We prove that bilinear fractional integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions. We also treat bilinear singular multiplier operators which preserve regularity and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed Lebesgue spaces.Bényi-Torres [11] showed that for m = 0 these bilinear Calderón-Zygmund operators also satisfy for s > 0 and 1/p 1 + 1/p 2 = 1/q < 1 the estimate J s P a 0 (f, g)
A new simple proof of the bilinear T(1) Theorem in the spirit of the proof of Coifman-Meyer of the celebrated result of David and Journé in the linear case is presented. This new proof is obtained independently of the linear T(1) Theorem by combining recent bilinear square function bounds and a paraproduct construction.
In this work, we give new sufficient conditions for Littlewood-Paley-Stein square function and necessary and sufficient conditions for a Calderón-Zygmund operator to be bounded on Hardy spaces H p with indices smaller than 1. New Carleson measure type conditions are defined for Littlewood-Paley-Stein operators, and the authors show that they are sufficient for the associated square function to be bounded from H p into L p . New polynomial growth BMO conditions are also introduced for Calderón-Zygmund operators. These results are applied to prove that Bony paraproducts can be constructed such that they are bounded on Hardy spaces with exponents ranging all the way down to zero.
This paper will be devoted to study the regularity and continuity properties of the following local multilinear fractional type maximal operators,where Ω is a subdomain in R n , Ω c = R n \Ω and B(x, r) is the ball in R n centered at x with radius r. Several new pointwise estimates for the derivative of the local multilinear maximal function M0,Ω and the fractional maximal functions Mα,Ω (0 < α < mn) will be presented. These estimates will not only enable us to establish certain norm inequalities for these operators in Sobolev spaces, but also give us the opportunity to obtain the bounds of these operators on the Sobolev space with zero boundary values.
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