The purpose of this paper is to characterize positive measure $\mu$ on the unit disk such that the analytic Morrey space $\mathcal{A}{{\mathcal{L}}_{p,\eta }}$ is boundedly and compactly embedded to the tent space

$$\mathcal{J}_{q,1-\frac{q}{p}\left( 1-\eta \right)}^{\infty }\left( \mu \right)$$

for the case $1\,\le \,q\,\le \,p\,<\,\infty$ respectively. As an application, these results are used to establish the boundedness and compactness of integral operators and multipliers between analytic Morrey spaces.

Let ${{T}_{\Omega }}$ be the singular integral operator with kernel $\left( \Omega \left( x \right) \right)/{{\left| x \right|}^{n}}$, where $\Omega $ is homogeneous of degree zero, has mean value zero, and belongs to ${{L}^{q}}\left( {{S}^{n-1}} \right)$ for some $q\,\in \,\left( 1,\,\infty \right]$. In this paper, the authors establish the compactness on weighted ${{L}^{p}}$ spaces and the Morrey spaces, for the commutator generated by $\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ function and ${{T}_{\Omega }}$. The associated maximal operator and the discrete maximal operator are also considered.

In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasimetric measure spaces, which generalizes some known results on ${{\mathbb{R}}^{n}}$.

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.

The relation between the fractional integral operator and the fractional maximal operator is investigated in the framework of Morrey spaces. Applications to the Fefferman–Phong and the Olsen inequalities are also included.

An a priori Campanato type regularity condition is established for a class of W1X local minimisers $\overline{u}$ of the general variational integral $\int_{\Omega} F(\nabla u(x))\,{\rm d}x$ where $\Omega \subset \mathbb{R}^n$ is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition $F(\xi)\leq c(1+|\xi|^p)$ for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space $\mathcal{L}^{p,\mu} (\Omega,\mathbb{R}^{N\times n})$ , µ < n, Campanato space $\mathcal{L}^{p,n}(\Omega,\mathbb{R}^{N\times n})$ and the space of bounded mean oscillation $ {\rm BMO}\Omega,\mathbb{R}^{N\times n})$ . The admissible maps $u\colon \Omega \to \mathbb{R}^N$ are of Sobolev class W1,p , satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W1BMO local minimisers is extended from Lipschitz maps to this admissible class.

We study the boundedness for multilinear fractional integrals on spaces as Morrey spaces and Lipschitz spaces.