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

A Nehari type theorem and applications to Toeplitz+Hankel operators on the Dirichlet space

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.

Spectral bounds for exit times on metric measure Dirichlet spaces and applications

Abstract We prove a generalized version of the Principle for Green's functions on bounded inner uniform domains in a wide class of Dirichlet spaces. In particular, our results apply to higher‐dimensional fractals such as Sierpinski carpets in , , as well as generalized fractal‐type spaces that do not have a well‐defined Hausdorff dimension or walk dimension. This yields new instances of the principle for these spaces. We also discuss applications to Schrödinger operators.

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.

Tolokonnikov’s Corona Theorem is used to obtain two results on cyclicity in Besov–Dirichlet spaces.

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.

Abstract To a conformal map $$\textrm{f}$$ from the disk $${\mathbb {D}}$$ into the complex plane onto a domain with rectifiable Ahlfors-regular boundary, we associate a new kind of Grunsky operator on the Hardy space of the unit disk. This is analogous to the classical Grunsky operator, which itself can be viewed as an operator on Bergman or Dirichlet space. We show that the pull-back of the Smirnov space of the complement of $$\textrm{f}({\mathbb {D}})$$ by $$\textrm{f}$$ is the graph of the Grunsky operator. We also characterize those domains with rectifiable Ahlfors-regular boundaries such that the Grunsky operator is Hilbert-Schmidt. In particular, we show that if the Grunsky operator is Hilbert-Schmidt, then $$\textrm{f}({\mathbb {D}})$$ is a Weil-Petersson quasidisk. The formulations of the results and proofs make essential use of a geometric treatment of Smirnov space as a space of half-order differentials.

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.

The Cauchy Transform on the Bergman–Dirichlet Spaces

Toeplitz operators on the Dirichlet space and the Brown Halmos operator identity

Abstract In this paper, we introduce the notion of $$L^p$$ L p -Green-tight measures of $$L^p$$ L p -Kato class in the framework of symmetric Markov processes. The class of $$L^p$$ L p -Green-tight measures of $$L^p$$ L p -Kato class is defined by the p-th power of resolvent kernels. We first prove that under the $$L^p$$ L p -Green tightness of the measure $$\mu $$ μ , the embedding of extended Dirichlet space into $$L^{2p}(E;\mu )$$ L 2 p ( E ; μ ) is compact under the absolute continuity condition for transient Markov processes, which is an extension of recent seminal work by Takeda. Secondly, we prove the coincidence between two classes of $$L^p$$ L p -Green-tightness, one is originally introduced by Zhao, and another one is invented by Chen. Finally, we prove that our class of $$L^p$$ L p -Green-tight measures of $$L^p$$ L p -Kato class coincides with the class of $$L^p$$ L p -Green tight measures of Kato class in terms of Green kernel under the global heat kernel estimates. We apply our results to d-dimensional Brownian motion and rotationally symmetric relativistic $$\alpha $$ α -stable processes on $$\mathbb {R}^d$$ R d .

Abstract In this note, we give a new necessary condition for the boundedness of the composition operator on the Dirichlet-type space on the disc, via a two dimensional change of variables formula. With the same formula, we characterize the bounded composition operators on the anisotropic Dirichlet-type spaces $\mathfrak {D}_{\vec {a}}(\mathbb {D}^2)$ induced by holomorphic self maps of the bidisc $\mathbb {D}^2$ of the form $\Phi (z_1,z_2)=(\phi _1(z_1),\phi _2(z_2))$ . We also consider the problem of boundedness of composition operators $C_{\Phi }:\mathfrak {D}(\mathbb {D}^2)\to A^2(\mathbb {D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the Dirichlet space of the bidisc.

Quantum differentials of spectral triples, Dirichlet spaces and discrete group

We present an explicit formula for the orthogonal projection onto the subspace of analytic polynomials of degree at most n in the local Dirichlet space D μ , where the positive measure µ consists of a finite number of Dirac measures located at points on the unit circle T . This result has two key aspects: first, while it is known that polynomials are dense in D μ , this approach offers a concrete linear approximation scheme within the space. Second, due to the orthogonality of the polynomials involved, the scheme is qualitative, as the distance of an arbitrary function f ∈ D μ to the projected subspace is explicitly determined.

Blaschke sequences and zero sets for Dirichlet spaces with superharmonic weights

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.