Grand Besov–Bourgain–Morrey spaces and their applications to boundedness of operators

Abstract

Let $1<q\le p \le r\le \infty$ and $\tau \in (0,\infty ]$. Besov–Bourgain–Morrey spaces ${\mathcal {M}}\dot{B}^{p,\tau }_{q,r}({\mathbb {R}}^n)$ in the special case where $\tau =r$, extending what was introduced by J. Bourgain, have proved useful in the study related to the Strichartz estimate and the non-linear Schrödinger equation. In this article, by cleverly mixing the norm structures of grand Lebesgue spaces and Besov–Bourgain–Morrey spaces and adding an extra exponent $\theta \in [0,\infty )$, the authors introduce a new class of function spaces, called generalized grand Besov–Bourgain–Morrey spaces ${\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)$. The authors explore their various real-variable properties including pre-dual spaces and the Gagliardo–Peetre and the ± interpolation theorems. Via establishing some equivalent quasi-norms of ${\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)$ related to Muckenhoupt $A_1({\mathbb {R}}^n)$-weights, the authors then obtain an extrapolation theorem of ${\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)$. Applying this extrapolation theorem, the Calderón product, and the sparse family of dyadic grids of ${\mathbb {R}}^n$, the authors establish the sharp boundedness on ${\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)$ of the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator.