Atomic Decompositions and John-Nirenberg Theorem of Grand Martingale Hardy Spaces with Variable Exponents

Abstract

Let θ ≥ 0 and p · be a variable exponent, and we introduce a new class of function spaces L p · , θ in a probabilistic setting which unifies and generalizes the variable Lebesgue spaces with θ = 0 and grand Lebesgue spaces with p · ≡ p and θ = 1 . Based on the new spaces, we introduce a kind of Hardy-type spaces, grand martingale Hardy spaces with variable exponents, via the martingale operators. The atomic decompositions and John-Nirenberg theorem shall be discussed in these new Hardy spaces.

The Boundedness of Doob’s Maximal and Fractional Integral Operators for Generalized Grand Morrey-Martingale Spaces

“…Remark 12. Recently, new results concerning the grand variable exponent Lebesgue spaces for martingales have emerged (see [33]).…”