Extrapolation to mixed norm spaces and applications

This paper establishes extrapolation theory to mixed norm spaces. By applying this extrapolation theory, we obtain the mapping properties of the Rubio de Francia Littlewood-Paley functions and the geometrical maximal functions on mixed norm spaces. As special cases of these results, we have the mapping properties on the mixed norm Lebesgue spaces with variable exponents and the mixed norm Lorentz spaces.

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}.

Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn

The main result of this paper is the embedding $$\begin{array}{} \displaystyle \mathcal{B}^{s,r}_\beta({\it\Omega})\hookrightarrow \mathcal{B}^{s_1,r_1}_{\beta+(n-1)\big(\frac 1s-\frac 1{s_1}\big)}({\it\Omega}), \end{array}$$ 0 < r ≤ r1 ≤ ∞, 0 < s ≤ s1 ≤ ∞, β > –1, of harmonic functions mixed norm spaces on a smoothly bounded domain Ω ⊂ ℝn. We also extend a result on boundedness, in mixed norm, of a maximal function-type operator from the case of the unit disc and the unit ball to general domains in ℝn.