The John–Nirenberg inequality #
Let Q₀ be a dyadic cube in ℝⁿ = ι → ℝ for a finite type ι, and let f : ℝⁿ → E be
integrable on Q₀ for an additive Haar measure μ, with dyadic mean oscillation at most M
on Q₀: for every dyadic cube
Q ⊆ Q₀,
⨍_Q ‖f - f_Q‖ ≤ M, where f_Q = ⨍_Q f.
The John–Nirenberg inequality says that f then deviates from its average exponentially
rarely:
|{x ∈ Q₀ : ‖f x - f_{Q₀}‖ > s}| ≤ 2 exp (-(log 2) s / (2ⁿ⁺¹ M)) |Q₀|
for every s (TauCeti.measure_lt_norm_sub_setAverage_le). Consequently exp (σ ‖f - f_{Q₀}‖)
is integrable on Q₀, with average bounded in terms of σ M and n alone, whenever
exp (2ⁿ⁺¹ σ M) < 2 (TauCeti.setLIntegral_exp_mul_norm_sub_setAverage_le). So a function of
bounded mean oscillation is exponentially integrable, far beyond the L¹ control its definition
provides. This is what lets Moser's proof of the Harnack inequality cross from bounds on negative
powers of a positive supersolution to bounds on positive powers: the logarithm of such a
supersolution has bounded mean oscillation.
Both statements hold on every cube, not only on dyadic ones. In the sup norm of ι → ℝ, for
r ≥ 0 the cube ∏ᵢ [cᵢ - r, cᵢ + r] is the closed ball closedBall c r. If f has mean
oscillation at most M on every closed ball inside it, then the same two bounds hold for any
additive Haar measure on closedBall c r
(TauCeti.measure_lt_norm_sub_setAverage_closedBall_le,
TauCeti.setLIntegral_exp_mul_norm_sub_setAverage_closedBall_le). For 0 < r this follows from
the dyadic case by the affine change of variables carrying the unit dyadic cube onto
closedBall c r, which rescales Lebesgue measure by a constant and so preserves averages.
In positive dimension, r ≤ 0 makes the closed ball null. In dimension zero every function is
constant, so the inequalities also hold, including on the singleton closed ball of radius zero.
In ℝⁿ = EuclideanSpace ℝ ι, with any additive Haar measure, the inequality holds on Euclidean
balls up to a loss in the radius: if f has mean oscillation at most M on every Euclidean ball
inside B(x₀, R), then ∫_{B'} exp (σ ‖f - f_{B'}‖) ≤ C |B'| on B' = B(x₀, R / (2√n)) whenever
σ M ≤ A, for constants A > 0 and C depending only on n
(TauCeti.exists_setLIntegral_exp_mul_norm_sub_setAverage_ball_le). This follows from the cube
case, since a cube of half-side ρ and the Euclidean ball of radius 2√n ρ about its centre have
comparable measures. For real f it gives Moser's crossover estimate
(∫_{B'} exp (σ f)) (∫_{B'} exp (-σ f)) ≤ C |B'|²
(TauCeti.exists_setLIntegral_exp_mul_mul_setLIntegral_exp_neg_mul_ball_le).
Exponential integrability controls every power: for each 0 < p there is C, depending only on
n and p, with ⨍_B ‖f - f_B‖ ^ p ≤ C ‖f‖_{BMO} ^ p on every ball B
(TauCeti.exists_setLAverage_enorm_sub_setAverage_rpow_le_mul_bmoSeminorm_rpow). This Lᵖ form of
the John–Nirenberg inequality is how a function of bounded mean oscillation enters Hölder's
inequality, as in the commutator estimates for singular integrals; in particular such a function
lies in Lᵖ of every ball for every finite p (TauCeti.MemBMO.memLp_restrict_ball).
The proof iterates the Calderón–Zygmund decomposition. At height 2M, the Calderón–Zygmund
cubes of ‖f - f_{Q₀}‖ restricted to Q₀ lie inside Q₀, cover at most half of it, and carry
averages within 2ⁿ⁺¹ M of f_{Q₀}, while ‖f - f_{Q₀}‖ ≤ 2M almost everywhere off them. Each
step of size 2ⁿ⁺¹ M in the level therefore halves the measure of the superlevel set
(TauCeti.measure_lt_enorm_sub_setAverage_le_inv_two_pow). The decomposition argument is proved
for Lebesgue measure and transferred to any additive Haar normalization using invariance of
averages under positive finite scaling.
Main declarations #
TauCeti.measure_lt_enorm_sub_setAverage_le_inv_two_pow: the set where‖f - f_{Q₀}‖exceeds2ⁿ⁺¹ N Mhas measure at most2⁻ᴺ |Q₀|.TauCeti.measure_lt_norm_sub_setAverage_le: the John–Nirenberg inequality, exponential decay of the distribution function of‖f - f_{Q₀}‖.TauCeti.setLIntegral_exp_mul_norm_sub_setAverage_le: exponential integrability.TauCeti.measure_lt_norm_sub_setAverage_closedBall_le,TauCeti.setLIntegral_exp_mul_norm_sub_setAverage_closedBall_le: the same two statements on an arbitrary cubeclosedBall c r.TauCeti.exists_setLIntegral_exp_mul_norm_sub_setAverage_ball_le: exponential integrability on Euclidean balls.TauCeti.exists_setLIntegral_exp_mul_norm_sub_setAverage_ball_le_of_bmoSeminorm_le: the same on every ball for a locally integrable function with bounded BMO seminorm.TauCeti.exists_setLIntegral_exp_mul_mul_setLIntegral_exp_neg_mul_ball_le: Moser's crossover estimate for real functions of bounded mean oscillation on Euclidean balls.TauCeti.exists_setLAverage_enorm_sub_setAverage_rpow_le_mul_bmoSeminorm_rpow: theLᵖform of the John–Nirenberg inequality,⨍_B ‖f - f_B‖ ^ p ≤ C ‖f‖_{BMO} ^ p.TauCeti.MemBMO.memLp_restrict_ball: functions of bounded mean oscillation are locally in everyLᵖ,p < ∞.
References #
- F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426.
- L. Grafakos, Modern Fourier Analysis, Theorem 3.1.6 and the corollary following it (the
Lᵖform). - E. Stein, Harmonic Analysis, Chapter IV, §1.3.
- J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591.
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Section 8.6 (the crossover estimate in the proof of the weak Harnack inequality).
The dyadic John–Nirenberg inequality for any additive Haar measure: the superlevel
set at 2ⁿ⁺¹ N M has measure at most 2⁻ᴺ times the cube measure.
The John–Nirenberg inequality on a dyadic cube, for any additive Haar measure. If the
mean oscillation on each dyadic subcube is at most M, then its superlevel sets have the
exponential measure bound 2 exp (-(log 2) s / (2ⁿ⁺¹ M)) μ Q₀.
Exponential integrability on a dyadic cube, for any additive Haar measure. If the mean
oscillation on each dyadic subcube is at most M and ρ = exp (2ⁿ⁺¹ σ M) < 2, the exponential
integral is at most (1 + 2ρ / (2 - ρ)) μ Q₀.
Arbitrary cubes #
For r ≥ 0, in the sup norm of ι → ℝ the closed ball closedBall c r is the closed cube
∏ᵢ [cᵢ - r, cᵢ + r]. For 0 < r, the affine bijection y ↦ 2r y + (c - r) carries the unit
dyadic cube [0, 1)ⁿ onto this cube up to a null set and its dyadic subcubes onto closed balls
inside closedBall c r up to null sets, and it rescales Lebesgue measure by the constant (2r)ⁿ.
Averages are invariant under it, so the dyadic John–Nirenberg inequality for f composed with
this map gives the inequality on every cube of positive radius. In positive dimension the closed
ball is null for r ≤ 0, so both bounds hold trivially; in dimension zero every nonempty closed
ball is the singleton space and agrees with every dyadic cube.
The John–Nirenberg inequality on a cube, for any additive Haar measure. Let f be
integrable on the closed sup-norm ball closedBall c r ⊆ ℝⁿ, with mean oscillation
⨍_B ‖f - f_B‖ ≤ M on every closed ball B ⊆ closedBall c r. Then for every s,
μ {x ∈ closedBall c r : ‖f x - f_{closedBall c r}‖ > s} ≤ 2 exp (-(log 2) s / (2ⁿ⁺¹ M)) μ (closedBall c r).
Exponential integrability of functions of bounded mean oscillation on a cube, for any
additive Haar measure. Let f be integrable on the closed sup-norm ball closedBall c r ⊆ ℝⁿ,
with mean oscillation ⨍_B ‖f - f_B‖ ≤ M on every closed ball B ⊆ closedBall c r.
If ρ = exp (2ⁿ⁺¹ σ M) < 2, then
∫_{closedBall c r} exp (σ ‖f - f_{closedBall c r}‖) ∂μ ≤ (1 + 2ρ / (2 - ρ)) μ (closedBall c r).
Euclidean balls #
In ℝⁿ = EuclideanSpace ℝ ι, with any additive Haar measure μ, suppose f has mean
oscillation at most M on every Euclidean ball inside B(x₀, R). Let s = √n and let Q₀ be
the cube of half-side r₀ = R / (2s) centred at x₀. For a cube Q ⊆ Q₀ of half-side ρ,
its centre is at sup distance at most r₀ - ρ from x₀, hence at Euclidean distance at most
s(r₀ - ρ). Thus the Euclidean ball B of radius 2sρ about that centre contains Q and lies
in B(x₀, R), since 2sρ + s(r₀ - ρ) ≤ 2sr₀ = R. Since B lies in the cube of half-side
2sρ about the same centre,
μ B ≤ (2s)ⁿ μ Q, and comparing mean oscillations on Q ⊆ B shows that f has mean oscillation
at most 2 (2s)ⁿ M on every cube inside Q₀. The John–Nirenberg inequality on Q₀, read through
the measurable equivalence ofLp between EuclideanSpace ℝ ι and ι → ℝ (which pushes μ
forward to a positive finite multiple of volume, and therefore preserves averages), then gives
exponential integrability on Q₀, and hence on the Euclidean ball B(x₀, r₀) ⊆ Q₀. This ball
contains the cube of half-side r₀ / s, so μ Q₀ ≤ sⁿ μ B(x₀, r₀), and the averages of f over
Q₀ and over B(x₀, r₀) differ by at most sⁿ times the mean oscillation on Q₀.
The John–Nirenberg inequality on Euclidean balls. There are constants A > 0 and C,
depending only on the dimension n, such that the following holds for every additive Haar
measure μ on ℝⁿ = EuclideanSpace ℝ ι. Let f : ℝⁿ → E be integrable on the ball B(x₀, R),
with mean oscillation ⨍_B ‖f - f_B‖ ≤ M on every ball B ⊆ B(x₀, R). If σ M ≤ A, then on the
concentric ball B' = B(x₀, R / (2√n)),
∫_{B'} exp (σ ‖f - f_{B'}‖) ≤ C |B'|.
The John–Nirenberg inequality for functions of bounded mean oscillation. There are
constants A > 0 and C, depending only on the dimension n, such that the following holds
for every additive Haar measure μ on ℝⁿ = EuclideanSpace ℝ ι. If f : ℝⁿ → E is locally
integrable with ‖f‖_{BMO} ≤ M and σ M ≤ A, then on every ball B,
∫_B exp (σ ‖f - f_B‖) ≤ C |B|.
Moser's crossover estimate. There are constants A > 0 and C, depending only on the
dimension n, such that the following holds for every additive Haar measure μ on
ℝⁿ = EuclideanSpace ℝ ι. Let f : ℝⁿ → ℝ be integrable on the ball B(x₀, R), with mean
oscillation ⨍_B |f - f_B| ≤ M on every ball B ⊆ B(x₀, R). If |σ| M ≤ A, then on the
concentric ball B' = B(x₀, R / (2√n)),
(∫_{B'} exp (σ f)) (∫_{B'} exp (-σ f)) ≤ C |B'|².
For f = log u, with u positive, this compares the averages of u^σ and of u^(-σ) over B':
it is the step of Moser's proof of the Harnack inequality that links the bounds for positive
powers of a supersolution to those for its negative powers.
The Lᵖ form #
Since t ^ p ≤ (p / σ) ^ p exp (σ t) for t ≥ 0 and σ > 0, exponential integrability of
σ ‖f - f_B‖ at the rate σ = A / ‖f‖_{BMO} controls every power ‖f - f_B‖ ^ p, with a
constant of order ‖f‖_{BMO} ^ p.
The John–Nirenberg inequality in Lᵖ form. For every exponent 0 < p there is a
constant C, depending only on the dimension n ≥ 1 and on p, such that the following holds
for every additive Haar measure μ on ℝⁿ = EuclideanSpace ℝ ι. If f : ℝⁿ → E is locally
integrable, then on every ball B
⨍_B ‖f - f_B‖ ^ p ≤ C ‖f‖_{BMO} ^ p.
Functions of bounded mean oscillation are locally in every Lᵖ, p < ∞. On
ℝⁿ = EuclideanSpace ℝ ι, n ≥ 1, with an additive Haar measure, a function of bounded mean
oscillation lies in Lᵖ of every ball for every finite exponent p.