Abstract A variety of norm inequalities related to Bergman and Dirichlet spaces induced by radial weights is established. Some of the results obtained can be considered as generalizations of certain known special cases while most of the estimates discovered are completely new. In particular, a Littlewood-Paley estimate recently proved by Peláez and the second author (Peláez and Rättyä Adv. Math. , 391 , 70, 2021) is improved in part. The second objective of the paper is to apply the obtained norm inequalities to relate the growth of the maximum modulus of a conformal map f , measured in terms of a weighted integrability condition, to a geometric quantity involving the area of image under f of a disc centered at the origin. Our findings in this direction yield new geometric characterizations of conformal maps in certain weighted Dirichlet and Besov spaces.

On the compactness of a class of radial operators on the weighted Dirichlet spaces

Let D ( μ ) D(\mu ) denote a superharmonically weighted Dirichlet space on the unit disc D \mathbb {D} . We show that outer functions f ∈ D ( μ ) f\in D(\mu ) are cyclic in D ( μ ) D(\mu ) , whenever log ⁡ f \log f belongs to the Pick-Smirnov class N + ( D ( μ ) ) N^+(D(\mu )) . If f f has H ∞ H^\infty -norm less than or equal to 1, then cyclicity can also be checked via iterated logarithms. For example, we show that such outer functions f f are cyclic, whenever log ⁡ ( 1 + log ⁡ ( 1 / f ) ) ∈ N + ( D ( μ ) ) \log (1+ \log (1/f))\in N^+(D(\mu )) . This condition can be checked by verifying that log ⁡ ( 1 + log ⁡ ( 1 / f ) ) ∈ D ( μ ) \log (1+ \log (1/f))\in D(\mu ) . If f f satisfies a mild extra condition, then the conditions also become necessary for cyclicity.

Abstract In this article we present new proofs for the boundedness and the compactness on of the Rhaly matrices, also known as terraced matrices. We completely characterize when such matrices belong to the Schatten class , for . Finally, we apply our results to study the Hadamard multipliers in weighted Dirichlet spaces, answering a question left open by Mashreghi–Ransford.

Abstract This paper aims to study the $$\mathcal {Q}_s$$ Q s and F(p, q, s) Carleson embedding problems near endpoints. We first show that $$\mu $$ μ is an s-Carleson measure if and only if $$id: \mathcal {Q}_t \mapsto \mathcal {T}_{s, 2}^2(\mu )$$ i d : Q t ↦ T s , 2 2 ( μ ) is bounded for any $$0<t<s \le 1$$ 0 < t < s ≤ 1 . Using the same idea, we also prove a near-endpoint Carleson embedding for $$F(p, p\alpha -2, s)$$ F ( p , p α - 2 , s ) for $$\alpha >1$$ α > 1 . Our method is different from the previously known approach, which involves a delicate study of Carleson measures (or logarithmic Carleson measures) on weighted Dirichlet spaces. As some byproducts, the corresponding compactness results are established. Moreover, we completely characterize the boundedness and compactness of a class of g-operators generated by the analytic paraproducts acting on various F(p, q, s) spaces. Finally, we compare the near-endpoint Carleson embedding with the existing solutions of Carleson embedding problems proposed by Xiao, Pau, Zhao, Zhu, etc. Our results assert that a “tiny-perturbed" version of a conjecture on the $$\mathcal {Q}_s$$ Q s Carleson embedding problem due to Liu, Lou, and Zhu is true. We also answer an open question by Pau and Zhao on the F(p, q, s) Carleson embedding near endpoints.

We characterize bounded multiplication operators in weighted Dirichlet spaces that are power bounded, Cesàro bounded and uniformly Kreiss. Moreover, we show the equivalence in such spaces between mean ergodicity and Cesàro boundedness for multiplication operators. We perform the same study for adjoints of multiplication operators. As a particular example, we obtain a uniform mean ergodic multiplication operator in Dirichlet spaces that fails to be power bounded.

In this paper we consider the weighted Hardy space Hβ. This space which gives a generalization of some Hilbert spaces of analytic functions on the unit disk like, the Hardy space H , the weighted Bergman space Bν and the weighted Dirichlet space Dν, it plays a background to our contribution. Especially, we examine the extremal functions for the primitive operator Pf(z) := R [0,z] f(w)dw, where [0, z] = {tz, t ∈ [0, 1]}; and we deduce approximate inversion formulas for the operator P on the weighted Hardy space Hβ.

We study zero-free regions of the Riemann zeta function ζ related to an approximation problem in the weighted Dirichlet space D-2 which is known to be equivalent to the Riemann Hypothesis since the work of Báez-Duarte. We prove, indeed, that analogous approximation problems for the standard weighted Dirichlet spaces Dα when α∈(-3,-2) give conditions so that the half-plane {s∈C:ℜ(s)>-α+12} -\\frac{\\alpha +1}{2}\\}$$\\end{document}]]> is also zero-free for ζ. Moreover, we extend such results to a large family of weighted spaces of analytic functions ℓαp. As a particular instance, in the limit case p=1 and α=-2, we provide a new equivalent formulation of the Prime Number Theorem.

Abstract In this article, by the use of nth derivative characterization, we obtain several some sufficient conditions for all solutions of the complex linear differential equation $$ \begin{align*}f^{(n)}+A_{n-1}(z)f^{(n-1)}+\ldots+A_1(z)f'+A_0(z)f=A_n(z) \end{align*} $$ to lie in weighted Dirichlet spaces and derivative Hardy spaces, respectively, where $A_i(z) (i=0,1,\ldots ,n)$ are analytic functions defined in the unit disc. This work continues the lines of the investigations by Heittokangas, et al. for growth estimates about the solutions of the above equation.

The order bounded Stevi-Sharma operators between weighted Dirichlet spaces are characterized, which generalizes the previous result obtained by Lin and his colleagues.

In [BK] and [KT], the Weighted Dirichlet Spaces of weight α∈(0,1),D_α (D), were discussed. In this paper, we discuss containment relations between analytic spaces on D, in general, and presents special cases of materials covered on [BK]. We make particular emphases on the weighted Dirichlet space of weight 0.5. We produce an example of the containment relationships between the multiplier algebras of the weighted Dirichlet space of weight 0.5, and the space of bounded holomorphic functions on the open unit disc.

We investigate the spectra of composition operators acting on weighted Dirichlet spaces induced by non-automorphic analytic self-maps with fixed points in the unit disk. We still provide more detailed characterizations of the spectra of composition operators induced by univalent symbols and monomial symbols.

Weighted Dirichlet spaces that are de Branges-Rovnyak spaces with equivalent norms

Multiplication operators on weighted Dirichlet spaces

Carleson measures for Hardy-Sobolev spaces in the Siegel upper half-space

Generalized Cesàro operators on weighted Dirichlet spaces

Weighted composition operators that preserve p-frames

Abstract We investigate different geometrical properties, related to Carleson measures and pseudo-hyperbolic separation, of inhomogeneous Poisson point processes on the unit disk. In particular, we give conditions so that these random sequences are almost surely interpolating for the Hardy, Bloch or weighted Dirichlet spaces.

For a complex Borel measure μ \mu on the open unit disk, and for a weighted Dirichlet space H s \mathcal {H}_s with 0 > s > 1 0>s>1 , we characterize the boundedness of the measure induced Hankel type operator H μ , s : H s → H s ¯ H_{\mu ,s}: \mathcal {H}_s \to \overline {\mathcal {H}_s} , extending the results of Xiao [Bull. Austral. Math. Soc. 62 (2000), pp. 135–140] for the classical Hardy space H 2 = H 1 H^2=\mathcal {H}_1 , and of Arcozzi, Rochberg, Sawyer, and Wick [J. Lond. Math. Soc. (2) 83 (2011), pp. 1–18] for the classical Dirichlet space D = H 0 \mathcal {D}= \mathcal {H}_0 . Our approach relies on some recent results about weak products of complete Nevanlinna-Pick reproducing kernel Hilbert spaces. We also include some related results on Hankel measures, Carleson measures, and Toeplitz type operators on weighted Dirichlet spaces H s \mathcal {H}_s , 0 > s > 1 0>s>1 .

Abstract In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators $$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$ on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here, $\mu $ is a positive Radon measure, K is a $\mu $ -measurable function on the open unit disk $\mathbb {D}$ , and $\sigma _w(z)$ is the classical Möbius transform of $\mathbb {D}$ .