Martingale Hardy Spaces and Probabilistic Analysis

Martingale Hardy spaces form a bridge between classical harmonic analysis and probability theory by extending Hardy space structures to stochastic processes. Originating from the study of boundary behaviour of holomorphic functions, Hardy spaces in a probabilistic setting characterise martingales through maximal functions and square functions. Central results include Doob’s maximal inequalities, atomic decompositions and Burkholder–Davis–Gundy type equivalences, which together provide powerful tools for controlling the growth and oscillation of martingales. Recent advances have introduced variable‐exponent frameworks and hybrid function spaces—such as Orlicz–Lorentz–Karamata variants—allowing fine‐tuned analysis in inhomogeneous or non‐stationary contexts. These developments have found applications in Fourier analysis, signal processing and quantitative finance, where one requires sharp norm estimates, convergence results and bounds for fractional integral operators acting on stochastic integrals.

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Technical terms

Martingale: A stochastic process whose conditional expectation at each time equals its current value, capturing a “fair‐game” property.

Hardy space (H p): A function or process space defined by finiteness of certain maximal or square‐function norms, generalising L p spaces in harmonic analysis.

Atomic decomposition: A representation of elements in a Hardy space as sums of simple “atoms” with controlled size and cancellation properties.

Maximal operator: An operator assigning to each function or process the supremum of its averages or square functions, fundamental for establishing norm inequalities.

Orlicz–Lorentz–Karamata space: A family of function spaces parametrised by two growth functions, generalising Orlicz and Lorentz scales to capture variable integrability conditions.

Fractional integral operator: A linear operator extending classical integration to non‐integer orders, used to probe intermediate smoothness and long‐range interactions.

References

1. Orlicz-Lorentz-Karamata Hardy martingale spaces: inequalities and fractional integral operators. Fractional Calculus and Applied Analysis (2024).
2. Atomic Decompositions and John‐Nirenberg Theorem of Grand Martingale Hardy Spaces with Variable Exponents. Journal of Function Spaces (2022).
3. New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents. Fractional Calculus and Applied Analysis (2022).