Some Bilinear Estimates

Abstract

Abstract. We establish some estimates on the hyper bilinear Hilbert transform on both Euclidean space and torus. We also use a transference method to obtain a Kenig-Stein's estimate on bilinear fractional integrals on the n-torus.

A Bilinear Fractional Integral on Compact Lie Groups

“…In [6], among other things, Kenig and Stein established the boundedness of ß α ( f , g) from L r (R n ) × L q (R n ) to L p (R n ) with 1/p = 1/q + 1/r − α/n > 0. This result was also recently obtained in the n-torus T n , by using a transference method (see [2]). As T n is the n-dimensional Abelian compact Lie group, it is more interesting to obtain Kenig-Stein's theorem on a general compact Lie group.…”

Maximal bilinear Calderón--Zygmund operators of type $\omega(t)$ on non-homogeneous space

“…We also note that multilinear Calderón-Zygmund theory has been studied by many researchers, we can see [3,5,7,11,13]. Grafakos and Torres [6] investigated the boundedness of maximal multilinear Calderón-Zygmund operator on product of Lebesgue spaces.…”

“…for some A > 0 and (x, y 1 , y 2 ) ∈ (X ) 3 with x = y i for some i. Furthermore, we have the smoothness estimates…”

“…Definition 1.3. Let K(x, y 1 , y 2 ) be a locally integrable function defined away from the diagonal x = y 1 = y 2 in (X ) 3 and ω : [0, ∞) → [0, ∞) be a nondecreasing function. We say that K(x, y 1 , y 2 ) is a bilinear Calderón-Zygmund kernel of type ω(t) if it satisfies the following size estimate…”

Rough bilinear fractional integrals with variable kernels

“…The operators K 1,α and B 1,α were well studied in [6,8]. Some transference results of B 1,α on the n-torus were also obtained in [2] (see also [3] for a version on a compact Lie group). On the other hand, the study of the operators K Ω,α (as well as B Ω,α ) and its related operators with rough Ω were recently attracted several authors (see [4,13]).…”