### Abstract

Original language | English |
---|---|

Journal | Proceedings of the Edinburgh Mathematical Society |

Number of pages | 15 |

ISSN | 0013-0915 |

DOIs | |

Publication status | E-pub ahead of print - 25 Mar 2019 |

### Fingerprint

### Keywords

- Muckenhoupt condition
- BMO
- Calderon–Zygmund decomposition
- weights
- Garnett–Jones distance;
- Jones factorization

### Cite this

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**Muckenhoupt Class Weight Decomposition and BMO Distance to Bounded Functions.** / Nielsen, M; Sikic, Hrvoje.

Research output: Contribution to journal › Journal article › Research › peer-review

TY - JOUR

T1 - Muckenhoupt Class Weight Decomposition and BMO Distance to Bounded Functions

AU - Nielsen, M

AU - Sikic, Hrvoje

PY - 2019/3/25

Y1 - 2019/3/25

N2 - We study the connection between the Muckenhoupt Ap weights and bounded mean oscillation (BMO) for general bases for ℝ^d. New classes of bases are introduced that allow for several deep results on the Muckenhoupt weights–BMO connection to hold in a very general form. The John–Nirenberg type inequality and its consequences are valid for the new class of Calderón–Zygmund bases which includes cubes in ℝd, but also the basis of rectangles in ℝ^d. Of particular interest to us is the Garnett–Jones theorem on the BMO distance, which is valid for cubes. We prove that the theorem is equivalent to the newly introduced A_2-decomposition property of bases. Several sufficient conditions for the theorem to hold are analysed as well. However, the question whether the theorem fully holds for rectangles remains open.

AB - We study the connection between the Muckenhoupt Ap weights and bounded mean oscillation (BMO) for general bases for ℝ^d. New classes of bases are introduced that allow for several deep results on the Muckenhoupt weights–BMO connection to hold in a very general form. The John–Nirenberg type inequality and its consequences are valid for the new class of Calderón–Zygmund bases which includes cubes in ℝd, but also the basis of rectangles in ℝ^d. Of particular interest to us is the Garnett–Jones theorem on the BMO distance, which is valid for cubes. We prove that the theorem is equivalent to the newly introduced A_2-decomposition property of bases. Several sufficient conditions for the theorem to hold are analysed as well. However, the question whether the theorem fully holds for rectangles remains open.

KW - Muckenhoupt condition

KW - BMO

KW - Calderon–Zygmund decomposition

KW - weights

KW - Garnett–Jones distance;

KW - Jones factorization

U2 - 10.1017/S0013091519000038

DO - 10.1017/S0013091519000038

M3 - Journal article

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

ER -