A stability result on Muckenhoupt's weights

Publicacions Matem`atiques, Vol 42 (1998), 153–163. A STABILITY RESULT ON MUCKENHOUPT’S WEIGHTS Juha Kinnunen Abstract We prove that Muckenhoupt’s A1-weights satisfy a reverse H¨older inequality with an explicit and asymptotically sharp estimate for the exponent. As a by-product we get a new characterization of A1-weights. 1. Introduction and statement of results Muckenhoupt’s weights are important tools in harmonic analysis, partial differential equations and quasiconformal mappings. The selfimproving property of Muckenhoupt’s weights is probably one of the most useful results in the field. The surprising fact that the weights are more regular than they seem to be a priori was observed already by Muckenhoupt [16]. The same phenomenon was studied by Gehring in [6] where he introduced the concept of reverse H¨older inequalites and proved that they improve themselves. Later Coifman and Fefferman [3] showed that Muckenhoupt’s weights are exactly those weights which satisfy a reverse H¨older inequality. Since then reverse H¨older inequalities have had a vast number of applications in modern analysis. An excellent source for all the mentioned results and other properties of Muckenhoupt’s weights is the monograph [7]. We are interested in a stability question related to Muckenhoupt’s A1-class and reverse H¨older inequalities. Suppose that w:Rn→[0,∞] is a locally integrable function satisfying Muckenhoupt’s A1-condition, (1.1) 1 |B|ZB w(x)dx ≤cwess inf x∈Bw(x), Keywords. Muckenhoupt weight, reverse H¨older inequality. 1991 Mathematics subject classifications: 42B26. 154 J. Kinnunen for all balls B⊂Rnwith the constant cw≥1 independent of the ball B. Here |B|is the volume of B.Ifwbelongs to Muckenhoupt’s class A1,we denote w∈A 1 ; the smallest constant cwfor which (1.1) holds is called the A1-constant of w. Condition (1.1) can be expressed in terms of the Hardy-Littlewood maximal function, defined by Mw(x) = sup B 1 |B|ZB w(y)dy, where the supremum is over all balls B⊂Rncontaining the point x.It is easy to see [7, p. 389] that (1.1) is equivalent to the requirement that Mw(x)≤cww(x) almost everywhere with exactly the same cwas in (1.1). It is clear that (1.1) imposes a serious restriction on the function. If the A1-constant is one, then 0≤1 |B|ZB¡w(y)−ess inf x∈Bw(x)¢dy ≤ess inf x∈Bw(x)−ess inf x∈Bw(x)=0 and hence wis constant. We are interested in the regularity of A1-weights as the constant tends to one. It is well-known that A1-weights satisfy the reverse H¨older inequality (1.2) µ1 |B|ZB w(x)pdx¶1/p ≤c1 |B|ZB w(x)dx, for some p>1 and cindependent of the ball B. Using (1.2) and (1.1) we see that wp∈A 1and wis locally integrable to power p. The question is: how large can pbe? If the A1-constant is one, then the weight is essentially bounded and it seems reasonable to expect that the degree of the local integrablity increases as the A1-constant tends to one. Questions related to the stability of reverse H¨older inequalities have obtained considerable attention in the last two decades, see [1], [2], [9], [11], [12], [13], [14], [15], [17], [18], [19], [20] and [21]. Our contribution is twofold. First, we present a new and a simple method which gives an explicit and asymptotically optimal bound for p. Second, our proof leads to a new characterization of A1-weights (Corollary 2.11) which may be of independent interest. Now we are ready to present our main result. A stability result on Muckenhoupt’s weights 155 1.3. Theorem. If w∈A 1with the constant cw, then there is a constant νdepending only on the dimension such that wsatisfies the reverse H¨older inequality (1.2) whenever (1.4) 1 ≤p<1+ ν c w−1. In the one-dimensional case we may take ν= 1 in (1.4), see [2] and [11], but our proof generally yields a small ν. Our method also allows us to replace balls in the A1-condition by cubes. Observe that the bound (1.4) for the local integrability of the weight is arbitrarily large provided cwis close enough to one. We remark that using factorization results of [10] and [4], our method gives similar estimates for Muckenhoupt’s Ap-weights as well. In the one-dimensional case this has been studied by Neugebauer [17]. 2. Characterization of A1-weights We begin by showing that every A1-weight can be approximated by smooth A1-weights. 2.1. Lemma. Suppose that w∈A 1with the constant cwand let ϕ∈C∞ 0(Rn),ϕ≥0with RRnϕdx =1. Then w∗ϕ∈A 1with the constant cw. Proof: A direct calculation gives (2.2) 1 B(x, r)ZB(x,r) w∗ϕ(y)dy =1 B(x, r)ZB(x,r)ZRn w(y−z)ϕ(z)dz dy =ZRn ϕ(z)1 B(x−z,r)ZB(x−z,r) w(y)dy dz ≤cwZRn ϕ(z) ess inf y∈B(x−z,r)w(y)dz =cwZRn ϕ(z) ess inf y∈B(x,r)w(y−z)dz ≤cwess inf y∈B(x,r)ZRn w(y−z)ϕ(z)dz =cwess inf y∈B(x,r)w∗ϕ(y). This completes the proof. We record a well-known covering theorem. 156 J. Kinnunen 2.3. Besicovitch’s covering Theorem. Suppose that Eis a bounded subset of Rnand that Bis a collection of balls such that each point of Eis a center of some ball in B. Then there exists an integer N≥2(depending only on the dimension) and subcollections B1,... ,B N⊂Bof at most countably many balls such that the balls Bi,j, j=1,2,..., in each family Bi,i=1,2,... ,N, are pairwise disjoint and E⊂ N [ i=1 ∞ [ j=1 Bi,j. For the proof of Besicovitch’s covering Theorem we refer to [5, Theorem 1.1]. Some estimates for the constant Nare obtained in [8]. Now we show that A1-weights satisfy a reverse Chebyshev inequality. This observation is a crucial ingredient in the proof of Theorem 1.3. For short we denote Eλ={x∈Rn:w(x)>λ},λ>0, throughout the paper. 2.4. Lemma. Let B⊂Rnbe a ball and suppose that w:Rn→ [0,∞]is an A1-weight with the constant cw. Then there is a constant η, depending only on the dimension, so that (2.5) ZEλ∩B w(x)dx ≤¡cw+η(cw−1)¢λ|Eλ∩B|, whenever ess infx∈Bw(x)≤λ<∞. Proof: Fix a ball B⊂Rn. Suppose first that wis a continuous A1-weight with the constant cwand that λ≥infx∈Bw(x). Then Eλ is open and for every x∈Eλwe take the ball B(x, rx) where rxis the distance from xto the boundary of Eλ. Let B={B(x, rx):x∈E λ∩B}. The radii of the balls in Bare bounded, because B\Eλ6=∅. By Besicovitch’s covering Theorem, there are families Bi={Bi,j :j=1,2,...}, i=1,2,... ,N, of countably many balls, chosen from B, such that Eλ∩B= N [ i=1 ∞ [ j=1 Bi,j ∩B and the balls in every Bi,i=1,2,... ,N, are pairwise disjoint. We denote the union of the pairwise disjoint balls by Ei λ= ∞ [ j=1 Bi,j,i=1,2,... ,N. A stability result on Muckenhoupt’s weights 157 The balls Bi,j touch the boundary of Eλand, since wis continuous, using the A1-condition we get (2.6) 1 |Bi,j|ZBi,j w(x)dx ≤cinf x∈Bi,j w(x)≤cλ, i=1,2,... ,N, j =1,2,... The balls Bi,j are not, in general, contained in B, but there is a constant γ>0, depending only on the dimension, so that |Bi,j \B|≤γ|B i,j ∩B|,i=1,2,... ,N, j =1,2,... To see this, let Bi,j be the ball B(x, rx)⊂Eλwith x∈Eλ∩B. Then by geometry, there is a ball B(y,rx/2) ⊂B(x, rx)∩B. This gives us the estimate |B(x, rx)\B|≤|B(x, rx)|=2 n |B(y,rx/2)|≤2 n |B(x, rx)∩B|. Hence we may take γ=2 n . By observing that w(x)>λfor every x∈Bi,j and recalling (2.6) we see that ZBi,j ∩B w(x)dx ≤cwλ|Bi,j ∩B|+cwλ|Bi,j \B|−ZB i,j \B w(x)dx ≤cwλ|Bi,j ∩B|+(c w−1)λ|Bi,j \B| ≤¡cw+γ(cw−1)¢λ|Bi,j ∩B|, i=1,2,... ,N, j=1,2,... Since the balls in each Bi,i=1,2,... ,N, are pairwise disjoint, we arrive at (2.7) ZEi λ∩B w(x)dx = ∞ X j=1 ZBi,j ∩B w(x)dx ≤¡cw+γ(cw−1)¢λ|Ei λ∩B|,i=1,2,... ,N. Let µbe a measure. Then we use the elementary inequality (2.8) µ(Eλ∩B)= N X i=1 µ(Ei λ∩B)− N X k=2 µ(Fk λ∩B), 158 J. Kinnunen where Fk λ=[ {l1,... ,lk}⊂{1,... ,N}¡El1 λ∩···∩El k λ¢,k=2,3,... ,N. A simple computation using (2.8), (2.7) and the fact that w(x)>λin Fk λ∩B,k=2,... ,N, gives (2.9) ZEλ∩B w(x)dx = N X i=1 ZEi λ∩B w(x)dx − N X k=2 ZFk λ∩B w(x)dx ≤¡cw+γ(cw−1)¢λ N X i=1 |Ei λ∩B|−λ N X k=2 |Fk λ∩B| =¡cw+γ(cw−1)¢λ|Eλ∩B|+λ(1+γ)(cw−1) N X k=2 |Fk λ∩B| ≤¡cw+γ(cw−1)¢λ|Eλ∩B|+(N−1)(1 + γ)(cw−1)λ|Eλ∩B| =¡cw+η(cw−1)¢λ|Eλ∩B|, where η=Nγ +N−1 and λ≥infx∈Bw(x). The general case follows from a standard approximation argument using Lemma 2.1. Suppose that w∈A 1with the constant cw. Let ϕ∈C∞ 0(Rn), ϕ≥0 with RRnϕdx = 1. We define wε=w∗ϕε, where ϕε(x)=ε −n ϕ(x/ε) and ε>0. Lemma 2.1 shows that wεis a continuous A1-weight with the constant cwfor every ε>0. Using (2.9) we see that Z{wε>λ}∩B wε(x)dx ≤¡cw+η(cw−1)¢λ|{wε>λ}∩B|, inf x∈Bwε(x)≤λ<∞. Letting ε→0 we obtain (2.5). This completes the proof. 2.10. Remark. (1) Observe that the constant on the right side of (2.5) tends to one as cwtends to one. On the other hand, it blows up as cwincreases. (2) We also remark that inequalities of type (2.5) appear already in the proof of Theorem 4 in [3]. However, their approach does not seem to give the correct behaviour as cwtends to one. We observe that (2.5) gives a characterization of A1-weights. A stability result on Muckenhoupt’s weights 159 2.11. Corollary. Suppose that w:Rn→[0,∞]is a measurable function. Then w∈A 1if and only if there is a constant c, independent of the ball B, so that (2.12) ZEλ∩B w(x)dx ≤cλ|E λ∩B|,ess inf x∈Bw(x)≤λ<∞, for every ball B⊂Rn. Proof: Lemma 2.4 shows that every A1-weight satisfies (2.12). To see the reverse implication suppose that (2.12) holds and let Bbe a ball in Rn. Then ZB w(x)dx =ZB\Eλ w(x)dx +ZEλ∩B w(x)dx ≤λ|B\Eλ|+cλ|B∩E λ| ≤cλ|B|,ess inf x∈Bw(x)≤λ<∞. By inserting λ= ess infx∈Bw(x)weget 1 |B|ZB w(x)dx ≤cess inf x∈Bw(x), where the constant is independent of the ball and hence w∈A 1 . 2.13. Remark. In the one-dimensional case we may take the constant in (2.12) equal to the A1-constant of w, see [11]. Lemma 2.4 shows that wsatisfies the assumptions of the following sharp version Muckenhoupt’s Lemma 4 in [16]. See also Lemma 2 in [2]. The proof of the following lemma can be found in [11], but we present it here for the sake of completeness. 2.14. Lemma. Suppose that w:Rn→[0,∞]is a measurable function and let B⊂Rnbe a ball. If there are α≥0and c>1such that (2.15) ZEλ∩B w(x)dx ≤cλ|E λ∩B|,α≤λ<∞, then for every p,1<p<c/(c−1), we have (2.16) ZEα∩B w(x)pdx ≤c c−p(c−1)αp|Eα∩B|. 160 J. Kinnunen Proof: Let β>αand denote wβ= min(w, β). Then Z{wβ>λ}∩B w(x)dx ≤cλ|{wβ>λ}∩B|,α≤λ<∞. We multiply both sides by λp−2and integrate from αto ∞. This implies Z∞ α λp−2Z{wβ>λ}∩B w(x)dx dλ ≤cZ∞ α λp−1|{wβ>λ}∩B|dλ. Then we use the equality (2.17) ZEα∩B w(x)pdµ =pZ∞ α λp−1µ(Eλ∩B)dλ +αpµ(Eα∩B), where 0 <p<∞, with µreplaced by wdµ and preplaced by p−1, to get ZEα∩B wβ(x)pdx ≤ZEα∩B wβ(x)p−1w(x)dx =(p−1) Z∞ α λp−2Z{wβ>λ}∩B w(x)dx dλ +αp−1ZEα∩B w(x)dx ≤c(p−1) Z∞ α λp−1|{wβ>λ}∩B|dλ +cα p|E α∩B|. Next we estimate the first integral on the right side using (2.17) and find Z∞ α λp−1|{wβ>λ}| dλ =1 pµZEα∩B wβ(x)pdx −αp|Eα∩B|¶. Hence we obtain ZEα∩B wβ(x)pdx ≤cp−1 pZEα∩B wβ(x)pdx +c pαp|Eα∩B|. Choosing p>1 such that c(p−1)/p<1 and using the fact that all terms in the previous inequality are finite, we conclude ZEα∩B wβ(x)pdx ≤c c−p(c−1)αp|Eα∩B|. Finally, as β→∞, the monotone convergence theorem gives (2.16). This proves the lemma. 2.18. Remark. Both the bound for pand the constant in (2.16) are the best possible as is easily seen by taking Bto be the unit ball and w:Rn→[0,∞], w(x)=|x| n(1/c−1). A stability result on Muckenhoupt’s weights 161 3. Proof of Theorem 1.3 Let Bbe a ball in Rnand suppose that w∈A 1with the constant cw. Using (2.5) we see that ZEλ∩B w(x)dx ≤¡cw+η(cw−1)¢λ|Eλ∩B|,ess inf x∈Bw(x)≤λ<∞, where ηis the constant given by Lemma 2.4. This shows that wfulfills the assumptions of Lemma 2.14 and from (2.16) we conclude that ZB w(x)pdx =ZB\Eα w(x)pdx +ZB∩Eα w(x)pdx ≤αp|B\Eα|+cα p|B∩E α| ≤cα p|B|, whenever ess infx∈Bw(x)≤α<∞and 1≤p<1+ 1 (η+ 1)(cw−1). In particular, we get µ1 |B|ZB w(x)pdx¶1/p ≤c1 |B|ZB w(x)dx. The constant cdoes not depend on Band hence we may repeat the same reasoning in every ball Band we see that wsatisfies the reverse H¨older inequality for every p>1 such that (1.4) holds if we take ν=(η+1) −1. This completes the proof of Theorem 1.3. Acknowledgements. I would like to thank Michael Korey for making valuable comments on early versions of this paper. References 1. B. Bojarski, Remarks on the stability of reverse H¨older inequalities and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 89–94. 2. B. Bojarski, C. Sbordone and I. Wik, The Muckenhoupt class A1(R), Studia Math. 101(2) (1992), 155–163.