We prove some weighted endpoint estimates for some multilinear operators related to certain singular integral operators on Herz and Herz type Hardy spaces.

Applying discrete Calderón’s identity, we study weighted multi-parameter mixed Hardy space $H_{\mathrm{mix}}^p(\omega ,{ℝ}^{n_1}\times {ℝ}^{n_2})$. Different from classical multi-parameter Hardy space, this space has characteristics of local Hardy space and Hardy space in different directions, respectively. As applications, we discuss the boundedness on $H_{\mathrm{mix}}^p(\omega ,{ℝ}^{n_1}\times {ℝ}^{n_2})$ of operators in mixed Journé’s class.

A weighted theory describing Morrey boundedness of fractional integral operators and fractional maximal operators is developed. A new class of weights adapted to Morrey spaces is proposed and a passage to the multilinear cases is covered.

We establish weighted sharp maximal function inequalities for a linear operator associated to a singular integral operator with non-smooth kernel. As an application, we obtain the boundedness of a commutator on weighted Lebesgue spaces.

We discuss continuity properties of the Weyl product when acting on classical modulation spaces. In particular, we prove that $M^{p,q}$ is an algebra under the Weyl product when p ∈ [1,∞] and 1 ≤ q ≤ min(p,p’).

We discuss the concept of Sobolev space associated to the Laguerre operator $L_{\alpha }=-yd²/dy²-d/dy+y/4+\alpha ²/4y$, y ∈ (0,∞). We show that the natural definition does not agree with the concept of potential space defined via the potentials ${\left(L_{\alpha }\right)}^{-s}$. An appropriate Laguerre-Sobolev space is defined in order to achieve that coincidence. An application is given to the almost everywhere convergence of solutions of the Schrödinger equation. Other Laguerre operators are also considered.