Geometric-arithmetic averaging of dyadic weights

Geometric-arithmetic averaging of dyadic weights Jill Pipher Department of Mathematics Brown University Providence, RI 2912 jpipher@math.brown.edu Lesley A. Ward School of Mathematics and Statistics University of South Australia Mawson Lakes, SA 595 Australia lesley.ward@unisa.edu.au Xiao Xiao Department of Mathematics Brown University Providence, RI 2912 xxiao@math.brown.edu December 7, 29 Abstract The theory of (Muckenhoupt) weights arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing A p weights from a measurably varying family of dyadic A p weights. This averaging process is suggested by the relationship between the A p weight class and the space of functions of bounded mean oscillation. The same averaging process also constructs weights satisfying reverse Hölder (RH p ) conditions from families of dyadic RH p weights, and extends to the polydisc as well. 1 Introduction Several classes of functions are defined in terms of a property that the function must satisfy on each interval, with a uniform constant. Well known examples from harmonic analysis and complex analysis include Muckenhoupt s A p weights, the reverse-hölder weight classes RH p, 28 Mathematics Subject Classification. Primary: 42B35; Secondary: 42B25. Research supported by the NSF under grant number DMS91139. 1

the class of doubling weights, and the space BMO of functions of bounded mean oscillation. Such classes have strictly larger dyadic analogues, where the defining property is required to hold only on the dyadic intervals. Certain types of averaging provide a bridge between these dyadic counterparts and the original function classes. Specifically, these averages convert each suitable family of functions in the dyadic class to a single function in the smaller, nondyadic class. We can think of averaging as an improving operation, in this sense. An easily stated example is the following. If all translates of a function f defined on the unit circle T =, 1] are in dyadic BMO, or equivalently if f is in dyadic BMO on every translated grid of dyadic intervals on the circle, then the function f itself is in true BMO. This result is a special case of a theorem in GJ], applied to the identity f(x) = 1 τ t f(x + t) dt, where for t R the translation operator τ t is defined by τ t f( ) := f( t), and x + t is to be interpreted as x + t mod 1. Now, what if a function f on T can be written as the translation-average f(x) := 1 f t (x + t) dt of dyadic BMO functions {f t } t,1] that are not identical translates of each other? If they satisfy the hypotheses of GJ], then still f is in true BMO. However, the analogous statements can fail for A p weights, for RH p weights, and for doubling weights W]. In this paper we show that a different type of averaging works for both A p and RH p (Theorems 1 and 2). This is the geometric-arithmetic average defined by { 1 } Ω(x) := exp log ω t (x + t) dt, where {ω t } t,1] is a suitable family of weights in A d p or RH d p. We also observe that translation-averaging does work for both A p and RH p under the additional assumption that the functions ω t are doubling weights, not just dyadic doubling weights (Theorem 3). All these results generalize to the polydisc (Theorems 4 and 5). The paper is organized as follows. In Section 2, we state our geometric-arithmetic averaging results on the circle. In Section 3, we collect the definitions and background results used in the paper. Also, Lemma 1 in that section gives a unified characterization of weights in A p for 1 p, RH p for 1 < p, and their dyadic counterparts, in terms of conditions on the oscillation of their logarithms. We take some care throughout in tracing the dependence of the various constants. In Section 4, we prove geometric-arithmetic averaging for A d p and RH d p weights (Theorems 1 and 2). In Section 5, we prove translation-averaging for A d p and RH d p weights that are doubling (Theorem 3). In Section 6, we generalize our results to the polydisc (Theorems 4 and 5). 2

2 Geometric-arithmetic averaging on the circle In W], examples are constructed to show that, for {ω t } t,1] a measurably varying family of dyadic A d p weights (for arbitrary p with 1 p ) or dyadic RHp d weights (for arbitrary p with 1 < p < ) on the circle T =, 1], with uniformly bounded dyadic A d p or RHp d constants, the translation-average ω(x) := 1 ωt (x + t) dt is not necessarily a doubling weight. Therefore ω need not be in true A p, nor in true RH p. The main result of the current paper is that, by contrast, the geometric-arithmetic average Ω(x) := exp{ 1 log ωt (x + t) dt} always turns a measurably varying family of suitably normalized dyadic A d p weights into an A p weight (for arbitrary p with 1 p ), and a measurably varying family of suitably normalized dyadic RHp d weights into an RH p weight (for arbitrary p with 1 < p ). Theorem 1. Fix p with 1 p. Let {ω t } t,1] be a family of dyadic A p weights on the circle T, ω t A d p(t), such that (i) the mapping t ω t is measurable, (ii) an appropriate average of the logarithms of the weights ω t is finite: and 1 1 log ω t (x) dx dt <, (iii) the A d p constants A d p(ω t ) are uniformly bounded, independent of t, 1]. Then the geometric-arithmetic average { 1 Ω(x) = exp } log ω t (x + t) dt of the dyadic weights ω t belongs to A p on T. Moreover, the A p constant of Ω depends only on p and on the bound on the A d p constants of the ω t. Remark. There is a simple heuristic motivation for this result. The weights ω t are in A d p, so their logarithms log ω t are in BMO d. Therefore, as shown in GJ] (and later PW] for the one- and two-parameter settings and T] for the general multiparameter setting), the translation-average log Ω of the functions log ω t is in BMO, and so by the John Nirenberg Theorem JoNi] sufficiently small powers Ω δ of Ω are in A p. It remains to show that Ω itself is in A p. Remark. Hypothesis (ii) of the theorem is merely a normalization condition: the fact that each ω t (x) belongs to A d p already implies that log ω t (x) belongs to L 1 (dx). The analogous result to Theorem 1 holds for reverse-hölder weights. Theorem 2. Fix p with 1 < p. Let {ω t } t,1] be a family of dyadic RH p weights on the circle T, ω t RH d p (T), such that hypotheses (i) and (ii) of Theorem 1 hold, and 3

(iii ) the RH d p constants RH d p (ω t ) are uniformly bounded, independent of t, 1]. Then the geometric-arithmetic average Ω(x) = exp{ 1 log ωt (x + t) dt} of the dyadic weights ω t lies in RH p on T. The RH p constant of Ω depends only on p and on the bound on the RH d p constants of the ω t. These results also hold on T k, with constants that depend on the dimension k. In addition, they hold in the setting of the polydisc; see Section 6. We remark that it is not necessary for the integral in t to be taken over the whole interval, 1]. The proofs below go through without change when the integral is taken over an arbitrary subset E, 1] of positive measure. 3 Definitions and Tools In this section we collect useful material about doubling weights, the weight classes A p and RH p, and their relationship to BMO. For fuller accounts of the theory of A p and RH p weights, see for example GCRF], Gar], Gra], and CN]. Let T denote the unit circle, obtained by identifying the endpoints of the interval, 1]. In the definitions of our averages Ω and ω, x + t is to be interpreted as x + t mod 1. Denote the collection of dyadic subintervals I of the circle T by D = D, 1]: D := {, 1]} { I = j 2, j + 1 k 2 k ) k N, j {, 1,..., 2 k 1}}. Throughout the paper, denotes a general subinterval of T, while I, J, K and L denote dyadic subintervals of T. The functions we consider are real-valued. We use the symbol E to denote the Lebesgue measure of a set E, the symbol to E denote 1, and the symbol f E E E for the average value f of a function f on a set E. The E notation E F includes the possibility E = F. Definition 1. Let ω(x) be a nonnegative locally integrable function on the circle T. We say ω is a doubling weight with doubling constant C if for all intervals T ω(x) dx C ω(x) dx, where is the double of ; that is, is the interval with the same midpoint as and twice the length of. We say ω is a dyadic doubling weight with dyadic doubling constant C if the analogous inequality holds for all dyadic intervals I T, where Ĩ is the dyadic double of I: that is, Ĩ is unique dyadic interval of length Ĩ = 2 I that contains I. The A p weights were identified by Muckenhoupt M] as the weights ω for which the Hardy Littlewood maximal function is bounded from L p (dµ) to itself, where dµ = ω(x) dx. Here we 4

give the definitions of the classes A p and RH p on the circle T; the equivalent definitions hold on R and on (one-parameter) R n. We delay the corresponding definitions for the polydisc setting until Section 6. Definition 2. Let ω(x) be a nonnegative locally integrable function on the circle T. For real p with 1 < p <, we say ω is an A p weight, written ω A p, if A p (ω) := sup ( ω) ( ( ) ) 1/(p 1) p 1 1 <. ω For p = 1, we say ω is an A 1 weight, written ω A 1, if ( ) ( ) 1 A 1 (ω) := sup ω ess inf x ω(x) <. For p =, we say ω is an A-infinity weight, written ω A, if ( ) ( ( )) 1 A (ω) := sup ω exp log <. ω Here the suprema are taken over all intervals T. The quantity A p (ω) is called the A p constant of ω. The dyadic A p classes A d p for 1 p are defined analogously, with the suprema A d p(ω) being taken over only the dyadic intervals I T. Definition 3. Let ω(x) be a nonnegative locally integrable function on the circle T. For real p with 1 < p <, we say ω is a reverse-hölder-p weight, written ω RH p or ω B p, if RH p (ω) := sup ( ) 1/p 1 ω ( ω) p <. For p =, we say ω is a reverse-hölder-infinity weight, written ω RH or ω B, if ( ) ( RH (ω) := sup ess sup ω x ω) 1 <. Here the suprema are taken over all intervals T. The quantity RH p (ω) is called the RH p constant of ω. For 1 < p, we say ω is a dyadic reverse-hölder-p weight, written ω RHp d or ω Bp, d ( if the analogous condition sup I D ωp) 1/p ( I I ω) 1 < or supi D (ess sup x I ω) ( ω) 1 I < holds with the supremum being taken over only the dyadic intervals I T, and if in addition ω is a dyadic doubling weight. We define the RHp d constant RHp d (ω) of ω to be the larger of this dyadic supremum and the dyadic doubling constant. 5

The A p inequality (or the RH p inequality) implies that the weight ω is doubling, and the dyadic A p inequality implies that ω is dyadic doubling. However, the dyadic RH p inequality does not imply that ω is dyadic doubling, which is why the dyadic doubling assumption is needed in the definition of RH d p. The A p classes are nested and increasing with p, while the RH p classes are nested and decreasing with p. Moreover, A q (ω) A p (ω), for 1 p < q, and Also RH p (ω) RH q (ω), for 1 < p < q. A = A p = RH q, A 1 A p, RH RH p. p 1 q>1 p>1 p>1 The dyadic versions of the assertions in this paragraph also hold. The example w(x) = (log(1/ x )) 1 (for x near zero) cited in JoNe] shows that A 1 is a proper subset of p>1 A p. The example ω(x) = max{log(1/ x ), 1} given in CN] shows that RH is a proper subset of p>1 RH p. However, as noted in CN], if a weight ω is in A p for each p > 1 and if the constants A p (ω) are uniformly bounded, then ω A 1 ; and the corresponding statement holds for RH p and RH. As noted above, for a nonnegative locally integrable function ω, ω is in A ω is in A p for some p 1, ) ω is in RH q for some q (1, ). In the first equivalence the A constant depends only on the A p constant and on p, which in turn depend only on the A constant. Similarly, the A constant depends only on the RH q constant and on q, which depend only on the A constant. See for example Gra, Theorem 9.3.3] where the constants in these and other characterizations of A are carefully analyzed. The analogous statements hold for the dyadic classes A d, A d p, and RH d p. The classes of A p and RH p weights can be characterized by conditions on the oscillation of the logarithm of the weight, as follows. Lemma 1. Let ω be a nonnegative locally integrable function on T. Let ϕ := log ω. Then the following five statements hold. (a) ω is in A if and only if sup exp{ϕ(x) ϕ } dx <. (1) (b) For 1 < p <, ω is in A p if and only if inequality (1) holds and also { } (ϕ(x) ϕ ) sup exp dx <. (2) p 1 6

(c) ω is in A 1 if and only if inequality (1) holds and also (d) ω is in RH if and only if sup sup ϕ ess inf x ϕ(x)] <. (3) ] ess sup ϕ(x) ϕ <. (4) x (e) For 1 < p <, ω is in RH p if and only if sup exp{p(ϕ(x) ϕ )} dx <. (5) In each part, the value of A p (ω) or RH p (ω) depends on the value(s) of the supremum (suprema) in the characterization given and, when 1 < p <, also on p. Conversely, the value(s) of the supremum (suprema) depend on the value of A p (ω) or RH p (ω) and, when 1 < p <, also on p. Taking the suprema over only dyadic intervals I T, the dyadic analogues of parts (a) (e) hold for the dyadic classes A d, A d p, A d 1, RH d, and RH d p, except that in parts (d) and (e) one needs in addition to inequality (4) or (5) the extra hypothesis that ω is dyadic doubling. The dependence of the constants in the dyadic case is the same as in the continuous case. Parts (b) and (c) appear in Gar], Gra], and GCRF], for example, and part (d) is in CN, Cor 4.6]. Inequality (3) says that ω is in A 1 when its logarithm ϕ belongs to the space BLO of functions of bounded lower oscillation, while inequality (4) says that ω is in RH when ϕ belongs to BLO. Proof. Let C 1, C 2, C 3, C 4, and C 5 be the suprema in inequalities (1), (2), (3), (4), and (5) respectively. (a) It is immediate that A (ω) = C 1, since for each interval the A quantity is ( ) { } 1 ω(x) dx exp log w(x) dx = exp{ϕ(x) ϕ } dx. (b) We show that C 1 A p (ω), C 2 A p (ω) 1/(p 1), and A p (ω) C 1 C p 1 2. Let ( ) ] 1/(p 1) 1 ψ := log = ϕ ω p 1. 7

Let be an interval in T. Then by Jensen s inequality, ( ) ( exp{ϕ(x) ϕ } dx = w exp (p 1) ( ) w exp ψ(x) dx ( = w) A p (ω). ) ψ(x) dx ] p 1 ( ) ] 1/(p 1) p 1 1 ω Thus inequality (1) holds with C 1 = A p (ω). Similarly, by Jensen s inequality, { } ( ( ) 1/(p 1) ) ( ) ] 1/(p 1) (ϕ(x) ϕ ) 1 exp dx = exp ϕ p 1 ω ( ( ) 1/(p 1) ) p 1 ( ) ] 1/(p 1) 1 ω ω A p (ω) 1/(p 1). Thus inequality (3) holds with C 2 = A p (ω) 1/(p 1). For the converse, ( ) ( ( ) 1/(p 1) ) p 1 ( 1 ω = ω ) ( exp ϕ(x) dx ( = exp{ϕ(x) ϕ } dx C 1 C p 1 2, p 1 exp ψ(x) dx) e ϕ e (p 1)ψ ) ( ) p 1 exp{ψ(x) ψ } dx and thus A p (ω) C 1 C p 1 2. (c) We show that C 1 A 1 (ω), C 3 log A 1 (ω), and A 1 (ω) C 1 exp C 3. If ω is in A 1, then for each interval in T we have e ϕ(x) dx = w(x) dx A 1 (ω) ess inf w(x) A 1(ω)e ϕ. x It follows that e ϕ(x) ϕ dx A 1 (ω). Thus ϕ satisfies inequality (1) with constant C 1 A 1 (ω). By Jensen s inequality and the A 1 property, e ϕ w(x) dx A 1 (ω) exp { ess inf ϕ(x)}. (6) x 8

Therefore ϕ log A 1 (ω) + ess inf x ϕ(x). Thus ϕ satisfies inequality (3) with constant C 3 = log A 1 (ω). Now suppose that ϕ satisfies inequalities (1) and (3). Then for each interval, w(x) dx = e ϕ(x) dx C 1 e ϕ C 1 exp { C 3 + ess inf x ϕ(x)} = C 1 e C 3 ess inf x w(x). Thus ω satisfies the A 1 property with constant A 1 (ω) C 1 e C 3. (d) We show that C 4 log(rh (ω)a (ω)) and RH (ω) e C 4, and that the bound on C 4 depends only on RH (ω). Suppose ω is in RH. Then ω is in A, so inequality (1) holds with C 1 = A (ω). Further, ω is in every RH p for p (1, ), and A (ω) depends only on RH p (ω), while RH p (ω) RH (ω). Thus A (ω) depends only on RH (ω). Now for each interval in T, inequality (1) implies that ess sup ω(x) RH (ω) e ϕ(x) dx RH (ω)a (ω)e ϕ. x Taking logarithms, we see that ess sup ϕ(x) ϕ + log(rh (ω)a (ω)), x and so inequality (4) holds with C 4 = log(rh (ω)a (ω)). Conversely, if inequality (4) holds, then by Jensen s inequality ess sup ω(x) exp{c 4 + ϕ } e C 4 x exp ϕ(x) dx = e C 4 ω. Thus RH (ω) e C 4. (e) We show that C 5 RH p (ω)a (ω) and RH p (ω) C 1/p 5, and that C 5 depends only on p and on RH p (ω). In terms of ϕ, the RH p expression for a given interval is ( ) 1/p 1 ( ) 1/p ( 1 ω ( ω) p = exp{p(ϕ(x) ϕ )} dx exp{ϕ(x) ϕ } dx). Also, if ω is in RH p, then ω is in A and A (ω) depends only on p and on RH p (ω). It follows that ( 1/p exp{p(ϕ(x) ϕ )} dx) RH p (ω) exp{ϕ(x) ϕ } dx RH p (ω)a (ω). 9

Thus inequality (5) holds with C 5 RH p (ω)a (ω), and this bound depends only on p and on RH p (ω). By Jensen s inequality, exp{ϕ(x) ϕ } dx 1. Thus if inequality (5) holds, then ( ) 1/p ( exp{p(ϕ(x) ϕ )} dx ) 1 exp{ϕ(x) ϕ } dx C 1/p 5. Thus ω is in RH p and RH p (ω) C 1/p 5. The same arguments go through for the dyadic classes A d p and RH d p. Muckenhoupt s A p weights are closely related to functions of bounded mean oscillation. Definition 4. A real-valued function f L 1 (T) is in the space BMO(T) of functions of bounded mean oscillation on the circle if its BMO norm is finite: f := sup f(x) f dx <. T Dyadic BMO of the circle, written BMO d (T), is the space of functions that satisfy the corresponding estimate where the supremum is taken over all dyadic subintervals I D of, 1]. The dyadic BMO norm of f is denoted f d. Elements of BMO (or BMO d ) that differ only by an additive constant are equivalent; thus BMO and BMO d are subspaces of L 1 loc /R. For 1 p, if ω is in A p then ϕ := log ω is in BMO, with BMO norm depending only on the A p constant A p (ω). See for example GCRF, p.49]. The same is true for RH p weights. Specifically, we have the following result; we omit the proof. Lemma 2. Suppose 1 p. If ω is in A p then ϕ := log ω is in BMO. For 1 < p <, ϕ A p (ω) + (p 1)A p (ω) 1/(p 1). For p = 1, ϕ 2A 1 (ω). For p =, ϕ depends only on A (ω). For 1 < p, if ω is in RH p then ϕ := log ω is in BMO, with ϕ depending only on RH p (ω) and on p. The analogous statements hold in the dyadic setting. We use a characterization of the dyadic BMO functions on the circle in terms of the size of Haar coefficients. The Haar function h I associated with the dyadic interval I is given by h I (x) = I 1/2 if x is in the left half of I, h I (x) = I 1/2 if x is in the right half of I, and h I = otherwise. The Haar coefficient over I of f is f I = (f, h I ) := f(x)h I I(x) dx. The Haar series for f is f(x) := (f, h I ) h I (x), I D and the L 2 -norm of f is given in terms of the Haar coefficients by ] 1/2 f 2 = (f, h J ) 2. J D 1

It follows from the John Nirenberg Theorem JoNi] that for p 1, for f in L 1 (T) the expression ( ) 1/p f d,p := sup f(x) (f) I p dx I D I is comparable to the dyadic BMO norm f d. In particular, a function f L 1 (T) of mean value zero is in BMO d (T) if and only if there is a constant C such that for all I D, (f, h J ) 2 C I. (7) J I,J D Moreover, the smallest such constant C is equal to f 2 d,2. Since the sum in inequality (7) ranges over dyadic intervals only, there is no need to restrict the interval I itself to be dyadic. 4 Proofs of Theorems 1 and 2 We begin this section with three lemmas, which we then use to prove the geometric-arithmetic averaging result for both A p and RH p. Lemma 3 below gives an estimate on Haar expansions of BMO d functions. Lemmas 4 and 5, which rely on the estimates (8) and (9) from Lemma 3, will allow us to pass from the dyadic versions to the non-dyadic versions of the inequalities that characterize A p and RH p. Throughout this section we use the following notation. Let D n := {I D I = 2 n } be the collection of dyadic intervals of length 2 n, for n =, 1, 2,... Expanding each ϕ t in Haar series, we have ϕ(x) = 1 (ϕ t, h J )h J (x + t) dt = 1 J D n= (ϕ t, h J ) h J (x + t) dt = J D n ϕ n (x), n= so that ϕ n is the translation-average over t of the slices at scale 2 n of the Haar expansions for the functions ϕ t. Fix an interval T; this need not necessarily be dyadic. Split the sum for ϕ(x), at the scale of, into two parts ϕ A and ϕ B in which the dyadic intervals J are respectively small and large compared with : ϕ = ϕ A + ϕ B, ϕ A (x) := ϕ n (x), ϕ B (x) := ϕ n (x). n:2 n < n:2 n The following result is proved in the course of the proof of Theorem 2 of PW]. Lemma 3. Suppose that {ϕ t } t,1] is a family of dyadic BMO functions on T, ϕ t BMO d, such that 11

(i) the mapping t ϕ t is measurable, (ii) the BMO d constants ϕ t d are uniformly bounded, independent of t, 1], and (iii) for each t, 1], the function ϕ t has mean value zero on T. Then there are constants C A and C B depending on the bound on the BMO d constants ϕ t d, and independent of, such that for each interval T and for each point x, 1 ϕ A (x) 2 dx C A, (8) 1 ϕ B (x) ϕ B (x ) dx C B. (9) Lemma 4. Let β be a real number. Suppose {ω t } t,1] is a family of nonnegative locally integrable functions on T such that for all ϕ t := log ω t, hypotheses (i) (iii) of Lemma 3 hold. Let ϕ(x) = log Ω(x) := 1 log ωt (x + t) dt. Suppose there is a constant C d (β) such that for all t, 1] and for all dyadic intervals I T, exp β(ϕ t (x) ϕ t I) ] dx C d (β). (1) I Then there is a constant C(β), depending only on C d (β), such that exp β(ϕ(x) ϕ ) ] dx C(β) (11) for all intervals T. In fact, for many choices of β, hypothesis (ii) of Lemma 3 is implied by inequality (1), together with Lemmas 1 and 2. This is made clear in the proof of Theorems 1 and 2. Proof. We first establish an inequality controlling the exponentials of the Haar expansions of the ϕ t. By inequality (1), for each dyadic interval I T we have exp β ] (ϕ t, h J )h J (x) dx = exp β (ϕ t, h J )h J (x) β ] (ϕ t, h J )h J (x) dx I J I,J D I J D J I,J D = exp β(ϕ t (x) ϕ t I) ] dx I J D I C d (β). (12) We have used the elementary fact that the average f I of a function f L 1 (T) over an interval I containing the point x can be written as f I = (f, h J )h J (s) ds = (f, h J )h J (x). 12 J I,J D

Fix an interval T, not necessarily dyadic. For each x we have ϕ B(x) ϕ(s) ds ϕ B (x) ϕ(s) ds ϕ A (s) ds + ϕ B (x) ϕ B (s) ds C A + C B, by Cauchy Schwarz and the estimates (8) and (9) from Lemma 3. Therefore exp β(ϕ(x) ϕ ) ] dx = exp βϕ A (x) ] exp β(ϕ B (x) ϕ ) ] dx exp β( C A + C B ) ] exp βϕ A (x) ] dx. Thus it suffices to bound the quantity expβϕ A (x)] dx = exp = = 1 1 β 2 n < 1 exp β exp +t exp β 1 ] (ϕ t, h J )h J (x + t) dt dx J D n ] (ϕ t, h J )h J (x + t) dt dx J <,J D J <,J D β J <,J D ] (ϕ t, h J )h J (x + t) dx dt ] (ϕ t, h J )h J (x) dx dt. We have used Jensen s inequality and Tonelli s Theorem. In order to apply inequality (12), we want to replace the interval in the preceding expression by appropriate dyadic intervals. Fix t. There are two adjacent dyadic intervals 13

K and L such that + t K L and K = L < 2. Then 1 1 ] expβϕ A (x)] dx exp β (ϕ t, h J )h J (x) dx dt = 2 2 1 1 1 2 K +t { 1 K K L + 1 L K exp β L exp β exp β J <,J D J < K,J D J < K,J D J < L,J D C d (β) + C d (β) dt = 4 C d (β), by inequality (12). Thus exp β(ϕ(x) ϕ ) ] dx 4 C d (β) exp ] (ϕ t, h J )h J (x) dx dt ] (ϕ t, h J )h J (x) dx ] } (ϕ t, h J )h J (x) dx dt ( )] β CA + C B. Taking the supremum over all intervals T, we see that inequality (11) holds for ϕ = log Ω, with constant C(β) = 4 C d (β) exp β ( C A + C B )]. Lemma 5. Suppose {ω t } t,1] is a family of nonnegative locally integrable functions on T such that for all ϕ t := log ω t, hypotheses (i) and (iii) of Lemma 3 hold. As before, let ϕ(x) = log Ω(x) := 1 log ωt (x + t) dt. (a) Suppose there is a constant C3 d such that for all t, 1] and for all dyadic intervals I T, ϕ t I ess inf ϕ t (x) ] C3. d x I Then there is a constant C 3 depending only on C d 3 such that ϕ ess inf x ϕ(x)] C 3 for all intervals T. (b) Similarly, if there is a constant C4 d such that for all t, 1] and for all dyadic intervals I T, ess sup ϕ t (x) ϕi] t C d 4, x I then there is a constant C 4 depending only on C4 d such that ] ess sup ϕ(x) ϕ C4 x for all intervals T. 14

Proof. Observe that hypothesis (ii) of Lemma 3 follows from the assumption in part (a) or the assumption in part (b), together with Lemmas 1 and 2. In particular, the BMO d constants ϕ t d depend only on C3 d or C4. d (a) For each dyadic interval I T and for a.e. x I, C3 d ϕ t I ϕ t (x) = (ϕ t, h J )h J (x). (13) J I,J D Fix an interval T, not necessarily dyadic. For x consider the quantity ϕ ϕ(x) = ϕa (s) + ϕ B (s) ] ds ϕ A (x) + ϕ B (x) ] ] ] = ϕ A (s) ds + ϕ B (s) ds ϕ B (x) + ϕ A (x). } {{ } } {{ } } {{ } (I) (II) (III) We bound the terms (I), (II), and (III) separately. By Lemma 3, and 1/2 (I) ϕ A (s) ds] 2 C A (II) ϕ B (s) ϕ B (x) ds C B for all x. Also, for x and t, 1] there is a unique dyadic interval I x,t such that x + t I x,t and /2 I x,t <. Then ϕ A (x) = = n:2 n < 1 1 = C d 3 for a.e. x, by inequality (13). So 1 J <,J D J I x,t,j D J D n (ϕ t, h J )h J (x + t) dt (ϕ t, h J )h J (x + t) dt (ϕ t, h J )h J (x + t) dt ϕ ess inf x ϕ(x) C A + C B + C d 3. Thus inequality (3) holds with constant C 3 = C A + C B + C d 3. (b) In the case of RH, the same argument shows that inequality (4) holds with constant C 4 = C d 4 + C B + C A. 15

Proof of Theorems 1 and 2. We prove Theorems 1 and 2 together. Let K denote any one of the classes A p with 1 p, or RH p with 1 < p. For ω in K let K(ω) denote the corresponding A p constant or RH p constant. Let K d denote the corresponding dyadic class, and for ω t in K d let K d (ω t ) denote the corresponding constant. It follows from the definitions of A p and RH p that if ω is in K then for each constant λ > the weight λω is also in K, and K(λω) = K(ω). For each t, 1] let ω t be a weight in K d, and let ϕ t := log ω t. Let ϕ := log Ω. By hypothesis (iii) of Theorem 1 or (iii ) of Theorem 2, the constants K d (ω t ) are bounded independently of t, 1]. Without loss of generality, we may assume that for a.e. t, 1] the ϕ t have mean value zero. To see this, note that the mean value ϕ t T := T ϕt (x) dx is finite for a.e. t by hypothesis (ii). Let ϕ t := ϕ t ϕ t T. Then for x T, { 1 Ω(x) = exp { 1 = exp { 1 = exp } ϕ t (x + t) dt } { 1 ϕ t T dt exp 1 } ϕ t (x + t) dt } { 1 } log ω t (x) dx dt exp ϕ t (x + t) dt. Denote the second term on the right-hand side by Ω(x). The first term on the right-hand side is finite and nonzero by hypothesis (ii) of Theorems 1 and 2, and is positive. Thus Ω is in K if and only if Ω is in K. Moreover, K(Ω) = K( Ω). We show that hypotheses (i) (iii) of Lemma 3 hold here. The mapping t ϕ t = (log ω t ) ϕ t T is measurable, since t ωt is measurable by hypothesis. By the dyadic version of Lemma 2, the BMO d norms ϕ t d = ϕ t d are uniformly bounded, by a constant depending only on p and on the bound on the K d constants of the dyadic weights ω t. Each ϕ t has mean value zero, by construction. For convenience we now drop the tildes, writing ϕ t for ϕ t and Ω for Ω from here on. As an aside, we note that log Ω is in BMO (see GJ]), and so by the John Nirenberg Theorem the function Ω δ is in A p for δ > sufficiently small. We now prove that Ω itself is in A p. Case 1: K = A. By the dyadic version of Lemma 1(a), there is a constant C1 d depending only on the bound on the A d constants of the ω t such that each ϕ t satisfies inequality (1) with β = 1: exp{ϕ t (x) ϕ t I} dx C1. d I Take C d (1) = C1. d Then by Lemma 4, ϕ = log Ω satisfies inequality (11) with β = 1: there is a constant C(1) depending only on C d (1) such that exp{ϕ(x) ϕ } dx C(1). 16

Take C 1 = C(1). Lemma 1(a) now implies that Ω A, with A constant bounded by C 1. The dependence of the constants is illustrated in the upper row of Figure 1 (taking p = there). We see that A (Ω) depends only on the bound on the A d constants of the weights ω t. max t,1] Ad p(ω t ) dyadic Lemma 1(a) C d 1 = C d (β), with β = 1 Lemma 4, β = 1 C 1 p dyadic Lemma 1(b) C d 2 = C d (β), with β = 1 p 1 Lemma 4, β = 1 p 1 C 2 Lemma 1(b) A p (Ω) Figure 1: Dependence of the constants in the proof of Theorem 1, for the case K = A p with 1 < p <. Case 2: K = A p, 1 < p <. As for case 1, using Lemma 1(b) and Lemma 4 with both β = 1 and β = 1/(p 1). Figure 1 illustrates the dependence of the constants. We find that A p (Ω) depends only on p and on the bound on the A d p constants of the ω t. Case 3: K = A 1. As for case 1, using Lemma 1(c), Lemma 4 with β = 1, and Lemma 5. The constant A 1 (Ω) depends only on the bound on the A d 1 constants of the ω t. Case 4: K = RH. As for case 1, using Lemma 1(d) and Lemma 5. The constant RH (Ω) depends only on the bound on the RH d constants of the ω t. Case 5: K = RH p. As for case 1, using Lemma 1(e) and Lemma 4 with β = p. We find that RH p (Ω) depends only on p and on the bound on the RH d p constants of the ω t. This completes the proof of Theorems 1 and 2. Remark. An alternative proof of Theorem 1 for K = A p, with 1 < p, can be obtained as follows from the result for K = A 1, using factorization of A p weights Jon]. Suppose 1 p <. If ω 1 and ω 2 are A 1 weights, then ω = ω 1 ω 1 p 2 is an A p weight, with constant A p (ω) A 1 (ω 1 )A 1 (ω 2 ) p 1. Conversely, if ω A p, then there exist ω 1 and ω 2 in A 1 such that ω = ω 1 ω 1 p 2. The A 1 constants of ω 1 and ω 2 depend only on p and on the A p constants of ω, as noted in Gra, p.717]. The analogous results hold in the dyadic setting A d p. Lemma 6. If Theorem 1 holds for A 1, then it holds for every A p, 1 p. Proof. By the (dyadic) factorization theorem, for each t, 1] there exist ω t 1, ω t 2 A d 1 such that ω t = ω t 1(ω t 2) 1 p. Furthermore A 1 (ω t 1) and A 1 (ω t 2) are uniformly bounded, independent 17

of t, 1], by a constant depending on p and on the bound for the constants A p (ω t ). Then { 1 Ω(x) = exp } log ω t (x + t) dt { 1 = exp log ω1(x t + t) ( ω1(x t + t) ) ] } 1 p dt { 1 1 ( = exp log ω1(x t + t) dt + (1 p) ω t 1 (x + t) ) } dt = exp { 1 }] { 1 log ω1(x t + t) dt exp log ω t 2(x + t) dt}] 1 p. By hypothesis, both of the expressions in square brackets are in A 1. Therefore Ω is in A p as required. Furthermore, the A p constant of Ω depends only on the A d p constants of the weights ω t. The result for A follows immediately from the result for A p with 1 p <, using the observations on the dependence of the constants made before Lemma 1 above. 5 Translation-averaging of doubling weights It appears that the obstacle to the translation-average ω(x) = 1 ωt (x+t) dt of A d p weights ω t being in A p is that the assumption of the ω t being dyadic doubling is insufficient to guarantee that the translation-average is actually doubling W]. A natural conjecture is that if the translation-average of a given family of A d p weights is in fact doubling, then this doubling weight also belongs to true A p. As a step in this direction, we show that the presumably stronger assumption that each of the A d p weights ω t is doubling does imply that their translation-average ω is in A p. Theorem 3. Suppose {ω t } t,1] is a family of doubling weights on T, with doubling constants bounded by a constant C dbl independent of t, 1]. Suppose the mapping t ω t is measurable. Fix p with 1 p, and suppose each ω t is in A d p, with A d p constant bounded by a constant V p,d independent of t, 1]. Then the translation-average ω(x) := 1 ω t (x + t) dt belongs to A p on T, with A p constant depending only on p, V p,d, and C dbl. Similarly, for p with 1 < p, if each ω t is in RH d p and RH d p (ω t ) V p,d for all t, 1], then ω(x) belongs to RH p on T, with RH p (ω) depending on p, V p,d, and C dbl. Proof. A doubling weight assigns comparable mass, with a constant depending only on the doubling constant, to any given interval and to each of the dyadic intervals at scale that intersect. (This observation can fail if the weight is dyadic doubling but not doubling.) 18

As a consequence, for fixed p with 1 p, an A d p weight is doubling if and only if it is actually in A p. Moreover the A p constant depends only on the A d p constant and the doubling constant, which in turn depend only on the A p constant. The same is true for RH p, for 1 < p. For example, one finds that for A p with 1 < p <, the A p constant of ω t is bounded by V p := 2 2p 1 V p,d Cdbl 2, while for RH p with 1 < p <, the RH p constant of ω t is bounded by V p := 2 2/p V p,d Cdbl 2. Theorem 3 now follows easily for A p, 1 < p <, using Muckenhoupt s original identification of A p in terms of the boundedness of the Hardy Littlewood maximal function M, defined as usual by Mf(x) := sup x f(y) dy. In particular, for these p, a nonnegative locally integrable function ω is in A p if and only if there is a constant C such that for all locally integrable functions f, Mf(x) p ω(x) dx C f(x) p ω(x) dx. (14) R Moreover, if inequality (14) holds then ω A p and A p (ω) C, while if ω A p then inequality (14) holds with C depending on p and A p (ω). For RH p, 1 < p <, Theorem 3 follows from Minkowski s Integral Inequality and the observation above on comparable mass. The cases of A 1, A, and RH are also straightforward, and we omit the proofs. 6 Generalizations to the polydisc We extend the above results for A p (T) and RH p (T) to the setting of the polydisc. For ease of notation, the statements and proofs given below are expressed for the bidisc. However, they generalize immediately to the polydisc for arbitrarily many factors. The theory of product weights was developed by K.-C. Lin in his thesis L], while the dyadic theory was developed in Buckley s paper B]. The product A p and RH p weights and the product doubling weights, and their dyadic analogues, are defined exactly as in Definitions 1 3 in Section 3, with intervals in T being replaced by rectangles in T T. It follows that a product weight belongs to A p (T T) if and only if it belongs to A p (T) in each variable separately. To be precise, ω A p (T T) if and only if ω(, y) A p (T) uniformly for a.e. y T and ω(x, ) A p (T) uniformly for a.e. x T. In one direction this is a consequence of the Lebesgue Differentiation Theorem, letting one side of the rectangle shrink to a point. The converse uses the equivalence between ω A p (T T) and inequality (14) with M replaced by the strong maximal function S, p.83]. Further, the A p (T T) constant depends only on the two A p (T) constants, and vice versa. The analogous characterizations in terms of the separate variables hold for product RH p weights and for product doubling weights, and for the dyadic product A p, RH p, and doubling weights. Theorem 4. Fix p with 1 p. Let {ω s,t } s,t,1] be a family of dyadic A p weights on the boundary of the bidisc, ω s,t A d p(t T), such that 19 R

(i) the mapping (s, t) ω s,t is measurable, (ii) the appropriate averages of the logarithms of the weights ω s,t are finite: and 1 1 1 1 log ω s,t (x, y) dx dy ds dt <, (iii) the A d p(t T) constants A d p(ω s,t ) are uniformly bounded, independent of s, t, 1]. Then the product geometric-arithmetic average { 1 1 } Ω(x, y) := exp log ω s,t (x + s, y + t) ds dt lies in A p (T T), with A p constant depending on p and on the bound on the A d p constants of the dyadic weights ω s,t. Similarly, for fixed p with 1 < p, if each ω s,t is in RH d p (T T), if hypotheses (i) and (ii) hold, and if (iii ) the RH d p (T T) constants RH d p (ω s,t ) are uniformly bounded, independent of s, t, 1], then Ω(x, y) lies in RH p (T T), with RH p constant depending on p and on the bound on the RH d p constants of the ω s,t. Proof. The proof is by iteration of the one-parameter argument, relying on Lemma 1 and Theorems 1 and 2. We give the argument for K = A p, 1 < p <. The other cases follow similar iteration arguments and we omit the proofs. We show that, for a.e. fixed y, Ω(x, y) belongs to A p (T) in the variable x, with constant independent of y. The hypotheses of Theorem 1 follow immediately from our assumptions. In particular, s ω s,t (, y) is measurable for each t and y, and ω s,t (, y) belongs to A d p(t) in the first variable for all s, t and for a.e. y, with constants independent of s, t, and y. Fix such a y; for emphasis we ll denote it by y. The conclusion of Theorem 1 is that the function { 1 } Ω 1 (x, y ) := exp log ω s,t (x + s, y ) ds belongs to A p (T) in x, with constant independent of y. Let ϕ 1 (x, y ) := log Ω 1 (x, y ) = 1 log ω s,t (x + s, y ) ds. 2

By Lemma 1(b) there are constants C 1 and C 2, depending only on p and on the A d p(t T) constants of the ω s,t, such that { sup exp ϕ 1 (x, y ) ( ϕ 1 (, y ) ) } dx C 1, (15) { 1 sup exp ϕ 1 (x, y ) ( ϕ 1 (, y ) ) p 1 ] } dx C 2. (16) We show that the same inequalities hold when ϕ 1 is replaced by ϕ(x, y ) := log Ω(x, y ) = 1 1 log ω s,t (x + s, y + t) ds dt. Fix an interval. An application of Fubini s Theorem shows that ϕ(x, y ) ( ϕ(, y ) ) 1 1 = ϕ s,t (x + s, y + t) ds dt = 1 ϕ 1 (x, y + t) ( ϕ 1 (, y + t) ) 1 1 ] dt. ϕ s,t (x + s, y + t) ds dt dx Then by Jensen s inequality and Tonelli s Theorem, { exp ϕ(x, y ) ( ϕ(, y ) ) } dx 1 { exp ϕ 1 (x, y + t) ( ϕ 1 (, y + t) ) } dt dx = 1 C 1, { exp ϕ 1 (x, y + t) ( ϕ 1 (, y + t) ) } dx dt by inequality (15) with y replaced by y + t for a.e. t. The same argument can be used to verify that inequality (16) holds for ϕ(, y ) for a.e. y. Therefore by Lemma 1, Ω(x, y ) belongs to A p (T) in x for a.e. y, with uniform constants. In an identical fashion, we find that Ω(x, y) belongs to A p (T) in y for a.e. x, with uniform constants, which proves the theorem for K = A p, 1 < p <. Remark. As in the one-parameter case, there is an alternative proof of the geometricarithmetic averaging result (Theorem 4) for A p (T T) where 1 < p, relying on the A 1 (T T) case and the generalization to the bidisc setting Jaw] of the A p factorization theorem. Moreover, the product A p case can also be derived from the one-parameter result using the maximal function characterization of this weight class. For product doubling weights, we also have a result analogous to Theorem 3. 21

Theorem 5. Suppose {ω s,t } s,t,1] is a family of doubling weights on T T, with doubling constants bounded by a constant C dbl independent of s, t, 1]. Suppose the mapping (s, t) ω s,t is measurable. Fix p with 1 p, and suppose each ω s,t is in A d p(t T) with A d p(t T) constant bounded by a constant V p,d independent of s, t, 1]. Then the translation-average ω(x, y) := 1 1 ω s,t (x + s, y + t) ds dt belongs to A p (T T), with an A p constant depending only on p, V p,d, and C dbl. Similarly, for 1 < p, if the A d p(t T) assumption above is replaced by the assumption that each ω s,t is in RH p (T T) with RH d p (ω s,t ) V p,d for all s, t, 1], then ω(x, y) belongs to RH p (T T), with RH p (ω) depending on p, V p,d, and C dbl. The proof is by iteration of the one-parameter argument given for Theorem 3 above. References B] S.M. Buckley, Summation conditions on weights, Michigan Math. J. 4 (1993), 153 17. CN] D. Cruz-Uribe and C.J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc. 347 (1995), 2941 296. GCRF] J. Garcia-Cuerva and J.L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland, New York, NY, 1985. Gar] J.B. Garnett, Bounded analytic functions, Academic Press, Orlando, FL, 1981. GJ] J.B. Garnett and P.W. Jones, BMO from dyadic BMO, Pacific. J. Math. 99 (1982), no. 2, 351 371. Gra] Jaw] JoNi] JoNe] L. Grafakos, Classical and modern Fourier analysis, Pearson Education, Upper Saddle River, NJ, 24. B. Jawerth, Weighted inequalities for maximal operators: linearization, localization and factorization, Amer. J. Math. 18 (1986), 361 414. F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415 426. R. Johnson and C.J. Neugebauer, Homeomorphisms preserving A p, Rev. Mat. Iberoamericana 3 (1987), 249 273. Jon] P.W. Jones, Factorization of A p weights, Ann. of Math. 111 (198), 511 53. L] K.-C. Lin, Harmonic analysis on the bidisc, Ph.D. thesis, UCLA, 1984. 22

M] PW] S] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 27 226. J. Pipher and L.A. Ward, BMO from dyadic BMO on the bidisc, J. London Math. Soc. 77 (28), 524 544. E.M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, NJ, 1993. T] S. Treil, H 1 and dyadic H 1, in Linear and Complex Analysis: Dedicated to V. P. Havin on the Occasion of His 75th Birthday (ed. S. Kislyakov, A. Alexandrov, A. Baranov), Advances in the Mathematical Sciences 226 (29), AMS, 179 194. W] L.A. Ward, Translation averages of dyadic weights are not always good weights, Rev. Mat. Iberoamericana 18 (22), 379 47. 23