Weighted estimates for bilinear fractional integral operators: a necessary and sufficient condition for power weights

Weighted estimates for bilinear fractional integral operator on the Heisenberg group

“…• Recall that the condition obtained in Theorem 2 in [KF20] were α ≤ n − λ, β ≤ n − λ, and −n + λ ≤ α + β which certainly implies our conditions. Also it is easy to see that by suitably choosing ǫ > 0 we can construct triplets α = n − λ + ǫ, β = 0, γ = −n + λ which satisfy conditions of Theorem 1.2 but it is not covered by Theorem 2 in [KF20]. • A simple computation shows that our result also recovers the well-known result of Hoang and Moen, see Theorem 10.1 in [HM18].…”

“…The necessary conditions. In [KF20], some counter examples were constructed to conclude necessary conditions for the boundedness of BI λ on the real line. Here, we construct them on the Heisenberg group H n of any dimension.…”

“…is not yet known whether the condition A p,q,r is also necessary for the boundedness of BI α . Interestingly, if we only consider power weights then it was shown in [KF20] that it is possible to obtain both necessary and sufficient conditions on α, β, and γ such that BI λ is bounded from L p (|x| αp ) × L q (|x| βq ) to L r (|x| −γr ), in the particular case when γ satisfies γ = −α − β.…”

A note on the bilinear fractional integral operator acting on Morrey spaces

“…We will follow the idea used in [10, Theorem 2]. See also [17,Theorem 3.2] and [13,14] as well. We decompose…”

“…The second-named author was supported by Grant-in-Aid for Scientific Research (C) (16K05209), the Japan Society for the Promotion of Science. The authors are thankful to Professor Komori-Furuya Yasuo for his kind introcduction to [13,14].…”