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

We utilize the (A alpha p</mml:msubsup>,q)-Carleson measure to investigate the bounded small Hankel operator hf alpha</mml:msubsup> acting between the Bergman-type space A alpha p</mml:msubsup> and A alpha q</mml:msubsup><overbar></mml:mover> on the complex unit ball Bn for some alpha <= -1. For n >= 2, and f is an element of H(Bn), we have the conclusions: <list list-type="order">Suppose alpha <= -1, max{1,-2 alpha -2}<p<infinity and pN+alpha>-1. Then hf alpha</mml:msubsup>:A alpha p</mml:msubsup>-> A alpha p<overbar></mml:mover> is bounded if and only if |RNf(z)|pd<mml:msub>vpN+alpha (z) is an (A alpha p,p)-Carleson measure.Suppose -n-1<<alpha><= -1, max{1,-2 alpha -2}<p<q<<infinity> and <disp-formula id="UM0001"><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gcov_a_2226060_um0001.gif"></graphic><mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mml:mtr><mml:mtd>pN>max{<mml:mfrac>q-pp</mml:mfrac>(n+1+alpha)-2(alpha +1),<mml:mfrac>-p(alpha +1)p-1</mml:mfrac>}.</mml:mtd></mml:mtr></mml:mtable></disp-formula> Then hf alpha :A alpha p -> A alpha q<overbar></mml:mover> is bounded if and only if |RNf(z)|qd<mml:msub>vqN+alpha (z) is an (A alpha p,q)-Carleson measure.Suppose 1<q<p<<infinity>, alpha <= -1, N be a positive integer such that qN+alpha>-1. <list list-type="alpha-lower">If |RNf(z)|qd<mml:msub>vqN+alpha (z) is an (A alpha p,q)-Carleson measure, then hf alpha :A alpha p -> A alpha q<overbar></mml:mover> is bounded.Let kappa=<mml:mfrac>pqp-q</mml:mfrac> and kappa=<mml:mfrac>kappa kappa -1</mml:mfrac>. If alpha+(p-1)kappa ' N>-1 and hf alpha :A alpha p -> A alpha q<overbar></mml:mover> is bounded, then f is an element of A alpha kappa with f<mml:msub>A alpha kappa less than or similar to hf alpha .If |RNf(z)|qd<mml:msub>vqN+alpha (z) is an (A alpha p,q)-Carleson measure, then f is an element of A alpha kappa. But, there is an f is an element of A alpha kappa such that |RNf(z)|qd<mml:msub>vqN+alpha <mml:mo stretchy="false">(z<mml:mo stretchy="false">) is not an <mml:mo stretchy="false">(A alpha p<mml:mo>,q<mml:mo stretchy="false">)-Carleson measure.