Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes

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http://hdl.handle.net/10138/307062

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Hänninen , T S 2017 , ' Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes ' , Israel Journal of Mathematics , vol. 219 , no. 1 , pp. 71-114 . https://doi.org/10.1007/s11856-017-1474-2

Title: Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes
Author: Hänninen, Timo S.
Contributor organization: Department of Mathematics and Statistics
Date: 2017-04
Language: eng
Number of pages: 44
Belongs to series: Israel Journal of Mathematics
ISSN: 0021-2172
DOI: https://doi.org/10.1007/s11856-017-1474-2
URI: http://hdl.handle.net/10138/307062
Abstract: We study the operator-valued positive dyadic operator T-lambda (f sigma) := Sigma(Q is an element of D) lambda(Q) integral(Q) f d sigma 1(Q,) where the coefficients {lambda(Q) : C -> D}(Q is an element of D) are positive operators from a Banach lattice C to a Banach lattice D. We assume that the Banach lattices C and D* each have the Hardy-Littlewood property. An example of a Banach lattice with the Hardy-Littlewood property is a Lebesgue space. In the two-weight case, we prove that the L-C(p) (sigma) -> L-D(q)(omega) boundedness of the operator T-lambda(. sigma) is characterized by the direct and the dual L-infinity testing conditions: parallel to 1(Q)T(lambda) (1(Q)f sigma)parallel to(LDq) ((omega)) less than or similar to parallel to f parallel to(LC infinity) ((Q,sigma)) sigma(Q)(1/p), parallel to 1(Q)T(lambda)* (1(Q)g omega)parallel to(LC*p') ((sigma)) less than or similar to parallel to g parallel to(LD*infinity) ((Q,omega)) omega(Q)(1/q'), Here L-C(p) (sigma) and L-D(q) (omega) denote the Lebesgue-Bochner spaces associated with exponents 1 <p In the unweighted case, we show that the L-C(p) (mu) -> L-D(p) (mu) boundedness of the operator T-lambda(. mu) is equivalent to the end-point direct L-infinity testing condition: parallel to 1(Q)T(lambda) (1(Q)f mu)parallel to(LD1) (mu) less than or similar to parallel to f parallel to(LC infinity) ((Q,mu)) mu(Q). This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way.
Subject: MAXIMAL OPERATORS
BELLMAN FUNCTIONS
BANACH-LATTICES
THEOREM
INTEGRALS
111 Mathematics
Peer reviewed: Yes
Usage restriction: closedAccess
Self-archived version: submittedVersion


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