Bourgain–Morrey Spaces Mixed with Structure of Besov Spaces

Bourgain–Morrey spaces $\mathcal {M}^p_{q,r}(\mathbb R^n)$, generalizing what was introduced by J. Bourgain, play an important role in the study related to the Strichartz estimate and the nonlinear Schrödinger equation. In this article, via adding an extra exponent $\tau $, the authors creatively introduce a new class of function spaces, called Besov–Bourgain–Morrey spaces $\mathcal {M}\dot {B}^{p,\tau }_{q,r}(\mathbb R^n)$, which is a bridge connecting Bourgain–Morrey spaces $\mathcal {M}^p_{q,r}(\mathbb R^n)$ with amalgam-type spaces $(L^q,\ell ^r)^p(\mathbb R^n)$. By making full use of the Fatou property of block spaces in the weak local topology of $L^{q'}(\mathbb R^n)$, the authors give both predual and dual spaces of $\mathcal {M}\dot {B}^{p,\tau }_{q,r}(\mathbb R^n)$. Applying these properties and the Calderón product, the authors also establish the complex interpolation of $\mathcal {M}\dot {B}^{p,\tau }_{q,r}(\mathbb R^n)$. Via fully using fine geometrical properties of dyadic cubes, the authors then give an equivalent norm of $\|\kern 1pt{\cdot }\kern 1pt\|_{\mathcal {M}\dot {B}^{p,\tau }_{q,r}(\mathbb R^n)}$ having an integral expression, which further induces a boundedness criterion of operators on $\mathcal {M}\dot {B}^{p,\tau }_{q,r}(\mathbb R^n)$. Applying this criterion, the authors obtain the boundedness on $\mathcal {M}\dot {B}^{p,\tau }_{q,r}(\mathbb R^n)$ of classical operators including the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator.