Classic Class. Singular Integral Operator And Its Operator, A Number Of Issues

This thesis studies the singular integral operators, Marcinkiewicz integrals, fractional operators and their commutators. First we recall the backgrounds on studying these operators and the methods and ideas from which they come and then pose the problem which we will be engaged into.1 Marcinkiewicz integral and its commutator1.1 Marcinkiewicz integralsIt is well-known that Littlewood-Paley g, g_λ~* and the area 5 functions play a very important role both in harmonic analysis and PDE's. In this context, Marcinkiewicz considered the expression μ(f)(x) defined bywhere F(x) = ∫_[0,x]f(t)dt. The Marcinkiewicz integral μ(f)(x) was introduced in order to give an analogue of the Littlewood-Paley g function without going into the interior of the disk for its definition. In 1953, Stein defined a generalization of the Marcinkiewicz integral to a higher dimension, which defined bywhichLet Ω be homogeneous of degree zero and satisfy the following Lip_α (0 < α≤ 1) condition, that is, suppose that Ω is continuous on the unit ball Σ in R~n and satisfiesand...