TY - JOUR
T1 - Explicit interpolation bounds between hardy space and L2
AU - Bui, H. Q.
AU - Laugesen, R. S.
N1 - Funding Information:
This paper was partially supported by a grant from the Simons Foundation (#204296 to Richard Laugesen).
PY - 2013/10
Y1 - 2013/10
N2 - Every bounded linear operator that maps H1 to L1 and L2 to L2 is bounded from Lp to Lp for each p ∈ (1, 2), by a famous interpolation result of Fefferman and Stein. We prove Lp-norm bounds that grow like O(1=(p - 1)) as p 1. This growth rate is optimal, and improves significantly on the previously known exponential bound O(21/(p-1)). For p ∈ (2, ∞), we prove explicit Lp estimates on each bounded linear operator mapping L ∞ to bounded mean oscillation (BMO) and L2 to L 2. This BMO interpolation result implies the H1 result above, by duality. In addition, we obtain stronger results by working with dyadic H1 and dyadic BMO. The proofs proceed by complex interpolation, after we develop an optimal dyadic 'good lambda' inequality for the dyadic #-maximal operator.
AB - Every bounded linear operator that maps H1 to L1 and L2 to L2 is bounded from Lp to Lp for each p ∈ (1, 2), by a famous interpolation result of Fefferman and Stein. We prove Lp-norm bounds that grow like O(1=(p - 1)) as p 1. This growth rate is optimal, and improves significantly on the previously known exponential bound O(21/(p-1)). For p ∈ (2, ∞), we prove explicit Lp estimates on each bounded linear operator mapping L ∞ to bounded mean oscillation (BMO) and L2 to L 2. This BMO interpolation result implies the H1 result above, by duality. In addition, we obtain stronger results by working with dyadic H1 and dyadic BMO. The proofs proceed by complex interpolation, after we develop an optimal dyadic 'good lambda' inequality for the dyadic #-maximal operator.
KW - BMO
KW - Bounded mean oscillation
KW - Interpolation of operators
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U2 - 10.1017/S1446788713000244
DO - 10.1017/S1446788713000244
M3 - Article
AN - SCOPUS:84890530416
SN - 1446-7887
VL - 95
SP - 158
EP - 168
JO - Journal of the Australian Mathematical Society
JF - Journal of the Australian Mathematical Society
IS - 2
ER -