The L² -Boundedness of the Variational Calderón–Zygmund Operators

In this paper, we verify the L²-boundedness for the jump functions and variations of Calderón–Zygmund singular integral operators with the underlying kernels satisfying ∫ε≤|x-y|≤NK(x,y)dy=∫ε≤|x-y|≤NK(x,y)dx=0∀0<ε≤N<∞,in addition to some proper size and smooth conditions. This result should be the first general criteria for the variational inequalities for kernels not necessarily of convolution type. The L²-boundedness assumption that we verified here is also the starting point of the related results on the (sharp) weighted norm inequalities appeared in many recent papers.