Extrapolation of vector-valued rearrangement operators II

作者：Mueller Paul F X

来源：Journal of the London Mathematical Society-Second Series, 2012, 85: 722-736.

DOI：10.1112/jlms/jdr063

We determine the extrapolation law of rearrangement operators acting on the Haar system in vector-valued H-p spaces: if 0 %26lt; q %26lt;= p %26lt; 2, then, %26lt;br%26gt;parallel to T-tau,T-q circle times Id(X)parallel to(q/(2-q))(q) %26lt;= A(p, q)parallel to T-tau,T-p circle times Id(X)parallel to(p/(2-p))(p) %26lt;br%26gt;For a fixed Banach space X, the extrapolation range 0 %26lt; q %26lt;= p %26lt; 2 is optimal. If, however, there exists 1 %26lt; p(0) %26lt; infinity, so that %26lt;br%26gt;parallel to T-tau,T-p0 circle times IdE parallel to(Lp0E) %26lt; infinity for each UMD space E, %26lt;br%26gt;then, for any 1 %26lt; p %26lt; infinity, %26lt;br%26gt;parallel to T-tau,T-p circle times Id(E)parallel to(LpE) %26lt; infinity, %26lt;br%26gt;for any UMD space E. (The value p(0) = 2 is not excluded.) We characterize Hilbert spaces in terms of vector-valued rearrangement operators. If %26lt;br%26gt;parallel to T-t,T-2 circle times Id(X)parallel to(L2Y) %26lt; infinity and parallel to T-tau,T-p parallel to(Lp) = infinity (for 1 %26lt; p not equal 2 %26lt; infinity), %26lt;br%26gt;then X is isomorphic to a Hilbert space.

• 出版日期2012-6