We explore Carleson measures given on the Hardy spaces defined over the bicomplex space and investigate the properties of the bicomplex Toeplitz operator. In the first section, we provide a comprehensive introduction and formulate the key definitions necessary for the subsequent analysis. In the second section, we present the results for 𝔻-Carleson measures and establish their relationship with bicomplex Hardy spaces. Finally, in the last section, we extend the concept of the Toeplitz operator to the bicomplex space, thus offering broader prospects.

In this paper, we explored the composition operators on the BMO-space of slice hyperholomorphic functions within the unit ball . Additionally, we also study Carleson measures on quaternionic Dirichlet-type spaces.

In this paper, we study the bicomplex version of the Paley-Weiner theorem and the Cauchy integral formula in the upper half-plane.

Let 𝔻 be the open unit disk in the complex plane. We characterize the boundedness and compactness of the sum of weighted differentiation composition operators (T_ψ, φ f)(z)=∑ _j=0^n(D^j_ψ _j, φf)(z)=∑ _j=0^nψ _j(z) f^(j) (φ (z)), z∈𝔻, where n∈ℕ_0 , ψ _j , j∈0,n , are holomorphic functions on 𝔻 , and φ , a holomorphic self-maps of 𝔻 , acting from Bergman spaces with admissible weights to Zygmund type spaces.

In this paper, we study the bicomplex version of weighted Bergman spaces and the composition operators acting on them. We also investigate the Bergman kernel, duality properties and Berezin transform. This paper is essentially based on the work of Zhu (Operator Theory in Function Spaces of Math. Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence, 2007).

In this paper, we characterize Carleson measures on the Dirichlet type space $$\mathcal D_{\alpha }^{p}$$ for different values of p, q and $$\alpha$$ and in terms of axially symmetric completion of a pseudohyperbolic disc.

In this paper we investigate some basic results on the slice regular Besov spaces of hyperholomorphic functions on the unit ball $\mathbb{B}.$ We also characterize the boundedness, compactness and find the essential norm estimates of composition operators between these spaces.

In this paper, we define the quaternionic Fock spaces F p? of entire slice hyperholomorphic functions in a quaternionic unit ball B in H: We also study growth estimates and various results of entire slice regular functions in these spaces. The work of this paper is motivated by the recent work of [5] and [26].

In this paper we define Bloch-type spaces of slice hyperholomorphic functions in the unit ball \(\mathbb {B}.\) We study the invariance of these spaces with slice regular Möbius transformation and also give some simple characterizations for boundedness and compactness of composition operators on the slice regular Bloch-type spaces on the unit ball \(\mathbb {B}.\) We also estimate the essential norm of these operators.

In this paper, we study the quaternionic counterpart of complex Fock spaces 𝔉_α^p ( 0<p<∞ and for some parameter α) of entire slice hyperholomorphic functions in an Euclidean unit ball 𝔹^n in ℍ^n. We also study some basic properties of these spaces.

In this paper we study the bicomplex version of weighted Hardy spaces. Further, we describe reproducing kernels for the bicomplex weighted Hardy spaces. In particular, we generalize some results which holds for the classical weighted Hardy spaces. We also introduce the notion of bicomplex C*-algebra and discuss some of its properties.

The authors characterize the order boundedness of weighted composition operators acting between Dirichlet type spaces.

In this paper we give a Carleson measure characterization for the compact composition operators between Dirichlet type spaces. We use this characterization to show that every compact composition operator on Dirichlet type spaces is compact on the Bloch space.

We generalize Gallardo-Gutiérrez and Partington’s results on BMOA to the Dirichlet-type spaces with similar strategy. That is, we use the generalized Nevanlinna counting function associated to the weight function to characterize the boundedness and compactness of weighted composition operators on the Dirichlet-type spaces.

In this work, we obtain trace ideal criteria for holomorphic weighted composition operators acting on the weighted Bergman spaces A λ 2 (ι) of a bounded symmetric domain ι in ℂn.

In this work we characterize boundedness and compactness of weighted composition operators acting between Dirichlet type spaces by using Carleson measures. We also find essential norm estimates for these operators.

In this paper, we study boundedness, compactness and the essential norm of a class of weighted composition operators on weighted Bergman spaces.

In this paper we consider linear operators M ψ C ϕ D and M ψ DC ϕ acting between weighted Bergman spaces, where M ψ , C ϕ and D are multiplication, composition and differentation operators respectively.Our goal is to characterize those holomorphic self maps ϕ of D for which M ψ DC ϕ and M ψ C ϕ D acts boundedly and compactly between weighted Bergman spaces.

In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator W φψ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten p-class weighted composition operators.