On multilinear fractional strong maximal operator associated with rectangles and multiple weights

Abstract

In this paper, the multilinear fractional strong maximal operator M R,α associated with rectangles and corresponding multiple weights A ( p,q),R are introduced. Under the dyadic reverse doubling condition, a necessary and sufficient condition for two-weight inequalities is given. As consequences, we first obtain a necessary and sufficient condition for one-weight inequalities. Then, we give a new proof for the weighted estimates of multilinear fractional maximal operator M α associated with cubes and multilinear fractional integral operator I α , which is quite different and simple from the proof known before.

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

“…Proof. The inclusion relationship A ∞,R (R n ) ⊂ RD (d) (R n ) has been proved in Proposition 4.2 [2]. Thus, it suffices to show that there exists some weight w ∈ RD…”

“…Definition 2.4 (Class of A ( p,q),R , [2]). Let1 < p 1 , · · · , p m < ∞, 1 p = 1 p 1 + · · · + 1 pm , and q > 0.…”

“…Furthermore, Hagelstein and Parissis [13] proved that the asymptotic estimate for weighted Tauberian constant associated to rectangles is equivalent to w ∈ A ∞,R , which gives a new characterization of the class A ∞,R . Still more recently, Cao, Xue and Yabuta [2] introduced the multilinear fractional strong maximal operator M R,α and multiple weights A ( p,q),R associated with rectangles as follows:…”

“…Motivated by the works in [2], [11] and [21], in this paper, we will investigate the boundedness of multilinear strong and fractional strong maximal operators in the setting of rectangles and in the setting of more general basis. We are mainly concerned with the end-point behavior, characterizations of two weighted norm inequalities and vector-valued norm inequalities.…”

Sharp Weighted Bounds for Multilinear Fractional Type Operators Associated with Bergman Projection

“…To prove Theorem 6, the following Carlson embedding Theorem (see, for examples, [10,14,15]) will be needed. And we give the definition of the Carlson sequence first.…”