Abstract
In this paper, we consider the martingale Hardy spaces defined with the help of the mixed L_{overrightarrow{p}}-norm. Five mixed martingale Hardy spaces will be investigated: H_{overrightarrow{p}}^{s}, H_{overrightarrow{p}}^S, H_{overrightarrow{p}}^M, mathcal {P}_{overrightarrow{p}}, and mathcal {Q}_{overrightarrow{p}}. Several results are proved for these spaces, like atomic decompositions, Doob’s inequality, boundedness, martingale inequalities, and the generalization of the well-known Burkholder–Davis–Gundy inequality.
Highlights
The mixed Lebesgue spaces were introduced in 1961 by Benedek and Panzone [2]
The boundedness of operators on mixed-norm spaces has been studied for instance by Fernandez [29] and Stefanov and Torres [36]
Torres and Ward [38] gave the wavelet characterization of the space L−→p (Rn)
Summary
The mixed Lebesgue spaces were introduced in 1961 by Benedek and Panzone [2]. They considered the Descartes product ( , F, P) of the probability spaces ( i , Fi , Pi ), where. Musielak–Orlicz–Hardy spaces were studied in Yang et al [44] These results were investigated for martingale Hardy spaces in Jiao et al [26,27] and Xie et al. We will develop a similar theory for mixed-norm martingale Hardy spaces. Herz [19] and Weisz [40] gave one of the most powerful techniques in the theory of martingale Hardy spaces, the so-called atomic decomposition. Using the atomic decompositions and Doob’s inequality, the boundedness of general σ -subadditive operators from H−→sp to L−→p , from P−→p to L−→p and from Q−→p to L−→p can be proved (see Theorems 7 and 8). With the help of these general boundedness theorems, several martingale inequalities will be proved in Sect. We would like to thank the referee for reading the paper carefully and for his/her useful comments and suggestions
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call