Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 175–188

POINCARÉ INEQUALITIES FOR POWERS AND PRODUCTS OF ADMISSIBLE WEIGHTS

Jana Björn

Linköping University, Department of Mathematics SE-581 83 Linköping, Sweden; jabjo@mai.liu.se

Abstract. Our main result is that if μ is an s-admissible measure in R^{n} and v ∈ A_{p}(dμ), then the measure dν = v dμ is ps-admissible. A two-weighted version of this result is also proved. It is further shown that every strong A_{∞} -weight w in R^{n}, n ≥ 2, is n/(n − 1)-admissible, that its power w^{1−1/n} is 1-admissible and that the weights w^{1−p/n} with 1 < p < n are q -admissible for some q < p. A counterexample showing that we cannot take q = 1 in general is also given. Finally, a new class of p-admissible weights is described.

1. Introduction

In Fabes–Kenig–Serapioni [4] four conditions sufficient for extending Moser’s iteration technique to weighted degenerate equations were singled out. Later, in Heinonen–Kilpeläinen–Martio [12] such weights were called “p-admissible” and a rich potential theory was developed for them. Recently, Hajłasz, Heinonen, Koskela and Semmes showed that the conditions described in Fabes–Kenig–Serapioni [4] can be reduced to only two, see Theorem 2 in Hajłasz–Koskela [10] and Theorem 5.2 in Heinonen–Koskela [13]. Thus, a non-negative locally integrable function w in R^{n} is a p-admissible weight with 1 ≤ p < ∞ if and only if the measure μ associated with w through dμ = w dx, where dx denotes integration with respect to the Lebesgue measure, satisfies the following two conditions:

Doubling condition: 0 < w < ∞ a.e. in R^{n} and there is a constant C > 0 such that

μ(2B) < Cμ(B)

for all balls B ⊂ R^{n}, where 2B denotes the ball concentric with B and with twice the radius.

2000 Mathematics Subject Classification: Primary 46E35.

The results of this paper were obtained while the author was visiting the University of Michigan, Ann Arbor, on leave from the Linköping University. The research was supported by postdoctoral grants from the Swedish Natural Science Research Council and the Knut and Alice Wallenberg Foundation.

Weak (1, p)-Poincaré inequality: There exist constants C > 0 and λ ≥ 1 such that

\overline{∫}_{B} |u − u_{B,μ}| dμ ≤ Cr(\overline{∫}_{λB} |∇u|^{p} dμ)^{1/p}

holds whenever B is a ball with radius r and u is, say, a locally Lipschitz function on λB. Here and in what follows, u_{B,μ} = \overline{∫}_{B} u dμ and the symbol \overline{∫} stands for the mean-value integral

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

A measure μ satisfying the above conditions will also be called p-admissible. The Hölder inequality implies that every p-admissible measure is also p′ -admissible for all p′ > p. Note also that by Theorem 1 in Hajłasz–Koskela [10], a weak (1, p)- Poincaré inequality for a doubling measure in R^{n} implies a strong (1, p)-Poincaré inequality with λ = 1.

Many interesting examples of p-admissible weights are provided by weights from the Muckenhoupt class A_{p}, see e.g. Chapter 15 in Heinonen–Kilpeläinen– Martio [12]. Non-A_{p} examples of p-admissible weights have been given in e.g. Chanillo–Wheeden [2] and Franchi–Gutiérrez–Wheeden [5]. In Chapter 15 in Heinonen–Kilpeläinen–Martio [12] it is shown for 1 < p < n that (1 − p/n)- powers of Jacobians of quasiconformal mappings in R^{n}, n ≥ 2, are p-admissible. This result was extended to strong A_{∞} -weights by Heinonen and Koskela [13], viz. they prove the following theorem in the case 1 < p < n. We will provide a new proof of this theorem which covers also the case p = 1.

Theorem 1. Let w be a strong A_{∞} -weight in R^{n}, n ≥ 2, and 1 ≤ p < n. Then the weight w^{1−p/n} is p-admissible.

Strong A_{∞} -weights were introduced in David–Semmes [3] and further studied in e.g. Semmes [17] and [18]. For a doubling measure μ consider the function δ(x, y) = μ(B_{xy})^{1/n}, where B_{xy} denotes the smallest closed ball containing both x and y. The doubling condition of μ implies that δ is a quasi-metric, i.e. it is symmetric, vanishes only if x = y and satisfies the weak triangle inequality δ(x, y) ≤ C(δ(x, z) + δ(z, y)). As mentioned in David–Semmes [3], an argument by Gehring [9] shows that if there exists a metric d in R^{n} such that

(1)  C^{−1}d(x, y) ≤ δ(x, y) ≤ C d(x, y)

for some C > 0 and all x, y ∈ R^{n}, then the measure μ is absolutely continuous with respect to the Lebesgue measure and its Radon–Nikodym derivative w satisfies the reverse Hölder inequality

(2)  (\overline{∫}_{B} w(x)^{r} dx)^{1/r} ≤ C \overline{∫}_{B} w(x) dx

for some constants C > 0, r > 1 and every ball B ⊂ R^{n}.

Definition 2. Weights satisfying the reverse Hölder inequality (2) are called A_{∞} -weights and those satisfying (1) for some metric d in R^{n} are strong A_{∞} - weights.

The above mentioned argument shows that every strong A_{∞} -weight is an A_{∞} -weight. Note also that every A_{1} -weight is a strong A_{∞} -weight and that Jacobians of quasiconformal mappings in R^{n}, n ≥ 2, are strong A_{∞} -weights, see e.g. David–Semmes [3].

We shall use the notation v ∈ A_{p}(dμ), 1 ≤ p < ∞, if for some C > 0 and all balls B ⊂ R^{n},

\overline{∫}_{B} v dμ < \{ \begin{matrix}C(\overline{∫}_{B} v^{1/(1−p)} dμ)^{1−p} & for p > 1, \\ C ess inf_{B} v & for p = 1. \end{matrix}

Equivalently, we can consider all cubes in R^{n} with sides parallel to the coordinate axes. If μ is the Lebesgue measure, then we write A_{p} rather than A_{p}(dx). For various properties of A_{p} -weights see e.g. García-Cuerva–Rubio de Francia [8] and Torchinsky [21]. We shall need the fact that w is an A_{∞} -weight if and only if w ∈ A_{p} for some p < ∞.

Heinonen–Koskela’s proof of Theorem 1 is based on the following result due to Franchi–Gutiérrez–Wheeden [5] (here stated in terms of the usual gradient rather than the λ-gradient considered in [5]). For a similar result on spaces of homogeneous type see Corollary 3.2 in Franchi–Pérez–Wheeden [7]. From now on, the open ball in R^{n} with centre x and radius r will be denoted B(x, r).

Theorem 3. Let w be a strong A_{∞} -weight in R^{n}, n ≥ 2, 1 ≤ p < q < ∞ and v ∈ A_{p}(w^{1−1/n} dx). Put dμ = vw^{1−1/n} dx and let ν be a doubling measure absolutely continuous with respect to the Lebesgue measure, satisfying the condition

(3)  \frac{r′}{r} (\frac{ν(B′)}{ν(B)})^{1/q} ≤ C(\frac{μ(B′)}{μ(B)})^{1/p}

for all balls B = B(x, r) and B′ = B(x′, r′) in R^{n} such that B′ ⊂ cB with some fixed c > 1. Then the pair (ν, μ) admits the two-weighted (q, p)-Poincaré inequality

(4)  (\overline{∫}_{B} |u − u_{B,ν}|^{q} dν)^{1/q} ≤ Cr(\overline{∫}_{B} |∇u|^{p} dμ)^{1/p}.

Remark. Note that by Chanillo–Wheeden [1] the condition (3) is essentially necessary for the Poincaré inequality (4) to be valid.

In this paper we give a short proof of the following analogue of Theorem 3 in the setting of admissible measures. Theorem 3 itself follows from Proposition 5 and Theorem 7 below.

Theorem 4. Let μ be an s-admissible measure, 1 ≤ s < ∞, and v ∈ A_{p}(dμ), 1 ≤ p < ∞. Then the measure ν given by dν = v dμ is ps-admissible.

As (1−1/n)-powers of strong A_{∞} -weights are 1-admissible by e.g. Theorem 1, Theorem 4 provides a simple proof of the following one-weighted (ν = μ) special case of Theorem 3.

Proposition 5. Let w be a strong A_{∞} -weight in R^{n}, n ≥ 2, and v ∈ A_{p}(w^{1−1/n} dx), 1 ≤ p < ∞. Then the weight vw^{1−1/n} is p-admissible.

Contrary to Theorem 3, Theorem 4 allows us to consider weights which are not strong A_{∞} -weights. For example, we can prove the following generalization of a result due to Chanillo–Wheeden [2].

Proposition 6. Let 2 ≤ k ≤ n and write the points in R^{n} as x = (x′, x′′), where x′ ∈ R^{k} and x′′ ∈ R^{n−k}. Let 1 ≤ p < ∞, v ∈ A_{p}, a_{1}, a_{2}, …, a_{m} ∈ R^{k} and γ_{j} ≥ 0, j = 0, 1, …, m. Then the weight

v(x)(1 + |x′|)^{γ_{0}} ∏_{j=1}^{m}(\frac{|x′ − a_{j}|}{1 + |x′ − a_{j}|})^{γ_{j}}

is p-admissible.

Two-weighted versions of Theorem 4 and Propositions 5 and 6 follow immediately from the following theorem.

Theorem 7. Let μ be a p-admissible measure. Let 1 ≤ p < q < ∞ and let ν be a doubling measure satisfying the condition (3) for all balls B = B(x, r) and B′ = B(x′, r′) such that B′ ⊂ B. Then the pair (ν, μ) admits the two-weighted (q, p)-Poincaré inequality (4).

Remark. A reader familiar with Poincaré inequalities on metric spaces easily verifies that Theorem 4 (with the same proof) and Theorem 7 (with a weak twoweighted (q, p)-Poincaré inequality in the conclusion and a slightly modified proof) are valid in the setting of doubling metric measure spaces. We will not dwell on this generalization in this paper.

Let us also mention some consequences of Proposition 5 and the reverse Hölder inequality (2), which improve Theorem 1.

Corollary 8. Let w ∈ A_{q} be a strong A_{∞} -weight in R^{n}, n ≥ 2, satisfying the reverse Hölder inequality (2) with r > 1 and let 1 − 1/n ≤ σ ≤ r. Then the weight w^{σ} is p-admissible for all

p ≥ \frac{n(σ(q − 1) + 1)}{q(n − 1) + 1}.

In particular, every strong A_{∞} -weight in R^{n}, n ≥ 2, is n/(n − 1)-admissible.

Corollary 9. Let w be a strong A_{∞} -weight in R^{n}, n ≥ 2, and 1 < p < n. Then the weight w^{1−p/n} is q -admissible for all

q ≥ \frac{n(r − 1) + p}{n(r − 1) + 1},

where r is the exponent from the reverse Hölder inequality (2). In particular, w^{1−p/n} is q -admissible for some q < p.

If moreover w^{−t} ∈ A_{1} for some t > 0, then it can be derived from Lemma 3.17 in Semmes [17] that w^{s} = w^{s−1} w is a strong A_{∞} -weight for all 0 < s < 1, and hence the weight w^{1−p/n} = (w^{(n−p)/(n−1)})^{1−1/n} with 1 ≤ p ≤ n is 1-admissible, by e.g. Theorem 1. On the other hand it is shown in the counterexample to Question 4.1 in Semmes [17], that there are strong A_{∞} -weights (even Jacobians of quasiconformal mappings) such that w^{s} is not a strong A_{∞} -weight for any 0 < s < 1 and consequently the above argument cannot be applied. In fact, the following occurs.

Proposition 10. There exists a strong A_{∞} -weight w in R^{2} such that none of the weights w^{1−p/2} , 1 < p < 2, is 1-admissible.

Finally, note that the well-known p-admissibility of A_{p} -weights is a direct consequence of Theorem 4 and the 1-admissibility of the Lebesgue measure.

Acknowledgement. The author is grateful to the referee for valuable suggestions and comments.

2. The proofs

Let us first recall a simple consequence of the reverse Hölder inequality (2). If 0 < s < 1, the Hölder inequality with w = w^{θ}w^{1−θ} and θ = s(r − 1)/(r − s) < 1 yields

(\overline{∫}_{B} w^{r} dx)^{1/r} ≤ C \overline{∫}_{B} w dx ≤ C(\overline{∫}_{B} w^{s} dx)^{θ/s}(\overline{∫}_{B} w^{r} dx)^{1−θ/s}

and division by the last factor on the right-hand side gives

(5)  (\overline{∫}_{B} w^{r} dx)^{1/r} ≤ C^{1/θ}(\overline{∫}_{B} w^{s} dx)^{1/s}.

From now on, the letter C will denote a positive constant whose exact value is unimportant and may change even within a line. We shall also use the notation a ≃ b if a/C ≤ b ≤ Ca holds for some C.

Proof of Theorem 1. It suffices to consider the case p = 1. If p > 1, then (5) and the Hölder inequality show that w^{(1−p)/n} ∈ A_{p}(w^{1−1/n} dx) and Theorem 4 then implies that the weight w^{1−p/n} = w^{(1−p)/n}w^{1−1/n} is p-admissible.

First, note that by (5), w^{1−1/n} is an A_{∞} -weight and hence the measure dμ = w^{1−1/n} dx is doubling. Let also dν = w dx. The (1, 1)-Poincaré inequality for w^{1−1/n} is a consequence of the following Poincaré type inequality for strong A_{∞} -weights by David and Semmes [3],

(6)  \overline{∫}_{B} \overline{∫}_{B} |u(ξ) − u(η)| dν(ξ) dν(η) ≤ C \frac{μ(2B)}{ν(B)^{1−1/n}} \overline{∫}_{2B} |∇u(ξ)| dμ(ξ),

which holds for all balls B in R^{n} and all locally Lipschitz functions u. In order to prove that w^{1−1/n} is 1-admissible we have to show that the measure ν on the left-hand side can be replaced by μ and that the factor in front of the integral on the right-hand side is comparable to r. This is done using an argument as in Franchi–Hajłasz [6]:

Let B = B(x_{0}, r) be a ball in R^{n}, u_{B,ν} = \overline{∫}_{B} u dν and let for x ∈ B and k = 0, 1, …,

B_{k}(x) = B(x, 2^{1−k}r)  and  u_{k} = \overline{∫}_{B_{k}(x)} u dν.

Then the doubling property of ν and the fact that u_{k} → u(x), as k → ∞, imply

\begin{aligned}|u(x) − u_{B,ν}| & ≤ |u_{0} − u_{B,ν}| + ∑_{k=0}^{∞} |u_{k+1} − u_{k}| \\ & ≤ |u_{0} − u_{B,ν}| + C ∑_{k=0}^{∞} \overline{∫}_{B_{k}(x)} \overline{∫}_{B_{k}(x)} |u(ξ) − u(η)| dν(ξ) dν(η). \end{aligned}

Note that B ⊂ B_{0}(x) and hence |u_{B,ν} − u_{0}| ≤ C \overline{∫}_{B_{0}(x)} |u − u_{0}| dν, which can be included in the above sum. The Poincaré type inequality (6) then implies

|u(x) − u_{B,ν}| ≤ C ∑_{k=0}^{∞} \frac{μ(2B_{k}(x))}{ν(B_{k}(x))^{1−1/n}} \overline{∫}_{2B_{k}(x)} |∇u(ξ)| dμ(ξ).

The quotient μ(2B_{k}(x))/ν(B_{k}(x))^{1−1/n} does not exceed C|B_{k}(x)|^{1/n} ≃ 2^{−k}r, by the doubling property of μ and the Hölder inequality, and we obtain

|u(x) − u_{B,ν}| ≤ Cr ∑_{k=0}^{∞} 2^{−k} \overline{∫}_{2B_{k}(x)} |∇u(ξ)| dμ(ξ).

Averaging both sides over the ball B with respect to μ gives

(7) \overline{∫}_{B} |u − u_{B,ν}| dμ ≤ Cr ∑_{k=0}^{∞} 2^{−k} \overline{∫}_{B} ∫_{R^{n}} \frac{χ_{2B_{k}(x)}(ξ)}{μ(2B_{k}(x))}|∇u(ξ)| dμ(ξ) dμ(x),

where χ denotes the characteristic function of a set. Note that 2B_{0}(x) ⊂ 5B, χ_{2B_{k}(x)}(ξ) = χ_{2B_{k}(ξ)}(x) and that the doubling property of μ implies μ(2B_{k}(x)) ≃ μ(2B_{k}(ξ)) whenever ξ ∈ 2B_{k}(x). It follows that

\frac{χ_{2B_{k}(x)}(ξ)}{μ(2B_{k}(x))} ≤ Cχ_{5B}(ξ)\frac{χ_{2B_{k}(ξ)}(x)}{μ(2B_{k}(ξ))},

which inserted into (7) together with the Fubini theorem and the doubling property of μ yields

\overline{∫}_{B} |u − u_{B,ν}| dμ ≤ Cr \overline{∫}_{5B} |∇u| dμ.

The required (1, 1)-Poincaré inequality for μ then follows from the inequality

(8)  \overline{∫}_{B} |u − u_{B,ν}| dμ ≤ 2 \overline{∫}_{B} |u − u_{B,μ}| dμ.

Proof of Theorem 4. We shall assume p > 1, the case p = 1 is treated similarly. If B is a ball then the Hölder inequality and the fact that v ∈ A_{p}(dμ) yield

\begin{aligned}μ(B) & = ∫_{B} v^{−1/p}v^{1/p} dμ ≤ (∫_{2B} v^{1/(1−p)} dμ)^{1−1/p}(∫_{B} v dμ)^{1/p} \\ & ≤ Cμ(2B) ν(2B)^{−1/p} ν(B)^{1/p}. \end{aligned}

The doubling condition μ(2B) ≤ Cμ(B) then implies ν(2B) ≤ Cν(B), i.e. the measure ν is doubling.

By Theorems 3.2 and 3.3 in Hajłasz–Koskela [11] or Lemma 5.15 in Heinonen– Koskela [14], the weak (1, s)-Poincaré inequality for μ is equivalent to the validity of the inequality

|u(x) − u(y)| ≤ C|x − y|(M_{μ,λ|x−y|} |∇u|^{s}(x) + M_{μ,λ|x−y|} |∇u|^{s}(y))^{1/s}

for some C > 0, λ ≥ 1, μ-a.e. x, y ∈ R^{n} and all locally Lipschitz functions u. Here M_{μ,λ|x−y|} |∇u|^{s}(x) is the maximal function defined for g ∈ L_{loc}^{1}(R^{n}, μ) by

M_{μ,R} g(x) = sup_{0<ϱ<R} \overline{∫}_{B(x,ϱ)} g dμ.

If we can show that for some C > 0,

(9)  M_{μ,R} g(x) ≤ C(M_{ν,R} g^{p}(x))^{1/p},

then another application of Lemma 5.15 in Heinonen–Koskela [14] or Theorem 3.3 in Hajłasz–Koskela [11] implies that ν admits the weak (1, ps)-Poincaré inequality and the theorem follows. In order to prove (9), let B be a ball. The Hölder inequality gives

\begin{aligned}\overline{∫}_{B} g dμ & ≤ (\overline{∫}_{B} g^{p}v dμ)^{1/p}(\overline{∫}_{B} v^{1/(1−p)} dμ)^{1−1/p} \\ & = (\overline{∫}_{B} g^{p} dν)^{1/p}(\overline{∫}_{B} v dμ)^{1/p}(\overline{∫}_{B} v^{1/(1−p)} dμ)^{1−1/p}. \end{aligned}

As v ∈ A_{p}(dμ), the product of the last two factors is bounded by a constant independent of B and (9) follows by taking supremum over all balls B with radius 0 < ϱ < R and centre x.

Proof of Proposition 6. We can assume that the points a_{0} = 0, a_{1}, a_{2}, …, a_{m} are distinct and that p > 1. The case p = 1 is treated similarly. Let

w(x) = w̃(x′) = (1 + |x′|)^{γ_{0}} ∏_{j=1}^{m}(\frac{|x′ − a_{j}|}{1 + |x′ − a_{j}|})^{γ_{j}}.

First, we claim that v ∈ A_{p}(w dx). Indeed, let Q = Q′ × Q′′ be a cube in R^{k} × R^{n−k} with sides parallel to the coordinate axes. Partitioning the cube Q′ into (6(m + 1))^{k} equally sized cubes we find a subcube Q̃ of Q′ with sidelength comparable to that of Q′, such that the sidelength h of Q̃ satisfies 2h ≤ d_{j} = dist(Q̃, a_{j}) for all j = 0, 1, …, m. Consider the weights defined on R^{k} by

ṽ_{1}(x′) = ∫_{Q′′} v(x′, x′′) dx′′  and  ṽ_{2}(x′) = ∫_{Q′′} v(x′, x′′)^{1/(1−p)} dx′′.

One easily verifies that both ṽ_{1} and ṽ_{2} satisfy the doubling condition on R^{k} with the same doubling constants as v and v^{1/(1−p)} , respectively. Lemma 6.3 in Strömberg–Wheeden [20] then implies that both ṽ_{1}w̃ and ṽ_{2}w̃ are doubling weights in R^{k}. Consequently, we have

∫_{Q} v(x)w(x) dx (∫_{Q} v(x)^{1/(1−p)}w(x) dx)^{p−1}

= ∫_{Q′} ṽ_{1}(x′)w̃(x′) dx′ (∫_{Q′} ṽ_{2}(x′)w̃(x′) dx′)^{p−1}

≤ C ∫_{Q} ṽ_{1}(x′)w̃(x′) dx′ (∫_{Q} ṽ_{2}(x′)w̃(x′) dx′)^{p−1}

and as d_{j} ≤ |x′ − a_{j}| ≤ (1 + \sqrt{k} /2)d_{j} for all x′ ∈ Q̃ and j = 0, 1, …, m, this is comparable to

∫_{Q} ṽ_{1}(x′) dx′ (∫_{Q} ṽ_{2}(x′) dx′)^{p−1}[(1 + d_{0})^{γ_{0}} ∏_{j=1}^{m}(\frac{d_{j}}{1 + d_{j}} )^{γ_{j}}]^{p}.

The A_{p} -condition for v implies

\begin{aligned}∫_{Q} ṽ_{1}(x′) dx′ (∫_{Q} ṽ_{2}(x′) dx′)^{p−1} & ≤ ∫_{Q} v(x) dx (∫_{Q} v(x)^{1/(1−p)} dx)^{p−1} \\ & ≤ C|Q|^{p} ≤ C|Q̃ × Q′′|^{p}. \end{aligned}

Altogether, using once again |x′ − a_{j}| ≃ d_{j} for x′ ∈ Q̃, we obtain

∫_{Q} v(x)w(x) dx (∫_{Q} v(x)^{1/(1−p)}w(x) dx)^{p−1}

≤ C[(1 + d_{0})^{γ_{0}} ∏_{j=1}^{m}(\frac{d_{j}}{1 + d_{j}} )^{γ_{j}}]^{p}|Q̃ × Q′′|^{p}

≤ C(∫_{Q′′} ∫_{Q} w̃(x′) dx′ dx′′)^{p}

≤ C(∫_{Q} w(x) dx)^{p},

i.e. v ∈ A_{p}(w dx).

The proposition now follows from Theorem 4 if we show that the weight w is 1-admissible. To this end, it is easily verified that the product

∏_{j=1}^{m}(\frac{|x′ − a_{j}|}{1 + |x′ − a_{j}|})^{kα_{j}},

with α_{j} > 0, is comparable to the Jacobian of the k -dimensional quasiconformal mapping

f(x′) = \{ \begin{matrix}a_{j} + (\frac{|x′ − a_{j}|}{ϱ})^{α_{j}}(x′ − a_{j}) & if |x′ − a_{j}| ≤ ϱ, \\ x′ & otherwise, \end{matrix}

where ϱ > 0 is fixed so that the balls {x′ ∈ R^{k} : |x′ − a_{j}| ≤ ϱ}, j = 0, 1, …, m, are pairwise disjoint. Similarly, the factor (1 + |x′|)^{kα_{0}} with α_{0} > 0 is comparable to the Jacobian of the quasiconformal mapping

g(x′) = \{ \begin{matrix}x′ & if |x′| ≤ 1, \\ |x′|^{α_{0}}x′ & otherwise. \end{matrix}

As |f(x′)| ≃ |x′| for all x′ ∈ R^{k}, we obtain by choosing α_{j} = γ_{j}/(k − 1) that

w̃(x′) ≃ J(x′)^{1−1/k},

where J denotes the Jacobian of the quasiconformal mapping g∘f. By Theorem 1 and the fact that Jacobians of quasiconformal mappings in R^{k}, k ≥ 2, are strong A_{∞} -weights, w̃ is 1-admissible in R^{k}. The 1-admissibility of w in R^{n} then follows from the following lemma which is easily proved using the Fubini theorem, cf. Lemma 2 in Lu–Wheeden [15].

Lemma 11. Let μ_{1} and μ_{2} be doubling measures on R^{n_{1}} and R^{n_{2}} , respectively, admitting the (1, p)-Poincaré inequality with p ≥ 1. Then the product measure μ = μ_{1} × μ_{2} on R^{n_{1}} × R^{n_{2}} is doubling and admits the (1, p)-Poincaré inequality.

Proof of Theorem 7. The argument is similar to the proof of Theorem 5.3 in Hajłasz–Koskela [11]. Let B = B(x_{0}, r) be a ball in R^{n} and let u be a Lipschitz function on B. We can assume that u_{B,μ} = 0 and that λ = 1 in the (1, p)- Poincaré inequality for μ. The (1, p)-Poincaré inequality for μ then implies as in the proof of Theorem 1 that for every x ∈ B,

|u(x)| ≤ C ∑_{j=0}^{∞} r_{j}(\overline{∫}_{B_{j}} |∇u|^{p} dμ)^{1/p},

where B_{j} = B(x_{j}, r_{j}), r_{j} = 2^{−j}r and each x_{j} lies on the segment [x_{0}, x] at the distance 2^{−j}|x − x_{0}| from x. The condition (3) applied to the balls B_{j} and B then yields

|u(x)| ≤ \frac{Crν(B)^{1/q}}{μ(B)^{1/p}} ∑_{j=0}^{∞} \frac{1}{ν(B_{j})^{1/q}} (∫_{B_{j}} |∇u|^{p} dμ)^{1/p}.

Next, we write the above sum as Σ′ + Σ′′, where the summation in Σ′ and Σ′′ is over j < j_{0} and j ≥ j_{0} , respectively (j_{0} will be chosen later). Note also that as B_{j+1} ⊂ B_{j} and the set B_{j} \ B_{j+1} contains a ball B′_{j} such that B_{j} ⊂ 7B′_{j}, the doubling property of ν implies ν(B_{j+1}) ≤ γν(B_{j}) for some γ < 1 independent of j. It follows that ν(B_{j}) ≥ γ^{j−j_{0}}ν(B_{j_{0}}) for j < j_{0} and ν(B_{j}) ≤ γ^{j−j_{0}}ν(B_{j_{0}}) for j ≥ j_{0} . Hence,

Σ′ = ∑_{j=0}^{j_{0}−1} \frac{1}{ν(B_{j})^{1/q}} (∫_{B_{j}} |∇u|^{p} dμ)^{1/p} ≤ \frac{C}{ν(B_{j_{0}})^{1/q}} (∫_{B} |∇u|^{p} dμ)^{1/p}

and

Σ′′ = ∑_{j=j_{0}}^{∞} \frac{1}{ν(B_{j})^{1/q}} (∫_{B_{j}} |∇u|^{p} dμ)^{1/p} ≤ Cν(B_{j_{0}})^{1/p−1/q}M(x)^{1/p},

where

M(x) = sup_{B′} \frac{1}{ν(B′)} ∫_{B′} |∇u|^{p} dμ

and the supremum is taken over all balls B′ ⊂ B containing x. Next, as ν(B)^{−1} ∫_{B} |∇u|^{p} dμ ≤ M(x), we can find j_{0} such that

ν(B_{j_{0}}) ≃ \frac{1}{M(x)} ∫_{B} |∇u|^{p} dμ

and inserting this into the above estimates of Σ′ and Σ′′ yields

|u(x)| ≤ \frac{Crν(B)^{1/q}}{μ(B)^{1/p}} (Σ′ + Σ′′) ≤ \frac{Crν(B)^{1/q}}{μ(B)^{1/p}} (∫_{B} |∇u|^{p} dμ)^{1/p−1/q}M(x)^{1/q}.

A standard argument using a Vitali type covering lemma (e.g. Lemma 5.5 in Heinonen–Koskela [14]) and the doubling property of ν shows that

ν(\{x ∈ B : M(x) ≥ τ\}) ≤ \frac{C}{τ} ∫_{B} |∇u|^{p} dμ,

cf. e.g. Chapter 1 in Stein [19]. Hence

ν(\{x ∈ B : |u(x)| ≥ t\}) ≤ \frac{Cr^{q}ν(B)}{t^{q}μ(B)^{q/p}} (∫_{B} |∇u|^{p} dμ)^{q/p}.

The rest of the proof is by Maz’ya’s truncation method [16] as in the proof of Lemma 5.15 in Heinonen–Koskela [14] or Theorem 2.1 in Hajłasz–Koskela [11]: We apply the above argument to the truncation of u given by

v(x) = min{2^{j}, max{u(x) − 2^{j}, 0}},

and conclude

2^{jq}ν(\{x ∈ B : u(x) ≥ 2^{j+1}\}) ≤ \frac{Cr^{q}ν(B)}{μ(B)^{q/p}} (∫_{\{x∈B:2^{j}<u(x)<2^{j+1}\}} |∇u|^{p} dμ)^{q/p}.

Summing up over all integers j then gives

∫_{{x∈B:u(x)>0}} |u|^{q} dν ≤ ∑_{j=−∞}^{∞} 2^{(j+2)q}ν({x ∈ B : u(x) ≥ 2^{j+1}})

≤ \frac{Cr^{q}ν(B)}{μ(B)^{q/p}} ∑_{j=−∞}^{∞}(∫_{\{x∈B:2^{j}<u(x)<2^{j+1}\}} |∇u|^{p} dμ)^{q/p}

≤ Cr^{q}ν(B)(\overline{∫}_{B} |∇u|^{p} dμ)^{q/p}.

The integral over {x ∈ B : u(x) < 0} is estimated similarly and the inequality (8) finishes the proof.

Proof of Corollary 8. Apply Proposition 5 to v = w^{σ−(1−1/n)} and

p = \frac{n(σ(q − 1) + 1)}{q(n − 1) + 1}.

The inequality (5) with s = 1 − 1/n, v^{1/(1−p)}w^{1−1/n} = w^{1/(1−q)} and w ∈ A_{q} imply that v ∈ A_{p}(w^{1−1/n} dx) and the claim follows.

Proof of Corollary 9. By Proposition 5 it suffices to show that w^{(1−p)/n} ∈ A_{q}(w^{1−1/n} dx), where

q = \frac{n(r − 1) + p}{n(r − 1) + 1}.

This follows from the inequality (5) with s = 1 − 1/n and the Hölder inequality.

Proof of Proposition 10. Let w_{1}: R^{2} → R be the weight from the counterexample to Question 4.1 in Semmes [17] and let

w(x_{1}, x_{2}) = w_{1}(x_{1}, x_{2}) + w_{1}(x_{1}, −x_{2}),

so that w(x_{1}, x_{2}) = w(x_{1}, −x_{2}). It is easily verified using the definition (1) that w is also a strong A_{∞} -weight. Let 1 < p < 2 and dμ = w^{1−p/2} dx. Define the functions u_{ε}: R^{2} → R, ε > 0, by

u_{ε}(x_{1}, x_{2}) = \{ \begin{matrix}\frac{x_{2}}{ε} & if |x_{2}| < ε, \\ sgn x_{2} & otherwise, \end{matrix}

and let B be the ball in R^{2} with center (\frac{1}{2}, 0) and radius \frac{1}{2} . Then u_{ε,B} = \overline{∫}_{B} u_{ε} dμ = 0 and

\overline{∫}_{B} |u_{ε} − u_{ε,B}| dμ = \overline{∫}_{B} |u_{ε}| dμ → 1, as ε → 0.

On the other hand, for 0 < ε < \frac{1}{2} ,

\begin{aligned}∫_{B} |∇u_{ε}| dμ & ≤ \frac{1}{ε} ∫_{−ε}^{ε} ∫_{0}^{1} w(x_{1}, x_{2})^{1−p/2} dx_{1} dx_{2} \\ & ≤ \frac{2}{ε} ∫_{−ε}^{ε} ∫_{0}^{1} w_{1}(x_{1}, x_{2})^{1−p/2} dx_{1} dx_{2}. \end{aligned}

It is shown in Semmes [17, p. 224], that for 0 < s < 1,

lim_{x_{2}→0} ∫_{0}^{1} w_{1}(x_{1}, x_{2})^{s/2} dx_{1} = 0.

Taking s = 2 − p now yields that

\overline{∫}_{B} |∇u_{ε}| dμ → 0, as ε → 0,

i.e. the weight w^{1−p/2} , 1 < p < 2, does not admit the 1-Poincaré inequality.

Poincaré inequalities for powers and products of admissible weights 187

References

[1] Chanillo, S., and R.L. Wheeden: Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions. - Amer. J. Math. 107, 1985, 1191– 1226.

[2] Chanillo, S., and R.L. Wheeden: Poincaré inequalities for a class of non-A_{p} weights. - Indiana Univ. Math. J. 41, 1992, 605–623.

[3] David, G., and S. Semmes: Strong A_{∞} -weights, Sobolev inequalities and quasi-conformal mappings. - In: Analysis and Partial Differential Equations, edited by C. Sadosky, Lecture Notes in Pure and Appl. Math. 122, pp. 101–111, Marcel Dekker, New York, 1990.

[4] Fabes, E.B., C.E. Kenig, and R.P. Serapioni: The local regularity of solutions of degenerate elliptic equations. - Comm. Partial Differential Equations 7, 1982, 77– 116.

[5] Franchi, B., C.E. Gutiérrez, and R.L. Wheeden: Weighted Sobolev–Poincaré inequalities for Grushin type operators. - Comm. Partial Differential Equations 19, 1994, 523–604.

[6] Franchi, B., and P. Hajłasz: How to get rid of one of the weights in a two weight Poincaré inequality? - Ann. Polon. Math. (to appear).

[7] Franchi, B., C.E. Pérez, and R.L. Wheeden: Self-improving properties of John– Nirenberg and Poincaré inequalities on spaces of homogeneous type. - J. Funct. Anal. 153, 1998, 108–146.

[8] García-Cuerva, J., and J.L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. - North Holland, Amsterdam, 1985.

[9] Gehring, F.W.: The L^{p} -integrability of the partial derivatives of a quasiconformal mapping. - Acta Math. 130, 1973, 265–277.

[10] Hajłasz, P., and P. Koskela: Sobolev meets Poincaré. - C. R. Acad. Sci. Paris Sér. I Math. 320, 1995, 1211–1215.

[11] Hajłasz, P., and P. Koskela: Sobolev met Poincaré. - Mem. Amer. Math. Soc. (to appear).

[12] Heinonen, J., T. Kilpeläinen, and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. - Oxford Univ. Press, Oxford, 1993.

[13] Heinonen, J., and P. Koskela: Weighted Sobolev and Poincaré inequalities and quasiregular mappings of polynomial type. - Math. Scand. 77, 1995, 251–271.

[14] Heinonen, J., and P. Koskela: Quasiconformal maps in metric spaces with controlled geometry. - Acta Math. 181, 1998, 1–61.

[15] Lu, G., and R.L. Wheeden: Poincaré inequalities, isoperimetric estimates and representation formulas on product spaces. - Indiana Univ. Math. J. 47, 1998, 123–151.

[16] Maz’ya, V.G.: A theorem on the multidimensional Schrödinger operator. - Izv. Akad. Nauk SSSR Ser. Mat. 28, 1964, 1145–1172 (in Russian).

[17] Semmes, S.: Bilipschitz mappings and strong A_{∞} -weights. - Ann. Acad. Sci. Fenn. Ser. A I Math. 18, 1993, 211–248.

[18] Semmes, S.: Some remarks about metric spaces, spherical mappings, functions and their derivatives. - Publ. Mat. 40, 1996, 411–430.

188 Jana Björn

[19] Stein, E.M.: Singular Integrals and Differentiability of Functions. - Princeton Univ. Press, Princeton, N.J., 1970.

[20] Strömberg, J.-O., and R.L. Wheeden: Fractional integrals on weighted H^{p} and L^{p} spaces. - Trans. Amer. Math. Soc. 287, 1985, 293–321.

[21] Torchinski, A.: Real-Variable Methods in Harmonic Analysis. - Academic Press, San Diego, Calif., 1986.

Received 30 June 1999