Adjoint of generalized Cesáro operators on analytic function spaces

In this article, we define a convolution operator and study its boundedness on mixed-norm spaces. In particular, we obtain a well-known result on the boundedness of composition operators given by Avetisyan and Stević in [K. Avetisyan and S. Stević, The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc, Appl. Math. Comput. 213 2009, 2, 304–311]. Also we consider the adjoint {\mathcal{A}^{b,c}} for {b>0} of two parameter families of Cesáro averaging operators and prove the boundedness on Besov mixed-norm spaces {B_{\alpha+(c-1)}^{p,q}} for {c>1}.

Geometric Properties of Cesàro Averaging Operators

Using positivity of trigonometric cosine and sine sums whose coefficients are generalization of Vietoris numbers, we find the conditions on coefficient {ak} to characterize the geometric properties of the corresponding analytic function f(z)=z+∑k=2∞akzk in the unit disc D. As an application, we also find geometric properties of generalized Cesàro-type polynomials.

Stochastic Lévy differential operators and Yang–Mills equations

The relationship between the Yang–Mills equations and the stochastic analogue of Lévy differential operators is studied. The value of the stochastic Lévy–Laplacian is found by means of Cèsaro averaging of directional derivatives on the stochastic parallel transport. It is shown that the Yang–Mills equations and the Lévy–Laplace equation for such Laplacian are not equivalent in contrast to the deterministic case. An equation equivalent to the Yang–Mills equations is obtained. The equation contains the Lévy divergence. It is proved that the Yang–Mills action functional can be represented as an infinite-dimensional analogue of the Direchlet functional of a chiral field. This analogue is also derived using Cèsaro averaging.

Boundedness of the Cesàro averaging operators on Dirichlet spaces

It is shown that the Cesàro averaging operators Cα, Re α > −1, introduced by Stempak, are bounded on the Dirichlet space Da if and only if a > 0, while the associated operators Aα are bounded on Da if and only if −1 < a < 2. This extends results of Galanopoulos, who considered the particular case α = 0 for 0 ≤ a ≤ 1.

Cesàro averaging operators on Hardy spaces

It is shown that the Cesàro averaging operatorℜα > – 1, satisfies an inequality which immediately implies that it is bounded on certain Hardy spaces including Hp, 0 < p < ∞. This answers an open question of Stempak, who introduced these operators and obtained their boundedness on Hp, 0 < p ≦ 2, for ℜα ≧ 0. The operator which is conjugate to on H2 is also shown to be bounded on Hp for 1 < p < ∞ and ℜα = – 1. This extends a result of Stempak who obtained this boundedness for 2 ≦ p≦ ∞ and ℜα ≧:0.

Cesàro averaging operators

We define a family of Cesàro operators , Reα≧0, and consider the question of their boundedness on Hp spaces. We also consider discrete versions of these operators acting on sequence spaces.