We prove that the uncentered Hardy–Littlewood maximal operator is discontinuous on ${BMO}(\mathbb {R}^n)$ and maps ${VMO}(\mathbb {R}^n)$ to itself. A counterexample to the boundedness of the strong and directional maximal operators on ${BMO}(\mathbb {R}^n)$ is given, and properties of slices of ${BMO}(\mathbb {R}^n)$ functions are discussed.

In this paper we define a space $VM{{O}_{P}}$ associated with a family $P$ of parabolic sections and show that the dual of $VM{{O}_{P}}$ is the Hardy space $H_{P}^{1}$ . As an application, we prove that almost everywhere convergence of a bounded sequence in $H_{P}^{1}$ implies weak $^{\star }$ convergence

We prove continuity in generalized parabolic Morrey spaces of sublinear operators generated by the parabolic Calderón—Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. As a consequence, we obtain a global -regularity result for the Cauchy—Dirichlet problem for linear uniformly parabolic equations with vanishing mean oscillation (VMO) coefficients.

In this paper we characterize the compactness of the commutator $\left[ b,\,T \right]$ for the singular integral operator on the Morrey spaces ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ . More precisely, we prove that if $b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$ , the $\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ -closure of $C_{c}^{\infty }\left( {{\mathbb{R}}^{n}} \right)$ , then $\left[ b,\,T \right]$ is a compact operator on the Morrey spaces ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ for $1\,<\,p\,<\,\infty $ and $0\,<\,\lambda \,<\,n$ . Conversely, if $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\left[ b,\,T \right]$ is a compact operator on the ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ for some $p\,\left( 1\,<\,p\,<\,\infty\right)$ , then $b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$ . Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $\left[ b,\,T \right]$ on ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.