveson space. We show that $d =\infty$ is necessary, even if we only demand that composition with $b$ induces a surjective map between the multiplier algebras.
We consider various Hilbert spaces of Dirichlet series whose norms are given by weighted $\ell^2$ norms of the Dirichlet coefficients. We characterize the multiplier algebras for some of these spaces.
Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series $D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the right half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be an $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| < \infty$ for every $D \in \mathcal{H}_\infty$. We study the problem of describing such sequences $(b_n)$ in terms of the asymptotic decay of the subsequence $(b_{p_j})$, where $p_j$ denotes the $j$th prime number. Given a multiplicative sequence $b=(b_n)$ we prove (among other results): $b$ is an …
In this work we undertake an extension of various aspects of the potential theory of Dirichlet forms from locally compact spaces to noncommutative C*-algebras with trace. In particular we introduce finite-energy states, potentials and multipliers of Dirichlet spaces. We prove several results among which the celebrated Deny's embedding theorem and the Deny's inequality, the fact that the carre' du champ of bounded potentials are finite-energy functionals and the relative supply of multipliers.
For a sequence $\mathbf w = \{w_j\}_{j = 2}^\infty$ of positive real numbers, consider the positive semi-definite kernel $κ_{\mathbf w}(s, u) = \sum_{j = 2}^\infty w_j j^{-s - \overline{u}}$ defined on some right-half plane $\mathbb H_ρ$ for a real number $ρ.$ Let $\mathscr H_{\mathbf w}$ denote the reproducing kernel Hilbert space associated with $κ_\mathbf w.$ Let \begin{equation*} δ_{\mathbf w} = \inf\Bigg\{\Re(s) : \sum\limits_{\substack{j \geqslant 2 \\ \tiny{\textbf{gpf}}(j) \leqslant p_n }} w_j j^{- s} < \infty ~\text{for all}~ n \in \mathbb Z_+\Bigg\}, \end{equation*} where $\{p_j\}_{j…
The purpose of this paper is to establish bilinear estimates in Besov spaces generated by the Dirichlet Laplacian on a domain of Euclidian spaces. These estimates are proved by using the gradient estimates for heat semigroup together with the Bony paraproduct formula and the boundedness of spectral multipliers.
It follows, from a generalised version of Paley-Wiener theorem, that the Laplace transform is an isometry between certain spaces of weighted $L^2$ functions defined on $(0, \infty)$ and (Hilbert) spaces of analytic functions on the right complex half-plane (for example Hardy, Bergman or Dirichlet spaces). We can use this fact to investigate properties of multipliers and multiplication operators on the latter type of spaces. In this paper we present a full characterisation of multipliers in terms of a generalised concept of a Carleson measure. Under certain conditions, these spaces of analytic …
We prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislaykov and the Kahane-Salem-Zygmund inequality. As a by-product we show various multiplier theorems for spaces of trigonometric polynomials on the $n$-dimensional torus $\mathbb{T}^n$ or Boolean cubes $\{-1,1\}^N$. Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application we show that various recent $\ell_1$-multiplier theorems for trigonome…
We study the multiplier algebras $A(\mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $\mathcal{H}$ on the ball $\mathbb{B}_d$ of $\mathbb{C}^d$. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of $A(\mathcal H)$ in terms of the complementary bands of Henkin and totally singular measures for $\operatorname{Mult}(\mathcal{H})$. This is applied to obtain several definitive results in interpolation. In part…
We prove results on complex interpolation of vector-valued Sobolev spaces over the half-line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space-valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel potential spaces, for which we present a new and simpler proof as well. We apply the results to characterize the fractional domain spaces of the first derivative operator on the half line.
The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Peláez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces. However, their characterizations for the boundedness are not complete. In this paper, we completely characterize the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces $D_α^p$ to $D_β^q$ ($-1<α,β$ and $0<p<q<\infty$), which essentially complete their works. Furthermore, we investigate the order boundedness of Volterra type operators between weighted Dirichlet s…
In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator…
We characterize model polynomials that are cyclic in Dirichlet-type spaces in the unit ball of $\mathbb{C}^n$, and we give a sufficient capacity condition in order to identify non-cyclic vectors.
We obtain a formula that relates the spherical moments of the multiplication tuple on a Dirichlet-type space to a complex moment problem in several variables. This can be seen as the ball-analogue of a formula originally invented by Richter. We capitalize on this formula to study Dirichlet-type spaces on the unit ball and joint $2$-isometries.
We prove a local Douglas formula for higher order weighted Dirichlet-type integrals. With the help of this formula, we study the multiplier algebra of the associated higher order weighted Dirichlet-type spaces $\mathcal H_{\pmbμ},$ induced by an $m$-tuple $\pmb μ=(μ_1,\ldots,μ_{m})$ of finite non-negative Borel measures on the unit circle. In particular, it is shown that any weighted Dirichlet-type space of order $m,$ for $m\geqslant 3,$ forms an algebra under pointwise product. We also prove that every non-zero closed $M_z$-invariant subspace of $\mathcal H_{\pmbμ},$ has codimension $1$ prope…
We extend results of Bénéteau, Condori, Liaw, Seco, and the author concerning cyclic vectors in Dirichlet-type spaces to the setting of the unit ball, identifying some classes of cyclic and non-cyclic functions, and noting the necessity of certain capacity conditions.
We give three characterizations of the Dirichlet-type spaces $D(μ)$. First we characterize $D(μ)$ in terms of a double integral and in terms of the mean oscillation in the Bergman metric, none of them involve the use of derivatives. Next, we obtain another characterization for $D(μ)$ in terms of higher order derivatives. Also, a decomposition theorem for $D(μ)$ is established.
In this paper, using a generalization of a Richter and Sundberg representation theorem, we give a new characterization of Carleson measures for the Dirichlet-type space $\mathcal D(μ)$ when $μ$ is a finite sum of point masses. A reproducing kernel thesis result is also established in this case.
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.
In this note a Fuglede type theorem is proved for Fourier multiplier operators on translation invariant Banach function spaces with order continuous norm over compact abelian groups.
In this paper we prove that if $Λ\in M_p(\mathbb R^N)$ and has compact support then $Λ$ is a weak summability kernel for $1<p<\infty$, where $M_p(\mathbb R^N)$ is the space of multipliers of $L^p(\mathbb R^N)$.
We prove a factorization theorem for reproducing kernel Hilbert spaces whose kernel has a normalized complete Nevanlinna-Pick factor. This result relates the functions in the original space to pointwise multipliers determined by the Nevanlinna-Pick kernel and has a number of interesting applications. For example, for a large class of spaces including Dirichlet and Drury-Arveson spaces, we construct for every function $f$ in the space a pluriharmonic majorant of $|f|^2$ with the property that whenever the majorant is bounded, the corresponding function $f$ is a pointwise multiplier.
We prove weighted restriction type estimates for Grushin operators. These estimates are then used to prove sharp spectral multiplier theorems as well as Bochner-Riesz summability results with sharp exponent.
For a function $φ$ in $L^2(0,1)$, extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates $φ(nx)$, $n=1,2,3,\ldots$, constitutes a Riesz basis or a complete sequence in $L^2(0,1)$. The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space $\cal H$ of Dirichlet series $f(s)=\sum_n a_nn^{-s}$, where the coefficients $a_n$ are square summable. It proves useful to model $\cal H$ as the $H^2$ space of the infinite-dimensional polydisk, or, which is the same, the $H^2$ space of the character space, whe…
Let $X$ be a space of homogeneous type and let $L$ be an injective, non-negative, self-adjoint operator on $L^2(X)$ such that the semigroup generated by $-L$ fulfills Davies-Gaffney estimates of arbitrary order. We prove that the operator $F(L)$, initially defined on $H^1_L(X)\cap L^2(X)$, acts as a bounded linear operator on the Hardy space $H^1_L(X)$ associated with $L$ whenever $F$ is a bounded, sufficiently smooth function. Based on this result, together with interpolation, we establish Hörmander type spectral multiplier theorems on Lebesgue spaces for non-negative, self-adjoint operators …
We establish endpoint estimates for a class of oscillating spectral multipliers on Lie groups of Heisenberg type. The analysis follows an earlier argument due to the second and fourth author but requires the detailed analysis of the wave equation on these groups due to Müller and Seeger. We highlight and develop the connection between sharp bounds for oscillating multipliers and the problem of determining the minimal amount of smoothness required for Mihlin-Hörmander multipliers, a problem that was solved for groups of Heisenberg type but remains open for other groups.
We obtain a general Marcinkiewicz-type multiplier theorem for mixed systems of strongly commuting operators $L=(L_1,...,L_d);$ where some of the operators in $L$ have only a holomorphic functional calculus, while others have additionally a Marcinkiewicz-type functional calculus. Moreover, we prove that specific Laplace transform type multipliers of the pair $(\mathcal{L},A)$ are of certain weak type $(1,1).$ Here $\mathcal{L}$ is the Ornstein-Uhlenbeck operator while $A$ is a non-negative operator having Gaussian bounds for its heat kernel. Our results include the Riesz transforms $A(\mathcal{…
We obtain restriction results of K. de Leeuw's type for maximal operators defined through multilinear Fourier multipliers of either strong or weak type acting on weighted Lebesgue spaces. We give some application of our development. In particular we obtain periodic weighted results for Coifman-Meyer, Hörmander and Hörmander-Mihlin type multilinear multipliers.
We completely describe spaces of multipliers of certain harmonic function spaces of Bergman type in R^n.This is the first sharp result of this kind for Bergman type mixed norm spaces of harmonic functions in the unit ball of R^n
Let $X$ be a Riemannian symmetric space of noncompact type and $T$ be a linear translation-invariant operator which is bounded on $L^p(X)$. We shall show that if $T$ is not a constant multiple of identity then there exist complex constants $z$ such that $zT$ is chaotic on $L^p(X)$ when $p$ is in the sharp range $2<p<\infty$. This vastly generalizes the result that dynamics of the (perturbed) heat semigroup is chaotic on $X$ proved in [15, 17].
In the present work, we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman-Meyer type, finite sum of products of Calderón-Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers $σ(ξ_1,\dots , ξ_m)$ and their derivatives vanish on the hyperplane $ξ_1+\cdots+ξ_m=0$.
