Small Hankel operators between Bergman-type spaces in the unit ball

Abstract

We utilize the ( A α p , q ) -Carleson measure to investigate the bounded small Hankel operator h f α acting between the Bergman-type space A α p and A α q ¯ on the complex unit ball B n for some α ≤ − 1 . For n ≥ 2 , and f ∈ H ( B n ) , we have the conclusions: Suppose α ≤ − 1 , max { 1 , − 2 α − 2 } < p < ∞ and -1 $ ]]> pN + α > − 1 . Then h f α : A α p → A α p ¯ is bounded if and only if | R N f ( z ) | p d v pN + α ( z ) is an ( A α p , p ) -Carleson measure. Suppose − n − 1 < α ≤ − 1 , max { 1 , − 2 α − 2 } < p < q < ∞ and \\max\\left\\{ \\frac {q-p}p (n+1+\\alpha)-2(\\alpha+1),\\frac{-p(\\alpha+1)}{p-1}\\right\\}. \\end{align*} $$]]> pN > max { q − p p ( n + 1 + α ) − 2 ( α + 1 ) , − p ( α + 1 ) p − 1 } . Then h f α : A α p → A α q ¯ is bounded if and only if | R N f ( z ) | q d v qN + α ( z ) is an ( A α p , q ) -Carleson measure. Suppose 1 < q < p < ∞ , α ≤ − 1 , N be a positive integer such that -1 $ ]]> qN + α > − 1 . If | R N f ( z ) | q d v qN + α ( z ) is an ( A α p , q ) -Carleson measure, then h f α : A α p → A α q ¯ is bounded. Let κ = pq p − q and κ ′ = κ κ − 1 . If -1 $ ]]> α + ( p − 1 ) κ ′ N > − 1 and h f α : A α p → A α q ¯ is bounded, then f ∈ A α κ with ‖ f ‖ A α κ ≲ ‖ h f α ‖ . If | R N f ( z ) | q d v qN + α ( z ) is an ( A α p , q ) -Carleson measure, then f ∈ A α κ . But, there is an f ∈ A α κ such that | R N f ( z ) | q d v qN + α ( z ) is not an ( A α p , q ) -Carleson measure.