TY - JOUR
T1 - On functions of bounded β-dimensional mean oscillation
AU - Chen, You Wei
AU - Spector, Daniel
N1 - Publisher Copyright:
© 2023 De Gruyter. All rights reserved.
PY - 2023
Y1 - 2023
N2 - In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q0 ⊂ ℝd → ℝ which are integrable on β-dimensional subsets of the cube Q0: 1 uBMOβ(Q0) := Qsup ⊂Q0 cinf ∈ℝ l(Q)β ∫ |u − c| dH∞β , Q where the supremum is taken over all finite subcubes Q parallel to Q0, l(Q) is the length of the side of the cube Q, and H∞β is the Hausdorff content. In the case β = d we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every β ∈ (0, d] one has a dimensionally appropriate analogue of the John–Nirenberg inequality for functions with bounded β-dimensional mean oscillation: There exist constants c, C > 0 such that H∞β ({x ∈ Q : |u(x) − cQ| > t}) ≤ Cl(Q)β exp(− uBMOctβ(Q0) ) for every t > 0, u ∈ BMOβ(Q0), Q ⊂ Q0, and suitable cQ ∈ ℝ. Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.
AB - In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q0 ⊂ ℝd → ℝ which are integrable on β-dimensional subsets of the cube Q0: 1 uBMOβ(Q0) := Qsup ⊂Q0 cinf ∈ℝ l(Q)β ∫ |u − c| dH∞β , Q where the supremum is taken over all finite subcubes Q parallel to Q0, l(Q) is the length of the side of the cube Q, and H∞β is the Hausdorff content. In the case β = d we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every β ∈ (0, d] one has a dimensionally appropriate analogue of the John–Nirenberg inequality for functions with bounded β-dimensional mean oscillation: There exist constants c, C > 0 such that H∞β ({x ∈ Q : |u(x) − cQ| > t}) ≤ Cl(Q)β exp(− uBMOctβ(Q0) ) for every t > 0, u ∈ BMOβ(Q0), Q ⊂ Q0, and suitable cQ ∈ ℝ. Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.
KW - Bounded mean oscillation
KW - Hausdorff content
KW - capacitary John–Nirenberg lemma
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U2 - 10.1515/acv-2022-0084
DO - 10.1515/acv-2022-0084
M3 - Article
AN - SCOPUS:85163866616
SN - 1864-8258
JO - Advances in Calculus of Variations
JF - Advances in Calculus of Variations
ER -