Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 355–366

HIGHER INTEGRABILITY WITH WEIGHTS

Juha Kinnunen

University of Jyväskylä, Department of Mathematics P.O. Box 35, SF-40351 Jyväskylä, Finland; jkinnune@jyu.fi.

Abstract. We present a new short proof for the classical Gehring lemma on higher integrability in the case when the Lebesgue measure is replaced by the doubling weight and the reverse Hölder inequality is replaced by the weak reverse Hölder inequality.

1. Introduction

A measurable function w is called a weight if it is locally Lebesgue integrable and 0 < w < ∞ almost everywhere in R^{n}. The weight w is naturally associated with the Borel measure

(1.1)  μ(E) = ∫_{E} w(x) dx.

It follows immediately from the definition of the weight that the measure μ and the Lebesgue measure are mutually absolutely continuous and hence we do not need to specify measure when we consider sets of measure zero, measurable sets or functions. We also identify the weight w and the measure μ using (1.1). The weight w is doubling if the measure μ satisfies the doubling condition

(1.2)  μ(2Q) ≤ Cμ(Q)

for every cube Q with the constant C ≥ 1 independent of Q. By a cube we always mean a bounded open cube in R^{n} with sides parallel to the coordinate axis and σQ denotes the cube with the same center as Q but the side length multiplied by the factor σ > 0. If w is a doubling weight, then for every σ > 0 there is a constant C > 0 depending only on σ such that μ(σQ) ≤ Cμ(Q).

Let Ω be a domain in R^{n} and 1 < p < ∞. A nonnegative function f ∈ L_{loc}^{p}(Ω; μ) is said to satisfy a reverse Hölder inequality in Ω, if

(1.3)  (\overline{∫} _{Q} f^{p} dμ)^{1/p} ≤ C\overline{∫} _{Q} f dμ,

1991 Mathematics Subject Classification: Primary 26D15; Secondary 42B25.

for every cube Q ⊂ Ω and the constant C ≥ 1 is independent of Q. Here

\overline{∫} _{Q} f dμ = \frac{1}{μ(Q)} ∫_{Q} f dμ

is the mean value of f over Q. It is well known that if f satisfies (1.3), then it is locally integrable to a power q > p. This local higher integrability result was first established by F. Gehring [Ge, Lemma 2] in the unweighted case w = 1.

Usually, however, condition (1.3) is too strong to hold for the gradients of the solutions of degenerate elliptic partial differential equations, see [HKM] and [S2]. In this case we only have a weak reverse Hölder inequality

(1.4)  (\overline{∫} _{Q} f^{p} dμ)^{1/p} ≤ C\overline{∫} _{2Q} f dμ

where the cube Q is such that 2Q ⊂ Ω and the constant C ≥ 1 is independent of the cube Q. By iterating as in [IN, Theorem 2] we see that the double cube on the right hand side can be replaced by any σQ with σ > 1. Clearly the reverse Hölder inequality (1.3) implies the weak reverse Hölder inequality (1.4) whenever the measure μ is doubling but not vice versa. Hence even in the unweighted case Gehring’s local higher integrability lemma is not available for us. The aim of this paper is to present a new proof for the following weighted version of Gehring’s lemma.

1.5. Theorem. Suppose that w is a doubling weight and that nonnegative f ∈ L_{loc}^{p}(Ω; μ), 1 < p < ∞, satisfies

(1.6)  (\overline{∫} _{Q} f^{p} dμ)^{1/p} ≤ C_{1}\overline{∫} _{2Q} f dμ

for each cube Q such that 2Q ⊂ Ω with the constant C_{1} ≥ 1 independent of the cube Q. Then there exist q > p so that

(\overline{∫} _{Q} f^{q} dμ)^{1/q} ≤ C_{2}(\overline{∫} _{2Q} f^{p} dμ)^{1/p},

where the constant C_{2} ≥ 1 is independent of the cube Q. In particular, f ∈ L_{loc}^{q}(Ω; μ).

The proof presented below contains some new ideas to deal with L^{p} -integrals. Especially the use of Stieltjes integrals in [Ge] is avoided. Several proofs are available in the unweighted case, see [BI, Theorem 4.2], [Gi, Theorem V.1.2],

[GM] and [S1]. The weighted case has been studied by E. Stredulinsky, see [S2, Theorem 2.3.3].

2. Local maximal functions

Let Q_{0} be a cube in R^{n}. Suppose that f ∈ L^{p}(Q_{0}; μ), 1 ≤ p < ∞, is a nonnegative function. The local maximal function M_{Q_{0}}^{p}f of f at the point x ∈ Q_{0} is defined by

M_{Q_{0}}^{p}f(x) = sup_{x∈Q⊂Q_{0}}(\overline{∫} _{Q} f^{p} dμ)^{1/p},

where the supremum is taken over all non-empty open subcubes Q of Q_{0} containing x. If p = 1 then we get the Hardy–Littlewood maximal function and in this case we write M_{Q_{0}}^{1}f = M_{Q_{0}}f. Since the set {M_{Q_{0}}^{p}f > t} is open for every t > 0, we see that the local maximal function is measurable. It follows immediately from the definition of the maximal function that it is nonnegative M_{Q_{0}}^{p}f ≥ 0, subadditive M_{Q_{0}}^{p}(f_{1} + f_{2}) ≤ M_{Q_{0}}^{p}f_{1} + M_{Q_{0}}^{p}f_{2} and homogeneous M_{Q_{0}}^{p}(λf) = λM_{Q_{0}}^{p}f, λ ≥ 0.

First we are going to show that the maximal function is of weak type (1,1). In the proof we need the following well known lemma.

2.1. Lemma. Let ℱ be a family of cubes in R^{n} such that ⋃_{Q∈ℱ} Q is bounded. Then there exists a countable (or finite) subfamily ℱ′ of ℱ such that cubes in ℱ′ are pairwise disjoint and

⋃_{Q∈ℱ} Q ⊂ ⋃_{Q∈ℱ′} 5Q.

An immediate consequence of the covering lemma is the following inequality.

2.2. Lemma. If w is a doubling weight and f ∈ L^{1}(Q_{0}; μ) is a nonnegative function, then

(2.3)  μ(\{M_{Q_{0}}f > t\}) ≤ \frac{C}{t} ∫_{Q_{0}} f dμ

for every t > 0. Here the constant C > 0 depends only on the doubling constant in (1.2) and the dimension n.

Proof. For each x ∈ {M_{Q_{0}}f > t} there exists a cube Q_{x} such that x ∈ Q_{x} ⊂ Q_{0} and

\overline{∫} _{Q_{x}} f dμ > t.

By the covering lemma 2.1 we may choose a countable family of pairwise disjoint cubes Q_{x_{k}}, k = 1, 2, ..., such that

{M_{Q_{0}}f > t} ⊂ ⋃_{x∈{M_{Q_{0}} f>t}} Q_{x} ⊂ ⋃_{k=1}^{∞} 5Q_{x_{k}}.

Therefore by the doubling condition (1.2),

\begin{aligned}μ(\{M_{Q_{0}}f > t\}) & ≤ ∑_{k=1}^{∞} μ(5Q_{x_{k}}) ≤ C ∑_{k=1}^{∞} μ(Q_{x_{k}}) \\ & < \frac{C}{t} ∑_{k=1}^{∞} ∫_{Q_{x_{k}}} f dμ = \frac{C}{t} ∫_{⋃_{k=1}^{∞} Q_{x_{k}}} f dμ ≤ \frac{C}{t} ∫_{Q_{0}} f dμ, \end{aligned}

which is the desired result.

Remark. Using the same assumptions as in the previous lemma we get

(2.4)  μ(\{M_{Q_{0}}f > t\}) ≤ 2\frac{C}{t} ∫_{\{f>t/2\}} f dμ.

In fact, let f = f_{1}+f_{2} where f_{1}(x) = f(x), if f(x) > t/2 and f_{1}(x) = 0 otherwise. Then by subadditivity and homogenity of the maximal function

M_{Q_{0}}f(x) ≤ M_{Q_{0}}f_{1}(x) + M_{Q_{0}}f_{2}(x) ≤ M_{Q_{0}}f_{1}(x) + t/2.

Using (2.3) we get

μ(\{M_{Q_{0}}f > t\}) ≤ μ(\{M_{Q_{0}}f_{1} > 1/2t\}) ≤ 2\frac{C}{t} ∫_{Q_{0}} f_{1} dμ = 2\frac{C}{t} ∫_{\{f>t/2\}} f dμ.

This proves inequality (2.4).

Next we recall the Calderón–Zygmund decomposition lemma for doubling weights, see [GCRF, Theorem II.1.14].

2.5. Lemma. Let w be a doubling weight and f ∈ L^{1}(Q_{0}; μ) be a nonnegative function. Then for every

t ≥\overline{∫} _{Q_{0}} f dμ,

there is a countable or finite family ℱ_{t} of pairwise disjoint dyadic subcubes of Q_{0} such that

(2.6)  t <\overline{∫} _{Q} f dμ ≤ Ct

for each Q ∈ ℱ_{t}, and

(2.7)  f(x) ≤ t  for a.e. x ∈ Q_{0} \ ⋃_{Q∈ℱ_{t}} Q.

The constant C depends on the doubling constant appearing in (1.2) and the dimension n.

The family ℱ_{t} is called the Calderón–Zygmund decomposition for f at the level t in Q_{0} . Calderón–Zygmund decomposition enables us to prove a kind of reverse inequality to (2.3). An unweighted version of the following lemma can be found in [BI, Lemma 4.2].

2.8. Lemma. Suppose that w is a doubling weight and that f ∈ L^{p}(Q_{0}; μ), 1 ≤ p < ∞, is a nonnegative function. Then for every t ≥ 0 with

\overline{∫} _{Q_{0}} f^{p} dμ ≤ t^{p},

we have

(2.9)  ∫_{{f>t}} f^{p} dμ ≤ Ct^{p}μ({M_{Q_{0}}^{p}f > t}).

The constant C is the same constant as in (2.6).

Proof. Let ℱ_{t} be the Calderón–Zygmund decomposition for f^{p} at the level t^{p}. Then by (2.7)

t^{p} <\overline{∫} _{Q} f^{p} dμ ≤ Ct^{p}

for each Q ∈ ℱ_{t}, and by (2.7)

f(x) ≤ t for a.e. x ∈ Q_{0} \ ⋃_{Q∈ℱ_{t}} Q.

Since by (2.7) almost every point in {f > t} belongs to some Q ∈ ℱ_{t} and cubes in ℱ_{t} are pairwise disjoint, we obtain

∫_{{f>t}} f^{p} dμ ≤ ∑_{Q∈ℱ_{t}} ∫_{Q} f^{p} dμ ≤ ∑_{Q∈ℱ_{t}} Ct^{p}μ(Q) = Ct^{p}μ( ⋃_{Q∈ℱ_{t}} Q).

On the other hand

⋃_{Q∈ℱ_{t}} Q ⊂ {M_{Q_{0}}^{p}f > t}

and the lemma follows.

3. Lemmas

Before proving the local higher integrability theorem 1.5, we shall state some results which will be needed in the proof. First of all we need a technical lemma based on Fubini’s theorem.

3.1. Lemma. Let ν be a measure and E ⊂ R^{n} be a set with ν(E) < ∞. If f is a nonnegative ν -measurable function on E, 0 < q < ∞ and 0 ≤ t_{0} < t_{1} < ∞, then

(3.2)  \begin{aligned}∫_{\{t_{0}<f≤t_{1}\}} f^{q} dν & = q ∫_{t_{0}}^{t_{1}} t^{q−1}ν(\{f > t\}) dt \\ & + t_{0}^{q}ν(\{f > t_{0}\}) − t_{1}^{q}ν(\{f > t_{1}\}). \end{aligned}

Proof. We make use of the well known formula

∫_{{t_{0}<f≤t_{1}}} f^{q} dν = q ∫_{0}^{∞} t^{q−1}ν({t_{0} < f ≤ t_{1}} ∩ {f > t}) dt,

which is a simple consequence of Fubini’s theorem. Now

∫_{0}^{∞} t^{q−1}ν({t_{0} < f ≤ t_{1}} ∩ {f > t}) dt

= ∫_{0}^{t_{0}} t^{q−1}ν({t_{0} < f ≤ t_{1}}) dt + ∫_{t_{0}}^{t_{1}} t^{q−1}ν({t < f ≤ t_{1}}) dt

= \frac{t_{0}^{q}}{q} ν(\{t_{0} < f ≤ t_{1}\}) + ∫_{t_{0}}^{t_{1}} t^{q−1}ν(\{t < f ≤ t_{1}\}) dt.

Since {t < f ≤ t_{1}} = {f > t} \ {f > t_{1}} and the measures of these sets are finite, ν({t < f ≤ t_{1}}) = ν({f > t}) − ν({f > t_{1}}), and consequently

∫_{t_{0}}^{t_{1}} t^{q−1}ν({t < f ≤ b}) dt

= ∫_{t_{0}}^{t_{1}} t^{q−1}ν({f > t}) dt − ν({f > b}) ∫_{t_{0}}^{t_{1}} t^{q−1} dt

= ∫_{t_{0}}^{t_{1}} t^{q−1}ν(\{f > t\}) dt − \frac{t_{1}^{q} − t_{0}^{q}}{q} ν(\{f > t_{1}\}).

Therefore equality (3.2) holds and the lemma follows.

The idea of the next lemma is similar to [Ge, Lemma 1], except that there the lemma is established in terms of Stieltjes integrals. Here we use the previous lemma, which produces a considerably shorter proof.

Higher integrability with weights 361

3.3. Lemma. Let f ∈ L^{p}(Q_{0}; μ), 1 < p < ∞, be a nonnegative function. If there is t_{0} ≥ 0 and C_{1} ≥ 1 such that

(3.4)  ∫_{{f>t}} f^{p} dμ ≤ C_{1}t^{p−1} ∫_{{f>t}} f dμ

for every t ≥ t_{0} , then for every q > p for which C_{1}(q − p)/(q − 1) < 1, we have

(3.5)  ∫_{Q_{0}} f^{q} dμ ≤ C_{2}t_{0}^{q−p} ∫_{Q_{0}} f^{p} dμ,

where C_{2} depends on C_{1} , q and p.

Proof. Let t_{0} ≥ 0 such that (3.4) holds for every t ≥ t_{0} . Clearly

(3.6)  \begin{aligned}∫_{Q_{0}} f^{q} dμ & = ∫_{\{f≤t_{0}\}} f^{q} dμ + ∫_{\{f>t_{0}\}} f^{q} dμ \\ & ≤ t_{0}^{q−p} ∫_{\{f≤t_{0}\}} f^{p} dμ + ∫_{\{f>t_{0}\}} f^{q} dμ. \end{aligned}

Next we shall estimate the second integral on the right hand side. Let t_{1} > t_{0} . Using equality (3.2) with q replaced by q − p and dν = f^{p} dμ we get

\begin{aligned}∫_{\{t_{0}<f≤t_{1}\}} f^{q} dμ & = (q − p) ∫_{t_{0}}^{t_{1}} t^{q−p−1} ∫_{\{f>t\}} f^{p} dμ dt \\ & + t_{0}^{q−p} ∫_{\{f>t_{0}\}} f^{p} dμ − t_{1}^{q−p} ∫_{\{f>t_{1}\}} f^{p} dμ. \end{aligned}

The assumption (3.4) yields

∫_{t_{0}}^{t_{1}} t^{q−p−1} ∫_{{f>t}} f^{p} dμ dt ≤ C_{1} ∫_{t_{0}}^{t_{1}} t^{q−2} ∫_{{f>t}} f dμ dt.

By using (3.2) again, now with q replaced by q − 1 and dν = f dμ, we obtain

\begin{aligned}∫_{t_{0}}^{t_{1}} t^{q−2} ∫_{\{f>t\}} f dμ dt & = \frac{1}{q − 1}(∫_{\{t_{0}<f≤t_{1}\}} f^{q} dμ \\ & − t_{0}^{q−1} ∫_{\{f>t_{0}\}} f dμ + t_{1}^{q−1} ∫_{\{f>t_{1}\}} f dμ), \end{aligned}

and consequently

∫_{\{t_{0}<f≤t_{1}\}} f^{q} dμ ≤ C_{1} \frac{q − p}{q − 1} ∫_{\{t_{0}<f≤t_{1}\}} f^{q} dμ

+ (1 − \frac{q − p}{q − 1})t_{0}^{q−p} ∫_{\{f>t_{0}\}} f^{p} dμ

+ (C_{1} \frac{q − p}{q − 1} − 1)t_{1}^{q−p} ∫_{\{f>t_{1}\}} f^{p} dμ.

Here we also used the trivial estimate

∫_{\{f>t_{1}\}} f dμ ≤ ∫_{\{f>t_{1}\}} f(\frac{f}{t_{1}} )^{p−1} dμ = t_{1}^{1−p} ∫_{\{f>t_{1}\}} f^{p} dμ.

Since

∫_{{t_{0}<f≤t_{1}}} f^{q} dμ ≤ t_{1}^{q}μ({t_{0} < f ≤ t_{1}}) ≤ t_{1}^{q}μ(Q_{0}) < ∞,

it can be subtracted from both sides to obtain

\begin{aligned}(1 − C_{1} \frac{q − p}{q − 1}) & ∫_{\{t_{0}<f≤t_{1}\}} f^{q} dμ ≤ \frac{p − 1}{q − 1}t_{0}^{q−p} ∫_{\{f>t_{0}\}} f^{p} dμ \\ & + (C_{1} \frac{q − p}{q − 1} − 1)t_{1}^{q−p} ∫_{\{f>t_{1}\}} f^{p} dμ. \end{aligned}

Now, choosing q > p such that C_{1}(q − p)/(q − 1) < 1, we get

\begin{aligned}∫_{\{t_{0}<f≤t_{1}\}} f^{q} dμ & ≤ C_{2}t_{0}^{q−p} ∫_{\{f>t_{0}\}} f^{p} dμ − t_{1}^{q−p} ∫_{\{f>t_{1}\}} f^{p} dμ \\ & ≤ C_{2}t_{0}^{q−p} ∫_{\{f>t_{0}\}} f^{p} dμ, \end{aligned}

where C_{2} = C_{2}(C_{1}, p, q) ≥ 1. Since the right hand side does not depend on t_{1} , letting t_{1} → ∞ we obtain

∫_{{f>t_{0}}} f^{q} dμ ≤ C_{2}t_{0}^{q−p} ∫_{{f>t_{0}}} f^{p} dμ.

Finally by (3.6) we arrive at

∫_{Q_{0}} f^{q} dμ ≤ C_{2}t_{0}^{q−p} ∫_{Q_{0}} f^{q} dμ

and the lemma follows.

4. Proof of Theorem 1.5

Let Q_{0} be a cube in Ω. First we shall construct a Whitney decomposition for Q_{0} . Denote

Q_{i} = (1 − 2^{−i}) Q_{0},

for i = 1, 2, …,. Next divide each Q_{i} into (2^{i+1} − 2)^{n} pairwise disjoint dyadic open cubes, which cover Q_{i} up to the measure zero. Denote this family by 𝒬_{i}. Define a new family of disjoint cubes 𝒞_{i} by

𝒞_{1} = 𝒬_{1},
𝒞_{i+1} = {Q ∈ 𝒬_{i+1} : Q ∩ Q_{i} = ∅ for every Q_{i} ∈ 𝒞_{i}},

for i = 2, 3, …,. Write 𝒞 = ⋃_{i=1}^{∞} 𝒞_{i}. Cubes in 𝒞 cover Q_{0} up to the measure zeroand they are pairwise disjoint. The enlarged cubes 2Q are still subsets of Q_{0} .

Denote

s_{0} = (\overline{∫} _{Q_{0}} f^{p} dμ)^{1/p}

and let s > 0 be such that s ≥ s_{0} . If Q ∈ 𝒞 , then

\overline{∫} _{Q} f^{p} dμ ≤ \frac{1}{μ(Q)} ∫_{Q_{0}} f^{p} dμ = \frac{μ(Q_{0})}{μ(Q)}\overline{∫} _{Q_{0}} f^{p} dμ ≤ a_{Q}s^{p}

where a_{Q} = μ(Q_{0})/μ(Q). Define a new function g by

g(x) = a_{Q}^{−1/p}f(x)

whenever x ∈ Q ∈ 𝒞 . Since cubes in 𝒞 cover Q_{0} up to the measure zero, g is a well defined function almost everywhere in Q_{0} . Define, for example, g to be zero elsewhere in Q_{0} . Fix Q ∈ 𝒞 . Then

\overline{∫} _{Q} g^{p} dμ ≤ s^{p}

and we can use inequality (2.9) to conclude

(4.1)  ∫_{{g>s}∩Q} g^{p} dμ ≤ C_{1}s^{p}μ({M_{Q}^{p}g > s}).

Next we estimate the right hand side of (4.1).

Suppose that x ∈ Q and Q_{x} is a subcube of Q containing x. Then the construction above guarantees that the twice enlarged cube 2Q_{x} is contained in the basic cube Q_{0} and hence the weak reverse Hölder inequality (1.6) implies

(4.2)  \begin{aligned}(\overline{∫} _{Q_{x}} g^{p} dμ)^{1/p} & = a_{Q}^{−1/p}(\overline{∫} _{Q_{x}} f^{p} dμ)^{1/p} \\ & ≤ C_{2}a_{Q}^{−1/p}\overline{∫} _{2Q_{x}} f dμ, \end{aligned}

where C_{2} is the constant in (1.6). Now it is easy to see that 2Q_{x} can intersect at most those cubes in 𝒞 which touch Q and hence there is a cube Q′ in 𝒞 which touches Q such that

g ≥ a_{Q′}^{−1/p}f

in 2Q_{x}. On the other hand, by the construction of the family 𝒞 , Q ⊂ 5Q′ and using the doubling property of μ we conclude that μ(Q) ≤ μ(5Q′) ≤ C_{3}μ(Q′) and consequently a_{Q′} ≤ C_{3}a_{Q}, where the constant C_{3} depends on the doubling constant in (1.2) and the dimension n. This implies

\begin{aligned}\overline{∫} _{2Q_{x}} f dμ & ≤ a_{Q′}^{1/p}\overline{∫} _{2Q_{x}} g dμ \\ & ≤ C_{3}^{1/p}a_{Q}^{1/p}\overline{∫} _{2Q_{x}} g dμ \end{aligned}

and by the inequality (4.2), we obtain

(\overline{∫} _{Q_{x}} g^{p} dμ)^{1/p} ≤ C_{4}\overline{∫} _{2Q_{x}} g dμ.

In other words M_{Q}^{p}g(x) ≤ C_{4}M_{Q_{0}}g(x) and hence

\{M_{Q}^{p}g > s\} ⊂ \{M_{Q_{0}}g > \frac{s}{C_{4}} \} ∩ Q.

Using (4.1) we see that

(4.3)  ∫_{\{g>s\}∩Q} g^{p} dμ ≤ C_{1}s^{p}μ(\{M_{Q_{0}}g > \frac{s}{C_{4}} \} ∩ Q).

Now (4.3) holds in every cube Q ∈ 𝒞 and by summing over all cubes in 𝒞 we obtain

(4.4)  ∫_{\{g>s\}} g^{p} dμ ≤ C_{1}s^{p}μ(\{M_{Q_{0}}g > \frac{s}{C_{4}} \}).

On the other hand formula (2.4) yields

(4.5)  μ(\{M_{Q_{0}}g > \frac{s}{C_{4}} \}) ≤ \frac{C_{5}}{s} ∫_{\{g>t\}} g dμ,

where t = s/(2C_{4}). Combining (4.4) and (4.5) we obtain

(4.6)  ∫_{{g>s}} g^{p} dμ = C_{6}s^{p−1} ∫_{{g>t}} g dμ.

On the other hand

(4.7)  \begin{aligned}∫_{\{t<g≤s\}} g^{p} dμ & = ∫_{\{t<g≤s\}} g^{p−1}g dμ \\ & ≤ s^{p−1} ∫_{\{g>t\}} g dμ \end{aligned}

and hence (4.6) and (4.7) yield

\begin{aligned}∫_{\{g>t\}} g^{p} dμ & = ∫_{\{t<g≤s\}} g^{p} dμ + ∫_{\{g>s\}} g^{p} dμ \\ & ≤ C_{7}t^{p−1} ∫_{\{g>t\}} g dμ, \end{aligned}

for every t ≥ t_{0} where t_{0} = s_{0}/(2C_{4}).

Now using the previous lemma we obtain q > p such that

∫_{Q_{0}} g^{q} dμ ≤ C_{8}t_{0}^{q−p} ∫_{Q_{0}} f^{p} dμ,

where C_{8} = C_{8}(p, q, C_{2}). Here we used the fact that g ≤ f in Q_{0} . By substituting t_{0} , we get

(\overline{∫} _{Q_{0}} g^{q} dμ)^{1/q} ≤ C_{9}(\overline{∫} _{Q_{0}} f^{p} dμ)^{1/p}.

The theorem now follows since for almost every x ∈ Q_{1} = 1/2Q_{0} , g(x) = C_{10}^{−1/p}f(x), where C_{10} is the doubling constant in (1.2), and hence

(\overline{∫} _{Q_{1}} f^{q} dμ)^{1/q} ≤ C_{10}^{1/p}(\frac{1}{μ(Q_{1})} ∫_{Q_{0}} g^{q} dμ)^{1/q}

≤ C_{10}^{1/p+1/q}(\overline{∫} _{Q_{0}} g^{q} dμ)^{1/q}

≤ C_{9}C_{10}^{1/p+1/q}(\overline{∫} _{Q_{0}} f^{p} dμ)^{1/p}

= C_{11}(\overline{∫} _{Q_{0}} f^{p} dμ)^{1/p},

where C_{11} = C_{11}(n, p, q, C_{2}).

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366 Juha Kinnunen

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Received 5 March 1993