The sharp maximal function and the Fefferman–Stein inequality #

The sharp maximal function of a locally integrable f is

f^# x = sup_{B ∋ x} ⨍_B ‖f - f_B‖, where f_B = ⨍_B f,

the supremum of the mean oscillations of f over the open balls B containing x. It is bounded exactly when f has bounded mean oscillation, and it measures the local oscillation of f rather than its size: it vanishes for constant f. The Fefferman–Stein inequality says that nevertheless, for 1 < p < ∞, the sharp maximal function controls the Lᵖ norm of f itself:

‖f‖_p ≤ C ‖f^#‖_p

with C depending only on the dimension n and on p, for every integrable f on ℝⁿ = ι → ℝ, n ≥ 1, with balls taken in the sup distance (TauCeti.eLpNorm_le_eLpNorm_sharpMaximalFunction). Some a priori integrability is needed, since f^# does not see constants. The inequality fails in dimension 0, where f^# vanishes identically.

This is the form in which mean oscillation estimates become Lᵖ estimates. In the W^{2,p} theory of elliptic equations with coefficients of vanishing mean oscillation, one bounds the mean oscillation of D²u on each small ball by comparison with a constant-coefficient equation; the Fefferman–Stein inequality, together with the Hardy–Littlewood maximal inequality, turns these pointwise bounds on (D²u)^# into the estimate of ‖D²u‖_p.

The argument #

The proof is the good-λ inequality of Fefferman and Stein, run on the Calderón–Zygmund cubes (TauCeti.calderonZygmundCubes) of ‖f‖. Write m(t) for the measure of the union of the Calderón–Zygmund cubes at height t.

The mean oscillation of f on a dyadic cube Q is at most 2ⁿ⁺¹ f^# x at every x ∈ Q (TauCeti.setLAverage_enorm_sub_setAverage_dyadicCube_le): Q lies in an open ball of 2ⁿ times its volume.
Let Q be a Calderón–Zygmund cube at height t, with parent R. The average of ‖f‖ over R is at most t, so ‖f_R‖ ≤ t, while every cube P at height 2t meeting Q lies in Q and has ⨍_P ‖f‖ > 2t, hence ⨍_P ‖f - f_R‖ > t. Summing over these disjoint P, the part of the height-2t cubes inside Q has measure at most t⁻¹ ∫_R ‖f - f_R‖, which is at most 2²ⁿ⁺¹ γ |Q| if f^# ≤ γ t somewhere in Q. Otherwise Q ⊆ {f^# > γ t}. Summing over Q,
m(2t) ≤ 2²ⁿ⁺¹ γ m(t) + |{f^# > γ t}|.
Integrating against p t^(p-1) dt and substituting t ↦ 2t gives ∫₀ᴺ p t^(p-1) m(t) dt ≤ 2^p 2²ⁿ⁺¹ γ ∫₀ᴺ p t^(p-1) m(t) dt + 2^p γ^(-p) ∫ (f^#)^p. Since f is integrable, m(t) ≤ ‖f‖₁ / t, so the integrals on the left are finite for p > 1, and the choice γ = 2^(-(p + 2n + 2)) lets the first term on the right be absorbed. As ‖f‖ ≤ t almost everywhere off the cubes at height t, the layer cake formula bounds ∫ ‖f‖^p by the limit of the left side as N → ∞.

Main declarations #

TauCeti.sharpMaximalFunction: the sharp maximal function, with TauCeti.setLAverage_enorm_sub_setAverage_le_sharpMaximalFunction, TauCeti.sharpMaximalFunction_le and TauCeti.lt_sharpMaximalFunction_iff as its interface, TauCeti.sharpMaximalFunction_const, TauCeti.sharpMaximalFunction_zero, TauCeti.sharpMaximalFunction_congr_ae and TauCeti.lowerSemicontinuous_sharpMaximalFunction.
TauCeti.sharpMaximalFunction_add_const, TauCeti.sharpMaximalFunction_const_smul and TauCeti.sharpMaximalFunction_add_le: the sharp maximal function ignores constants, is homogeneous and is subadditive.
TauCeti.setLAverage_enorm_sub_setAverage_dyadicCube_le: dyadic mean oscillation is controlled by the sharp maximal function.
TauCeti.lintegral_enorm_rpow_le_lintegral_sharpMaximalFunction_rpow and TauCeti.eLpNorm_le_eLpNorm_sharpMaximalFunction: the Fefferman–Stein inequality.

References #

C. Fefferman, E. M. Stein, H^p spaces of several variables, Acta Math. 129 (1972), 137–193, Theorem 5.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Chapter IV, §2.
L. Grafakos, Modern Fourier Analysis, 3rd ed., Section 3.4.
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Chapter 3, Section 2 (the sharp function theorem as the tool for coefficients of vanishing mean oscillation).