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

Let 1 < q ≤ p ≤ r ≤ ∞ and τ ∈ (0 , ∞ ] . Besov–Bourgain–Morrey spaces M B ˙ q , r p , τ (R n) in the special case where τ = 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 θ ∈ [ 0 , ∞) , the authors introduce a new class of function spaces, called generalized grand Besov–Bourgain–Morrey spaces M B ˙ q) , r , θ p , τ (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 M B ˙ q) , r , θ p , τ (R n) related to Muckenhoupt A 1 (R n) -weights, the authors then obtain an extrapolation theorem of M B ˙ q) , r , θ p , τ (R n) . Applying this extrapolation theorem, the Calderón product, and the sparse family of dyadic grids of R n , the authors establish the sharp boundedness on M B ˙ q) , r , θ p , τ (R n) of the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator. [ABSTRACT FROM AUTHOR]