On the boundedness of multilinear fractional strong maximal operators with multiple weights

Abstract

In this paper, we investigated the boundedness of multilinear fractional strong maximal operator MR,α associated with rectangles or related to more general basis with multiple weights A ( p,q),R . In the rectangles setting, we first gave an end-point estimate of MR,α, which not only extended the famous linear result of Jessen, Marcinkiewicz and Zygmund, but also extended the multilinear result of Grafakos, Liu, Pérez and Torres (α = 0) to the case 0 < α < mn. Then, in one weight case, we gave several equivalent characterizations between MR,α and A ( p,q),R , by applying a different approach from what we have used before. Moreover, a sufficient condition for the two weighted norm inequality of MR,α was presented and a version of vector-valued two weighted inequality for the strong maximal operator was established when m = 1. In the general basis setting, we further studied the properties of the multiple weights A ( p,q),R conditions, including the equivalent characterizations and monotonic properties, which essentially extended one's previous understanding. Finally, a survey on multiple strong Muckenhoupt weights was given, which demonstrates the properties of multiple weights related to rectangles systematically.Date: December 28, 2015.

The limiting weak type behaviors and The lower bound for a new weak $L\log L$ type norm of strong maximal operators

“…The strong boundedness, endpoint weak type boundedness and weighted boundedness has been given. Subsequently, similar results was extented to multilinear fractional strong maximal operator by Cao et al [5][6][7]. For more works about M (m) n , we refer the readers to [26,33,34].…”

On the regularity and continuity of the multilinear fractional strong maximal operators

“…For a fixed point 𝑥 = (𝑥 1 , 𝑥 2 ) ∈ ℝ 2 , we define the set ℛ( ⃗ 𝑓)(𝑥 1 , 𝑥 2 ) by 𝑓 are continuous at ⃗ 𝑟 ∈ (ℝ + ) 4 1 for almost every (𝑥 1 , 𝑥 2 ) ∈ ℝ 2 . The other cases are analogous.…”

“…Then, for any 𝑙 = 1, 2 and a.e. 𝑥 ∈ ℝ 2 , we have 𝐷 𝑙 ℳ 𝛼, ( ⃗ 𝑓)(𝑥) = 0 if there exists ⃗ 𝑟 ∈ ℛ( ⃗ 𝑓)(𝑥) ∩ ((ℝ + ) 4 1 ∪ (ℝ + ) 4 2 ∪ {(0, 0, 0, 0)});…”

“…By Lemma 2.2, we have that for a.e. 𝑥 ∈ ℝ 2 , if there exists ⃗ 𝑟 ∈ ℛ( ⃗ 𝑓)(𝑥) ∩ ((ℝ + ) 4 1 ∪ (ℝ + ) 4 2 ∪ {(0, 0, 0, 0)}), then ℳ 𝛼, ( ⃗ 𝑓)(𝑥) = 0. It follows that ∏ 𝑚 𝑖=1 |𝑓 𝑖 (𝑦)| = 0 for a.e.…”

“…When 𝛼 = 0, the operator ℳ 𝛼, is just the multilinear strong maximal operator ℳ  , which was first introduced by Grafakos et al [13] in 2011 when they investigated the weighted strong and endpoint estimates for ℳ  . Motivated by the work [13], Cao et al [3] introduced the operator ℳ 𝛼, and established the weighted strong and endpoint estimates for ℳ 𝛼, (see also [4]). It was pointed out in [3] that…”