Muckenhoupt Class Weight Decomposition and BMO Distance to Bounded Functions

M Nielsen, Hrvoje Sikic

Research output: Contribution to journalJournal articleResearchpeer-review

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Abstract

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.
Original languageEnglish
JournalProceedings of the Edinburgh Mathematical Society
Number of pages15
ISSN0013-0915
DOIs
Publication statusE-pub ahead of print - 25 Mar 2019

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Bounded Mean Oscillation
Decompose
Theorem
Rectangle
Regular hexahedron
Ap Weights
Muckenhoupt Weights
Valid
Class
Sufficient Conditions

Keywords

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

Cite this

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abstract = "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 Calder{\'o}n–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.",
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Muckenhoupt Class Weight Decomposition and BMO Distance to Bounded Functions. / Nielsen, M; Sikic, Hrvoje.

In: Proceedings of the Edinburgh Mathematical Society, 25.03.2019.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

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

AU - Nielsen, M

AU - Sikic, Hrvoje

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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.

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KW - Calderon–Zygmund decomposition

KW - weights

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