About The Exchange Operator And Some Estimation Of Linear Operator

The theme of this Ph.D thesis is the boundedness of operators in harmonic analysis, we mainly study the boundedness of some commutators and multilinear operators, and related applications. This paper contains six chapters.Chapter0introduce the relevant background of considered problems and the main results.In chapter1. we characterize the boundedness of the commutators generated by three maximal functions with a BMO function on generalized Morrey space respectively, these results extend the known conclusions; besides, we also give the boundedness of the maximal functions commutators with a Lipschitz function on Morrey spaces.In chapter2, we first give a new molecular decomposition of the local Hardy space hp. Using this decomposition, we prove that the strongly singular Calderon Zygmund operator is bounded from Hp to hp. then discuss the boundedness of the commutator of the strongly singular Calderon Zygmund operator on Hqp spaces.In chapter3. we prove that the multilinear operators is bounded on mod-ulation spaces by the famous Nikol’skij-Triebel inequality. As applications, we obtain the boundedness on the modulation spaces for the bilinear Hilbert trans-form, bilinear fractional integral etc. Also, in modulation spaces, we study the well posedness of the Cauchy problem for the fractional heat and wave equations with some new nonlinear terms.Chapter4is devoted to the boundedness of the multilinear operators on Bp,q8(Rn) and the boundedness on Bp,qs(Tn) of the multilinear operators on Tn.In the last chapter, we mainly discuss the bilinear fractional integral with rough kernel BΩ,α. After relaxed the conditions of the kernel and the index, we still get the boundedness of BΩ,α on Lp spaces. At the end, we consider the boundedness on Morrey spaces of BΩ,α.