### Abstract

Every bounded linear operator that maps H^{1} to L^{1} and L^{2} to L^{2} is bounded from L^{p} to L^{p} for each p ∈ (1, 2), by a famous interpolation result of Fefferman and Stein. We prove L^{p}-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(2^{1/(p-1)}). For p ∈ (2, ∞), we prove explicit L^{p} estimates on each bounded linear operator mapping L ^{∞} to bounded mean oscillation (BMO) and L^{2} to L ^{2}. This BMO interpolation result implies the H^{1} result above, by duality. In addition, we obtain stronger results by working with dyadic H^{1} and dyadic BMO. The proofs proceed by complex interpolation, after we develop an optimal dyadic 'good lambda' inequality for the dyadic #-maximal operator.

Original language | English (US) |
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Pages (from-to) | 158-168 |

Number of pages | 11 |

Journal | Journal of the Australian Mathematical Society |

Volume | 95 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1 2013 |

### Keywords

- BMO
- Bounded mean oscillation
- Interpolation of operators

### ASJC Scopus subject areas

- Mathematics(all)