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Keywords:
variable exponent; atomic decomposition; martingale Hardy space; fractional integral
Summary:
This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let $(\Omega , \mathcal {F}, \mathbb {P})$ be a probability space and $p(\cdot )\colon \Omega \rightarrow (0,\infty )$ be a $\mathcal {F}$-measurable function such that $0<\inf \nolimits _{x\in \Omega }p(x)\leq \sup \nolimits _{x\in \Omega }p(x)<\infty $. It is proved that a predictable martingale Hardy space $\mathcal P_{p(\cdot )}$ has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy space with variable exponents when the stochastic basis is regular.
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