Variation and oscillation for harmonic operators in the inverse Gaussian setting

<p style='text-indent:20px;'>We prove variation and oscillation <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish <inline-formula><tex-math id="M3">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-boundedness properties for weighted difference involving the semigroups under consideration.</p>

A New Characterization of Hardy Martingale Cotype Space

We give a new characterization of Hardy martingale cotype property of complex quasi- Banach space by using the existence of a kind of plurisubharmonic functions. We also characterize the best constants of Hardy martingale inequalities with values in the complex quasi-Banach space.

Boundedness of vector-valued martingale transforms on extreme points and applciations

Let Β1, Β2be a pair of Banach spaces andTbe a vector valued martingale transform (with respect to general filtration) which maps Β1-valued martingales into Β2-valued martingales. Then, the following statements are equivalent:Tis bounded fromintofor somep(or equivalently for everyp) in the range 1 <p< ∞;Tis bounded fromintoBMOB2;Tis bounded fromBMOB1intoBMOB2;Tis bounded frominto. Applications toUMDand martingale cotype properties are given. We also prove that the Hardy spacedefined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if Β has the Radon-Nikodým property.