UMD Property And Weak M-Type2Property On The Variable Exponent Lebesgue Spaces

This theory of UMD properties and M-Type2properties in both probabilityand analysis has a high value. The study of the UMD properties, ζ-convexityproperties and Hilbert transform makes the vital significance of UMD spaceresearch. In addition, when we consider the existence and uniqueness of solutions ofstochastic partial differential equation from Hilbert space extend to general Banachspace, not all Banach spaces fit into such a theory. Only under the condition ofBanach spaces have UMD and M-Type2properties, the solution of stochasticevolution equation is the existence and uniqueness. As a result, we discuss variableexponent space has UMD and M-Type2properties, which is of great value toresearch the geometric structure of the variable exponent space and stochastic partialdifferential equations on the variable exponent space.The main content is divided into two parts:The first part, considering variable exponent space is the UMD space and hascarried on the simple extension. First of all, according to the equivalent conditionsof UMD space, under the condition of martingale difference sequence bounded, weproved unconditional martingale difference sequence is almost surely convergence.Equivalent conditions of variable exponent space mode and norm of thetransformation relations, RN properties and convergence of Banach value randomvariable are applied to prove that variable exponent space is the UMD space. Then,we proved the variable exponent space of value in UMD space is the UMD space.Using this result, we proved variable exponential vector-valued Stein inequality.Finally, the boundedness of the Hilbert transformation operater from UMD spaceextend to variable exponent space value in UMD space.The second part, considering variable exponent space has weak M-Type2properties. First of all, according to variable exponent space has RN properties, weproved the convergence of martingale. Then, the closed graph theorem are applied toproved natural mapping of structure two spaces is continuous. Finally, we provedthat bounded martingale has M-Type2properties by using a set of Burkholder’smartingale inequality method.