Riesz Transform Of Non-Local Symmetric Markov Operators

The purpose of this master thesis is to study Riesz transform of gradient-type operators corresponding to non-local symmetric Markov processes and related problems.The thesis consists of three parts.In the first part,we consider symmetric Markov processes on metric measure space,and define the corresponding gradient-type operators.We establish the weak type(1,1)boundedness of gradient operators by using heat kernel estimates.This assertion is very useful to study the boundedness of Riesz transform for the Markov operators,and it also includes the settings for local Dirichlet forms and non-local Dirichlet forms.In the second part,we are concerned with modified gradient operators for nonlocal Dirichlet forms.The weighted Lp-estimates and tail estimates for modified gradient operators of heat kernels are established,where pseudo-gradient operators and upper bounds of heat kernel estimates are fully used.The main result applies to symmetric α-stable-like processes on d-sets.In the third part,we further study the weighted Lp-boundedness for LittlewoodPaley functions for non-local Dirichlet forms.For this,we use again pseudo-gradient operators and apply the Riesz-Thorin interpolation theorem.