Campanato's characterization of Hölder continuity #
Let μ be an additive Haar measure on a finite-dimensional real normed space E of dimension
n, and let 0 < α. For a locally integrable f : E → F, write f_{x,r} for its average over
closedBall x r, and call ⨍_{closedBall x r} ‖f - f_{x,r}‖ its mean oscillation on that
ball. Campanato's theorem says that f agrees almost everywhere with an α-Hölder function
exactly when its mean oscillation on balls of radius r is O(r^α). The Hölder representative
is the precise representative TauCeti.MeasureTheory.preciseRepresentative μ f, the limit of the
averages f_{x,r} as r → 0.
The hypothesis controls an integral average rather than an essential supremum, so it is much
weaker than the oscillation bound behind TauCeti.MeasureTheory.holderOnWith_preciseRepresentative.
This is the form in which energy estimates deliver regularity: Poincaré's inequality turns a
growth bound on ∫_{closedBall x r} ‖∇u‖ᵖ (Morrey's Dirichlet growth condition) into such a mean
oscillation bound, and the Campanato approach to the Schauder estimates compares a solution with
solutions of a constant-coefficient problem in the same integral norms.
Domains with the measure density property #
Estimates up to the boundary need the theorem on a domain Ω ⊆ E, with the balls replaced by
their intersections with Ω: all averages and integrals below are taken for μ.restrict Ω. The
theorem holds as soon as Ω has the measure density property: there is c > 0 with
c μ(B(x, r)) ≤ μ(B(x, r) ∩ Ω)
for x ∈ Ω and 0 < r up to twice the diameter of Ω (every r, if Ω is unbounded). The
classical examples are the bounded Lipschitz domains, by the interior cone condition. The whole
space has it with c = 1, and the whole-space theorems are the case Ω = univ. The Hölder
representative is the precise representative of f for μ.restrict Ω, which agrees with f
almost everywhere on Ω by Lebesgue's differentiation theorem for the locally finite measure
μ.restrict Ω (TauCeti.MeasureTheory.ae_eq_preciseRepresentative_of_hasBesicovitchCovering).
For exponents p ≥ 1 and λ, the Campanato seminorm of f on Ω is
[f]_{p,λ} = (sup_{x ∈ Ω, r > 0} r^(-λ) ∫_{B(x, r) ∩ Ω} ‖f - f_{x,r}‖^p)^(1/p),
and the Campanato space 𝓛^{p,λ}(Ω) consists of the functions for which it is finite. Following
Giaquinta–Martinazzi, who state their estimates for [f]_{p,λ}^p, the declaration
TauCeti.MeasureTheory.campanatoSeminormPow is the p-th power [f]_{p,λ}^p, the supremum itself,
without the p-th root. It is p-homogeneous rather than homogeneous, and its finiteness is
membership in the Campanato space. On a null-measurable set with the measure density property a
locally integrable function lies in 𝓛^{p,n + p α}(Ω) exactly when it agrees almost everywhere on
Ω with a function that is α-Hölder continuous on Ω
(TauCeti.MeasureTheory.exists_holderOnWith_ae_eq_iff_campanatoSeminormPow_ne_top), with explicit
bounds between the Hölder constant and [f]_{p,λ}^p in both directions. For λ = n the
Campanato space is the space of functions of bounded mean oscillation, which contains unbounded
functions; the characterization needs λ > n, that is α > 0.
The argument #
The proof is a telescoping argument. Comparing the averages over closedBall x r and
closedBall x (r / 2) costs at most 2ⁿ / c · M r^α, where 2ⁿ / c bounds the ratio of the
μ.restrict Ω-measures of the two balls (by TauCeti.norm_setAverage_sub_le_of_subset,
MeasureTheory.Measure.addHaar_real_closedBall_div_le and the measure density property).
Summing over dyadic radii shows that the averages f_{x,r} converge as r → 0, with
‖f_{x,r} - f*(x)‖ ≤ 2ⁿ / (c (1 - 2^(-α))) · M r^α, and comparing the averages over
closedBall y d ⊆ closedBall x (2 d), d = dist x y, then gives the Hölder bound. For the
Campanato seminorm, Hölder's inequality bounds the mean oscillation on B(x, r) ∩ Ω by
((μ(B(x, r) ∩ Ω))⁻¹ [f]_{p,λ}^p r^λ)^(1/p), and the measure density property bounds the measure
below by c μ(B(0, 1)) rⁿ.
Main declarations #
All declarations are in the TauCeti.MeasureTheory namespace.
setAverage_norm_sub_setAverage_le_of_holderOnWith: the mean oscillation of anα-Hölder function on a ball of radiusris at most a multiple ofr^α.ae_norm_sub_setAverage_le_of_holderOnWith: on a setΩwherefisα-Hölder,flies almost everywhere on a ball centred inΩwithin a multiple ofr^αof its average over the ball, both for the restriction of the measure toΩ.tendsto_setAverage_closedBall_restrict_preciseRepresentative_of_setAverage_norm_sub_le,tendsto_setAverage_closedBall_preciseRepresentative_of_setAverage_norm_sub_le: mean oscillation decaying liker^αatxmakes the averages converge atx, on a domain with the measure density property and on the whole space.norm_setAverage_closedBall_restrict_sub_preciseRepresentative_le,norm_setAverage_closedBall_sub_preciseRepresentative_le: the rate of that convergence.dist_preciseRepresentative_restrict_le_of_setAverage_norm_sub_le,dist_preciseRepresentative_le_of_setAverage_norm_sub_le: the local Hölder estimate between two points, from mean oscillation bounds on balls of radius comparable to their distance.norm_setAverage_closedBall_sub_setAverage_closedBall_le: mean oscillation decaying liker^αatxkeeps the averages over concentric balls withinO(r^α)of each other.holderOnWith_preciseRepresentative_of_setAverage_norm_sub_le: the Hölder bound on a set, from mean oscillation bounds at scales below a fixed radius and a bound on the averages at that radius.holderOnWith_preciseRepresentative_restrict_of_setAverage_norm_sub_le: Campanato's theorem on a domain with the measure density property.holderWith_preciseRepresentative_of_setAverage_norm_sub_le: Campanato's theorem on the whole space.exists_holderWith_ae_eq_iff_forall_setAverage_norm_sub_le: a locally integrable function agrees almost everywhere with anα-Hölder function if and only if its mean oscillation on balls of radiusrisO(r^α).campanatoSeminormPow: thep-th power[f]_{p,λ}^pof the Campanato seminorm on a set.campanatoSeminormPow_le_of_holderOnWith: Hölder functions lie in the Campanato space.setAverage_norm_sub_le_of_campanatoSeminormPow_le: the Campanato seminorm bounds the mean oscillation on a domain with the measure density property.exists_holderOnWith_ae_eq_iff_campanatoSeminormPow_ne_top: Campanato's characterization𝓛^{p,n + p α}(Ω) = C^{0,α}(Ω)on a domain with the measure density property.
References #
- S. Campanato, Proprietà di hölderianità di alcune classi di funzioni, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 175–188.
- M. Giaquinta, L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Theorem 5.5.
- Q. Han, F. Lin, Elliptic Partial Differential Equations, Theorem 3.1.
The mean oscillation of a Hölder function. If f is α-Hölder continuous on
closedBall x r with constant K, α > 0, and this ball has positive measure, then the mean
oscillation of f on the ball is at most K (2 r)^α.
The oscillation of a Hölder function about its average on a domain. If f is α-Hölder
continuous on a null-measurable set Ω with constant K, then almost everywhere on
closedBall x r, x ∈ Ω, for μ.restrict Ω, the function f lies within K (2 r)^α of its
average over that ball.
Campanato's theorem on a set with the measure density property #
Mean oscillation decay gives convergence of the averages, on a set with the measure density
property. Let α > 0, ρ > 0 and c > 0. Suppose that Ω has density at least c in the
closed balls around x of radius at most ρ, that is c μ(B(x, r)) ≤ μ(B(x, r) ∩ Ω), that f is
integrable on closedBall x ρ for μ.restrict Ω, and that its mean oscillation on closedBall x r
for μ.restrict Ω is at most M r^α for 0 < r ≤ ρ. Then the averages of f over the closed
balls around x for μ.restrict Ω converge to the precise representative of f for
μ.restrict Ω at x.
The rate of convergence of the averages, on a set with the measure density property. Let
α > 0 and c > 0. Suppose that Ω has density at least c in the closed balls around x of
radius at most ρ, that f is integrable on closedBall x ρ for μ.restrict Ω, and that its
mean oscillation on closedBall x r for μ.restrict Ω is at most M r^α for 0 < r ≤ ρ. Then
for 0 < r ≤ ρ the average of f over closedBall x r for μ.restrict Ω lies within
2ⁿ / (c (1 - 2^(-α))) · M r^α of the precise representative of f for μ.restrict Ω at x,
where n is the dimension.
The local Campanato estimate on a set with the measure density property. Let α > 0,
c > 0 and d = dist x y. Suppose that Ω has density at least c in closedBall x r for
0 < r ≤ 2 d and in closedBall y r for 0 < r ≤ d, that f is integrable on
closedBall x (2 d) for μ.restrict Ω, and that the mean oscillation of f for μ.restrict Ω is
at most M r^α on the same balls. Then the precise representatives of f for μ.restrict Ω at
x and y differ by at most 2 · 2^α · 2ⁿ / (c (1 - 2^(-α))) · M d^α, where n is the
dimension.
Campanato's theorem on a set with the measure density property. Let α > 0 and c > 0.
Suppose that Ω has density at least c in every closed ball centred in Ω of radius r > 0 at
most twice the diameter of Ω (every radius, if Ω is unbounded):
c μ(B(x, r)) ≤ μ(B(x, r) ∩ Ω). If f is locally integrable for μ.restrict Ω and its mean
oscillation for μ.restrict Ω on these balls is at most M r^α, then the precise representative of
f for μ.restrict Ω is α-Hölder continuous on Ω, with constant
2 · 2^α · 2ⁿ / (c (1 - 2^(-α))) · M, where n is the dimension. It agrees with f almost
everywhere on Ω (TauCeti.MeasureTheory.ae_eq_preciseRepresentative_of_hasBesicovitchCovering).
Campanato's theorem on the whole space #
Averages over concentric balls. Let α > 0. If f is integrable on closedBall x ρ
and its mean oscillation on closedBall x r is at most M r^α for 0 < r ≤ ρ, then the averages
of f over closedBall x ε and closedBall x r, 0 < ε ≤ r ≤ ρ, differ by at most
2ⁿ / (1 - 2^(-α)) · M r^α, where n is the dimension. The proof telescopes over dyadic
radii.
Mean oscillation decay gives convergence of the averages. Let α > 0 and ρ > 0. If f
is integrable on closedBall x ρ and its mean oscillation on closedBall x r is at most M r^α
for 0 < r ≤ ρ, then the averages of f over the closed balls around x converge to the
precise representative of f at x.
The rate of convergence of the averages. Let α > 0. If f is integrable on
closedBall x ρ and its mean oscillation on closedBall x r is at most M r^α for
0 < r ≤ ρ, then for 0 < r ≤ ρ the average of f over closedBall x r lies within
2ⁿ / (1 - 2^(-α)) · M r^α of the precise representative of f at x, where n is the
dimension.
The local Campanato estimate. Let α > 0 and d = dist x y. If f has mean oscillation
at most M r^α on closedBall x r for 0 < r ≤ 2 d and on closedBall y r for 0 < r ≤ d,
and is integrable on closedBall x (2 d), then the precise representatives of f at x
and y differ by at most 2 · 2^α · 2ⁿ / (1 - 2^(-α)) · M d^α, where n is the dimension.
Campanato's theorem on a set, below a fixed scale. Let α > 0 and ρ > 0. Suppose that
for every x ∈ S, f is integrable on closedBall x ρ, has mean oscillation at most M r^α on
closedBall x r for 0 < r ≤ ρ, and the norm of its average over closedBall x ρ is at most
B. Then the precise representative of f is α-Hölder continuous on S, with constant
2 · 2^α · (2ⁿ / (1 - 2^(-α)) · M + B ρ^(-α)), where n is the dimension.
Campanato's theorem. Let α > 0, and let f be locally integrable, with mean oscillation
on every closed ball of radius r > 0 at most M r^α. Then the precise representative of f
is α-Hölder continuous, with constant 2 · 2^α · 2ⁿ / (1 - 2^(-α)) · M, where n is the
dimension. It agrees with f almost everywhere
(TauCeti.MeasureTheory.ae_eq_preciseRepresentative).
Campanato's characterization of Hölder continuity. Let α > 0. A locally integrable
function agrees almost everywhere with an α-Hölder continuous function if and only if its mean
oscillation on closed balls of radius r > 0 is at most M r^α for some constant M.
Campanato spaces #
The p-th power [f]_{p,λ}^p of the Campanato seminorm of f on Ω with exponents p
and λ: the supremum, over the points x ∈ Ω and the radii r > 0, of
r^(-λ) ∫_{B(x, r) ∩ Ω} ‖f - f_{x,r}‖^p, where f_{x,r} = ⨍_{B(x, r) ∩ Ω} f
and B(x, r) is the closed ball, both the integral and the average being taken for
μ.restrict Ω. Following Giaquinta–Martinazzi, no p-th root is taken, so for p > 0 this
quantity scales by |a|^p, not |a|, when f is multiplied by a; the Campanato seminorm
itself is its p-th root. The Campanato space 𝓛^{p,λ}(Ω) consists of the functions for
which it is finite.
Instances For
The defining formula of campanatoSeminormPow, the supremum of the rescaled p-th power mean
oscillations over the points of Ω and the positive radii.
The Campanato seminorm is at most C exactly when each of the quantities of which it is the
supremum is.
The quantity attached to a point x ∈ Ω and a radius r > 0 is at most the Campanato
seminorm.
The Campanato seminorm only depends on the almost-everywhere class of f on Ω.
The Campanato seminorm on the empty set vanishes.
For p > 0, the Campanato seminorm of the zero function vanishes.
Hölder functions lie in the Campanato space. If f is α-Hölder continuous on a
null-measurable set Ω with constant K, then for p ≥ 0 the p-th power [f]_{p,λ}^p of its
Campanato seminorm with exponents p and λ = n + p α is at most (2^α K)^p μ(B(0, 1)), where
n is the dimension.
The Campanato seminorm bounds the mean oscillation. If Ω has density at least c > 0 in
closedBall x r, x ∈ Ω, and the p-th power [f]_{p,λ}^p of the Campanato seminorm of f
with exponents p ≥ 1 and λ = n + p α is at most M, then the mean oscillation of f on
closedBall x r for μ.restrict Ω is at most (M / (c μ(B(0, 1))))^(1/p) r^α, where n is the
dimension.
Campanato's characterization of Hölder continuity on a set with the measure density
property. Let α > 0, p ≥ 1 and c > 0, and let Ω be a null-measurable set with density
at least c in every closed ball centred in Ω of radius r > 0 at most twice the diameter of
Ω (every radius, if Ω is unbounded): c μ(B(x, r)) ≤ μ(B(x, r) ∩ Ω). A function f locally
integrable for μ.restrict Ω agrees almost everywhere on Ω with a function that is α-Hölder
continuous on Ω if and only if it belongs to the Campanato space 𝓛^{p,λ}(Ω) for
λ = n + p α, where n is the dimension.