Gehring's lemma for nondoubling measures

Michigan Math. J. 47 (2000) Gehring’s Lemma for Nondoubling Measures Joaqu i m M a rt í n & M a r i o M i l m a n 1. Introduction Let Q0 ⊂ R n be a fixed cube with sides parallel to the coordinate axes, let w be a strictly positive integrable function on Q0 , and let 1 < p < ∞. We shall say that p a positive function g ∈ Lw (Q0 ) belongs to RHp (w) (i.e., that g satisfies a reverse Hölder inequality) if there exists a C ≥ 1 such that, for every cube Q ⊂ Q0 with sides parallel to the coordinate axes, we have 1/p  Z Z C 1 ≤ g(x) p w(x) dx g(x)w(x) dx, w(Q) Q w(Q) Q R with w(Q) = Q w(x) dx. If the underlying measure µ := w(x) dx satisfies the doubling condition—that is, if there exists a constant c > 0 such that µ(B(x, 2r)) ≤ cµ(B(x, r))—then by Gehring’s lemma [7] there exists an ε > 0 such that g ∈ RHp+ε (w). For excellent accounts of the role that reverse Hölder inequalities play in PDEs, we refer to [9] and [11]. Recently there has been interest in extending the Calderón–Zygmund program to the context of nondoubling measures (cf. [1; 13; 14; 16; 20; 21] and the references therein). The purpose of this note is to prove Gehring’s lemma for nondoubling measures of the form µ := w(x) dx. Our main results are given in the next two theorems; for proofs, see Section 4. (When preparing the final version of this paper for publication we realized that Theorem 1 can be also obtained by a different method by means of combining Lemma 2.3 and Corollary 2.4 of [16] with Exercise 6.6 of [18].) Theorem 1. Let 1 < p < ∞, and let w be a positive integrable function on Q0 . Suppose that g ∈ RHp (w). Then there exists an ε > 0 such that g ∈ RHp+ε (w). Theorem 2 (see [13] for the corresponding R n version of this result; see [9] and the references therein for the doubling case). Let g, h be positive functions in p Lw (Q0 ) and suppose that there exists c > 1 such that, for all cubes Q ⊂ Q0 with sides parallel to the coordinate axes, we have Received April 27, 2000. Revision received July 10, 2000. The first author was partially supported by DGICYT, grant no. PB97-0986, and by Suport a Grups de Recerca 1997SGR 00185. 559 Joaqu i m M a rt í n & M a r i o M i l m a n 560  1 w(Q) Z Q 1/p ≤c g(x) w(x) dx Z 1 g(x)w(x) dx w(Q) Q 1/p  Z 1 h(x) p w(x) dx . +c w(Q) Q p (1.1) q Then there exist q > p and C = C(c, q) > 0 such that if g, h ∈ Lw (Q0 ) then, for every cube Q ⊂ Q0 with sides parallel to the coordinate axes, we have  1/q 1/p  Z Z 1 1 ≤C g(x) q w(x) dx g(x) p w(x) dx w(Q) Q w(Q) Q  1/q Z 1 q +C h(x) w(x) dx . (1.2) w(Q) Q Our methods are based on covering lemmas and interpolation theory. For doubling measures dµ := w(x) dx, the connection with interpolation is given by the fact that the maximal operator of Hardy and Littlewood associated with dµ, Z 1 |f (x)|w(x) dx, Mµ f (x) = sup Q 0 ⊃Q∋x µ(Q) Q satisfies Z 1 t ∗ ≈ f (s) ds = fw∗∗(t) (1.3) t 0 w (see Section 2) while—independently of doubling conditions—we always have (Mµ f )∗w (t) K(t, f ; L1w , L∞ ) (1.4) t (see [19, pp. 213–214]). In this case the inverse reiteration theorem of [5] (cf. our Theorem 4) immediately proves Theorem 1. If w is not doubling then (1.3) may p not hold (see [1]); in fact, the maximal operator may not be bounded on Lw (see [8]), although doubling conditions do not, of course, alter the interpolation theory of Lp spaces. Therefore, dealing with nondoubling measures using the K-method requires a different maximal operator. It turns out that a suitable maximal operator can be obtained through the use of packings [1]. (The idea of maximal operators associated with packings can be traced at least as far back as the classical paper of John and Nirenberg [10].) In order to explain in more detail what we do in this paper, let us start by recalling that a packing in Q0 is simply a finite or countably infinite collection of nonoverlapping cubes with sides parallel to the coordinate axes contained in Q0 . |π| For a given packing π = {Qi }i=1 in Q0 , we associate a linear operator S π defined by:  Z |π|  X 1 f (y)w(y) dy χQ i (x), f ∈ L1w (Q0 ) + L∞ (Q0 ). S π (f )(x) = w(Qi ) Q i i=1 fw∗∗(t) = (Here |π| = ∞ if the packing has infinitely many cubes.) We consider the maximal operator defined by Gehring’s Lemma for Nondoubling Measures 561 (Ff )w (t) = sup (S π (|f |))∗w (t), π gw∗ denotes the nonincreasing rearrangement of g with respect to the meawhere sure w(y) dy, and the supremum is taken over all packings. A characterization of the K-functional for the pair (L1w (R n ), L∞ (R n )) in terms of the maximal operator (Ff )w (t) was given in [1]. This characterization was exploited in [13] to prove the nonlocal version of Theorem 2. In order to prove local self-improving results of Gehring type using the K-method, we show the following complement to the global computations of [1]. Theorem 3. Let f ∈ L1w (Q0 ) + L∞ (Q0 ); then, for 0 < t < w(Q0 ) = R Q 0 w(y) dy, K(t, f ; L1w (Q0 ), L∞ (Q0 )) ≈ t(Ff )w (t), with constants of equivalence that are independent of f. The proof of this characterization relies on a modification of the Calderón– Zygmund decomposition for nondoubling measures that was recently obtained in [14] and [16]. The paper is organized as follows. In Section 2 we provide a rather concise review (but with detailed references) of the parts of the real method of interpolation we shall use in this paper. In Section 3 we give a brief but self-contained account of the Calderón–Zygmund decomposition for nondoubling measures obtained in [14] and [16], conveniently modified for our purposes, and then use it to prove the equivalence between maximal operators associated with packings and K-functionals. Then, in Section 4, we provide the proofs of Theorem 1 and Theorem 2. Finally, it is important to note here our belief that the methods we are developing are more interesting than the particular results obtained so far. For example, the interpolation method can be used to study Gehring-type self-improving results in a geometry-free context (see [12]). Moreover, our method can be used to study self-improving inequalities where the qualitative property whose improvement is sought is not necessarily integrability. We hope to return to this point elsewhere. Acknowledgment. We thank J. Soria for providing us with a copy of [16] and M. Korey and the referee for helpful suggestions to improve the presentation. 2. Background The main tools from real interpolation that we use are the K-functional and the reiteration theorem. Our main references will be [4; 5; 19], to which the reader is referred for further information. E = (X0 , X1 ) of quasi-normed spaces that are continuWe work with pairs X ously embedded into a common Hausdorff topological vector space. For a given E we can thus form the sum space 6(X) E = X0 + X1 and define for x ∈ pair X, E t > 0, the “K-functional” 6(X), E = inf{kx 0 kX0 + tkx1kX1 : x = x 0 + x1, x i ∈ Xi , i = 0, 1}. K(t, x; X) 562 Joaqu i m M a rt í n & M a r i o M i l m a n In the context of the pair (L1(R n ), L∞ (R n )), the splitting implicit in the computation of the corresponding K-functionals is closely related to the Calderón– Zygmund decomposition. In fact, following [17, p. 1 (4); 17, Thm. 3.2], if we split f using a Calderón–Zygmund decomposition f = bα +g α —where bα is the “bad” part, g α is the “good” part, and the usual parameter α of the Calderón–Zygmund decomposition is chosen to be (Mf )∗(t), where (Mf )∗ is the nondecreasing rearrangement of the maximal operator of Hardy–Littlewood—then we have K(t, f ; L1(R n ), L∞ (R n )) ≈ kb(Mf )∗(t) kL1(R n ) + tkg(Mf )∗(t) kL∞ (R n ) ≈ t(Mf )∗(t) (2.1) (cf. [4, p. 123]). The following elementary formula also holds [19, pp. 213–214]: Z t f ∗(s) ds. (2.2) K(t, f ; L1(R n ), L∞ (R n )) = 0 Comparing (2.1) and (2.2), we see that 1 (Mf ) (t) ≈ t ∗ Z t f ∗(s) ds. (2.3) 0 The equivalences (2.1) and (2.3) fail, in general, if we replace the Lebesgue measure dx by a nondoubling measure dµ(x) (see [1]). For more examples of computations of K-functionals, see [4]. E a (“real”) scale of interpolation spaces between them can For a given pair X, be constructed as follows. Given 0 < θ < 1 ≤ q ≤ ∞, define 1/q  Z ∞  E q dt E : E θ,q = x ∈ 6(X) (t −θK(t, x; X)) <∞ . X t 0 It turns out that many of the familiar scales of spaces used in analysis can be identified with suitable real interpolation scales. The process of identification of concrete spaces as interpolation spaces for a given pair hinges upon the computation of K-functionals, and it is usually greatly simplified by the following reiteration (or iteration) property (cf. [4, Thm. 2.4; 5, Thm. 3.5.3]): E θ 1,q1 )θ,q = X E τ,q , E θ 0 ,q 0 , X (X (2.4) where τ = (1 − θ)θ 0 + θθ 1 . A quantitative form of the reiteration formula (2.4) is given by Holmstedt’s formula [4, Thm. 2.1; 5, Thm. 3.6.1]: E θ 1,q1 ) ≈ E θ 0 ,q 0 , X K(t, f ; X Z + t 1/(θ 1−θ 0 ) E q0 (s −θ 0 K(s, f ; X)) 0 Z ∞ ds s E q1 (s −θ 1K(s, f ; X)) t 1/(θ 1−θ 0 ) 1/q 0 ds s 1/q1 . 563 Gehring’s Lemma for Nondoubling Measures The following endpoint version of Holmstedt’s formula (cf. [4, Cor. 2.3; 5, Cor. 3.6.2]) will be particularly useful here: E1 ) ≈ E θ 0 ,q 0 , X K(t, f ; X Z t 1/(1−θ 0 ) (s 0 −θ 0 E q 0 ds K(s, f ; X)) s 1/q 0 . (2.5) Reverse Hölder inequalities were formulated as “inverse” reiteration theorems in [15], where the following abstract form of Gehring’s lemma was obtained. Theorem 4. Let (A 0 , A1 ) be an ordered pair of Banach spaces (i.e. A1 ⊂ A 0 ), and suppose that f ∈ A 0 is such that there exist some constant c > 1, θ 0 ∈ (0, 1), and 1 ≤ p < ∞ such that, for every t ∈ (0, 1), K(t, f ; A θ 0 ,p;K , A1 ) ≤ ct K(t 1/(1−θ 0 ) , f ; A 0 , A1 ) . t 1/(1−θ 0 ) (2.6) Then there exists a θ 1 > θ 0 such that, for q ≥ p and 0 < t < 1, we have K(t, A θ 1,q;K , A1 ) ≈ t K(t 1/(1−θ 1 ) , f ; A 0 , A1 ) . t 1/(1−θ 1 ) The connection with the classical Gehring’s lemma can be seen from the following facts: Lp = (L1, L∞ )1/p ′,p, 1 < p < ∞; 1/p  Z t 1 K(t 1/p, f ; Lp, L∞ ) ∗ p , 1 ≤ p < ∞, f (s) ds ≈ t 1/p t 0 (Mp f )∗(t) ≈ K(t 1/p, f ; Lp, L∞ ) . t 1/p (2.7) (2.8) (2.9) For (2.7) and (2.8), see [5, Thm. 5.2.1]; while (2.9) follows from (2.1), the fact that Mp (f ) = (M(|f |p ))1/p by definition, and (2.8). For a more detailed discussion on the connection with Gehring’s lemma, see Section 4 and [15]. 3. The K-Functional for the Pair (L1w (Q0 ), L∞ (Q0 )) We start with the Calderón–Zygmund decomposition for nondoubling measures obtained in [14] and [16]. We briefly indicate a proof of a version that is convenient for our development in this note. Let Q0 be a fixed cube in R n with sides of length L that are parallel to the coordinate axes. For each x in the interior of Q0 we define the basis CQ 0 (x) = {Q x (r)}, 564 Joaqu i m M a rt í n & M a r i o M i l m a n where Q x (r) is the unique cube with side r that minimizes the distance from x of the center of Q x (r) (0 < r < L := side of Q0 ), so that Q x (r) ⊂ Q0 . We consider Z 1 |f (y)|w(y) dy. MQ (g)(x) = sup Q∈CQ 0 (x) w(Q) Q Lemma 1. Let g ∈ L1w (Q0 ) Rbe a nonnegative function. Let λ be a positive number such that λ > 1/w(Q0 ) Q 0 g(y)w(y) dy and the level set  λ = {x ∈ Q0 : MQ (g)(x) > λ} is not empty. Then there exists a quasi-disjoint family of cubes {Qj } contained in Q0 such that, for each j, Z 1 λ< g(y)w(y) dy ≤ 2λ (3.1) w(Qj ) Qj and, moreover, [ Qj a.e. (3.2) g(x) ≤ λ for x ∈ Q0 j In fact, we can write [ j Qj = B(n) [ [ Qi , (3.3) k=1 i∈F k where each family {Qi }i∈F k , k = 1, ..., B(n), is formed by pairwise disjoint cubes. Notes. Recall that a family P of cubes {Qj } is quasi-disjoint if there exists a universal constant C such that j χQj (x) ≤ C; B(n) is usually called the Besicovitch constant. Proof of Lemma 1. Since  λ is not empty, it follows that for any x ∈  λ we can find a cube Ax ∈ CQ 0 (x) such that Z 1 g(y)w(y) dy. λ< w(Ax ) Ax R Therefore, since the function hx (r) = 1/µ(Q x (r)) Q x (r) |f (y)| dµ(y) is continuous for each x ∈ int(Q0 ), we can select a cube Q x ∈ CQ 0 (x) satisfying Z 1 g(y)w(y) dy ≤ 2λ λ< w(Q x ) Q x with Q x ( Q0. The family {Q x } selected in this fashion covers  λ . From now on we follow verbatim the argument in [14; 16]. Thus, for any cube Q x we define the rectangle R x in R n as the unique rectangle in R n centered at x such that R x ∩ Q0 = Q x . It follows that the ratio of any two side lengths of R x is bounded by 2, and thus by Besicovitch’s covering lemma we can select a countable collection {Rj } of rectangles covering  λ and such that every point of  λ belongs to at most B(n) rectangles Rj . Replacing each Rj by its corresponding cube Qj , we obtain a family of cubes {Qj } with the properties that we need. Finally, (3.2) follows (as usual) by Lebesgue’s differentiation theorem. Gehring’s Lemma for Nondoubling Measures 565 The connection between packings and interpolation theory is given by the following local version of a result originally proved in [1] for R n . Theorem 5. Let f ∈ L1w (Q0 ) + L∞ (Q0 ). Then, for 0 < t < w(Q0 ), K(t, f ; L1w (Q0 ), L∞ (Q0 )) ≈ t(Ff )w (t), with constants of equivalence independent of f. Proof. For a given packing π, the operator f → S π (|f |) is obviously a norm-1 sublinear operator acting on the pair (L1w (Q0 ), L∞ (Q0 )); therefore, K(t, S π (|f |); L1w (Q0 ), L∞ (Q0 )) ≤ K(t, f ; L1w (Q0 ), L∞ (Q0 )). Combining this inequality with t(S π (|f |))∗w (t) ≤ K(t, S π (|f |); L1w (Q0 ), L∞ (Q0 )) and taking the supremum over all packings, we obtain t(Ff )w (t) ≤ K(t, f ; L1w (Q0 ), L∞ (Q0 )). To prove the converse we fix a constant 1 < c < 2 and, for a given 0 < t < w(Q0 ), consider the set   Z 1 |f (y)|w(y) dy > c(Ff )w (t) .  0 (t) = x ∈ Q0 : sup r>0 w(Q x (r)) Q x (r) If  0 (t) is empty then by Lebesgue’s theorem we have that f ∈ L∞ ; in fact, |f | ≤ c(Ff )w (t) a.e. Consequently, using the decomposition f = 0 + f yields K(t, f ; L1w (Q0 ), L∞ (Q0 )) ≤ tkf k∞ ≤ ct(Ff )w (t), as we wanted to show. Suppose now that  0 (t) is not empty. Note that, for 0 < t < w(Q0 ), c(Ff )w (t) > (Ff )w (t) = sup (S π (|f |))∗w (t) π  ∗ 1 |f (y)|w(y) dy χQ 0 (x) (t) ≥ w(Q0 ) Q 0 w   Z 1 |f (y)|w(y) dy χ [0,w(Q 0 ))(t) = w(Q0 ) Q 0 Z 1 |f (y)|w(y) dy. = w(Q0 ) Q 0  Z We can therefore apply Lemma 1 (with λ = c(Ff )w (t)) to obtain a family of cubes F = {Qj } such that Z 1 c(Ff )w (t) < |f (y)|w(y) dy ≤ 2c(Ff )w (t) on Qj (3.4) w(Qj ) Qj Joaqu i m M a rt í n & M a r i o M i l m a n 566 and |f | ≤ c(Ff )w (t) on Q0 with [ Qj = j B(n) [ [ [ B(n) [ Qi = k=1 i∈F k Qj (3.5) πk , (3.6) k=1 where each πk = {Qi : i ∈ F k }, k = 1, . . . , B(n), is a packing. The nearly optimal decomposition we need will then be given by f = fχ S Qj + fχQ 0 \ S Qj . In fact, ° ° S ° °fχ Q ° 1 = °fχ S B(n) S j L k=1 ≤ B(n) X i∈ F k ° ° Qi L1w (Q0 ) X w(Qi ) Z k=1 i∈F k w(Qi ) ≤ 2c(Ff )w (t) B(n) X k=1 Qi  |f |w dx X i∈F k  w(Qi ) (by (3.4)). We shall show in a moment that X w(Qi ) ≤ t. (3.7) i∈F k Assuming for now the validity of (3.7), we obtain  B(n) X X ° S ° °fχ Q ° 1 ≤ 2c(Ff )w (t) w(Q ) i j L k=1 ≤ 2c(Ff )w (t) B(n) X i∈F k t k=1 ≤ 2cB(n)t(Ff )w (t). Moreover, by (3.5) we have ° ° t °fχQ 0 \ S Qj °L∞ ≤ ct(Ff )w (t). Collecting estimates, we finally arrive at K(t, f ; L1w (Q0 ), L∞ (Q0 )) ≤ 3cB(n)t(Ff )w (t). In order to prove (3.7) we must show that, given π = {Qi } an arbitrary packing from the family F, we have X w(Qi ) ≤ t. Qi ∈ π Indeed, if P Qi ∈ π w(Qi ) > t then using (3.4) yields Gehring’s Lemma for Nondoubling Measures 567  Z 1 |f (y)|w(y) dy χQ i (z) w(Qi ) Q i Qi ∈ π  X  χQ i (z) > c(Ff )w (t) S π (|f |)(z) = X Qi ∈ π  X  χQ i (z) . > (Ff )w (t) Qi ∈ π S Therefore, for any z ∈ Q i ∈ π Qi we have S π (|f |)(z) > (Ff )w (t), and since ¡S
c we see that S π (|f |)(z) = 0 on Q i ∈ π Qi X w(Qi ), S π (|f |)∗w (t) > (Ff )w (t) for t < Qi ∈ π contradicting the definition of (Ff )w (t). 4. Proof of Theorems 1 and 2 In preparation for the proof of Theorems 1 and 2, let us introduce (following [1]) the functionals 1/p Z |π|  X 1 χQ i (x), |f (y)|p w(y) dy S π,p (f )(x) = µ(Qi ) Q i i=1 |π| which are associated with “packings” π = {Qi }i=1 ⊂ Q0 . If p = 1 then S π,1 coincides with S π , as defined in Section 2. 4.1. Proof of Theorem 1 Since we are dealing with families of disjoint cubes it follows readily that g ∈ RHp (w) implies that, for any packing π, we have S π,p (g)(x) ≤ cS π (g)(x). Taking nondecreasing rearrangements in the previous inequality with respect to the measure dµ = w(x) dx, we obtain (S π,p (g))∗w (t) ≤ c(S π (g))∗w (t), t > 0 (π any packing). Therefore, taking the supremum over all packings yields sup (S π,p (g))∗w (t) ≤ c sup (S π (g))∗w (t). π (4.1) π By Theorem 5, we know that K(t, g; L1w (Q0 ), L∞ (Q0 )) ≈ t sup (S π (g))∗w (t). (4.2) π On the other hand (cf. [1]), by well-known general considerations it follows from (4.2) that Joaqu i m M a rt í n & M a r i o M i l m a n 568 K(t 1/p, g; Lpw (Q0 ), L∞ (Q0 )) ≈ (K(t, |g|p ; L1w (Q0 ), L∞ (Q0 ))1/p  1/p ≈ t sup (S π (g p ))∗w (t) π = t 1/p sup (S π,p (g))∗w (t). (4.3) π Multiplying (4.1) by t 1/p, we have t 1/p sup (S π,p (g))∗w (t) ≤ ct 1/p sup (S π (g))∗(t) π π and thus arrive at the K-functional estimate ′ K(t 1/p, g; Lpw (Q0 ), L∞ (Q0 )) ≤ ct −1/p K(t, g; L1w (Q0 ), L∞ (Q0 )) or, equivalently, K(t, g; Lpw (Q0 ), L∞ (Q0 )) ≤ ct 1−pK(t p, g; L1w (Q0 ), L∞ (Q0 )). (4.4) Now we can apply Theorem 4 (cf. [15, Thm. 1]) to conclude that there exists a q > p such that K(t, g; Lqw (Q0 ), L∞ (Q0 )) ≤ ct 1−q/pK(t q, g; Lpw (Q0 ), L∞ (Q0 )). (4.5) Thus, in view of the well-known formula K(t, h; Lrw (Q0 ), L∞ (Q0 )) ≃ Z tr 0 1/r h∗w (s) r ds (cf. (2.8)), we have that (4.5) is equivalent to  Z t 1/q 1/p  Z t 1 1 ≤C gw∗ (s) q ds gw∗ (s) p ds t 0 t 0 (4.6) for 0 < t < w(Q0 ). Observe that 1/r  1/r  Z Z w(Q0 ) 1 1 r ∗ r = g(x) w(x) dx gw (s) ds ; w(Q0 ) Q0 w(Q0 ) 0 therefore we see that (4.6) gives  1/q 1/p  Z Z 1 1 q p ≤C g(x) w(x) dx g(x) w(x) dx . w(Q0 ) Q0 w(Q0 ) Q0 4.2. Proof of Theorem 2 Let 1 ≤ p < ∞ and denote by Pp f the Hardy operator defined on locally integrable functions by  Z t 1/p 1 |f (s)|p ds , t > 0. Pp f (t) = t 0 If p = 1, we let Pp f = Pf. We shall need the following lemma from [3], which we prove here for the sake of completeness. Gehring’s Lemma for Nondoubling Measures 569 Lemma 2 (cf. [3, Prop. 2.1]). Let f be a nonincreasing function, and let 1 < p < ∞. Then 1/p  Z t 1/p  Z t p − 1 1/p p p Pf (s) ds ≤ f (s) ds p 0 0  1/p Z t 1 + f (s) ds. (4.7) t (1−p)/p p 0 Proof. Because f is decreasing, Z x f (s) ds ≥ xf (x). 0 It follows that p−1 Z x p Z x d ≥ px p−1f (x) p ; f (s) ds f (s) ds = pf (x) dx 0 0 integrating, we have Z x 1 ps p−1f (s) p ds. xp 0 Further integration and Fubini yield Z t  Z t Z t p p−1 p p 1−p p s f (s) ds , f (s) ds − t Pf (s) ds ≥ p −1 0 0 0 (Pf (x)) p ≥ which implies that Z t 0 1/p  Z t 1/p  p −1/p Pf (s) p ds ≤ f (y) p dy p −1 0 1/p Z t s p−1f (s) p ds . + t (1−p)/p (4.8) 0

1/p ¡R t ¡R t ≤ (1/p)1/p 0 f (s) ds (cf. [19, Thm. 3.11]), Finally, since 0 s p−1f (s) p ds we obtain (4.7). Now let us proceed with the proof of Theorem 2. We shall follow closely the argument in [13]. As in the proof of Theorem 1, we see that (1.1) implies for any packing π that S π,p (g)(x) ≤ c(S π (g)(x) + S π,p (h)(x)). Taking nondecreasing rearrangements with respect to the measure dµ = w(x) dx and taking the supremum over all packings yields   sup (S π,p (g))∗(2t) ≤ c sup (S π (g))∗(t) + sup (S π,p (h))∗(t) . π Multiplying this inequality by t at the K-functional estimate π 1/p π and then using (4.3) and Theorem 5, we arrive K(t 1/p, g; Lpw (Q0 ), L∞ )
¡ ′ ≤ c t −1/p K(t, g; L1w (Q0 ), L∞ ) + K(t 1/p, h; Lpw (Q0 ), L∞ ) . (4.9) Joaqu i m M a rt í n & M a r i o M i l m a n 570 Using Holmstedt’s reiteration formula (cf. (2.5)), we write Z t 1/p
¡ −1/p ′ 1 ∞ p ds 1/p p ∞ s K(s, g, Lw (Q0 ), L ) K(t , g; Lw (Q0 ), L ) ≃ s 0   Z t  1/p K(s, g, L1w (Q0 ), L∞ ) p ds . ≃ s 0 Inserting this expression into (4.9) gives  1/p  Z t K(s, g; L1w (Q0 ), L∞ ) p ds s 0 K(t, g; L1w (Q0 ), L∞ ) t  1/p  Z t K(s, h; L1w (Q0 ), L∞ ) p 1/p 1 ds . + ct t 0 s ≤ ct 1/p In terms of Hardy operators, we have    K(t, g; L1w (Q0 ), L∞ ) K(·, g, L1w (Q0 ), L∞ ) p (t) ≤ c Pp · t   K(·, h; L1w (Q0 ), L∞ ) (t). + cPp · Applying Lq(0, w(Q0 ))-norms to the previous inequality then yields ° ° °  1 ∞ ° 1 ∞ ° ° ° ° °Pp K(·, g, Lw (Q0 ), L ) ° ≤ c° K(·, g, Lw (Q0 ), L ) ° ° ° ° ° · · q q °  ° 1 ° K(·, h; Lw (Q0 ), L∞ ) ° ° . ° + c°Pp ° · q Now, since °  °  1 ∞ ° 1 ∞ p ° °p ° ° ° ° °Pp K(·, g, Lw (Q0 ), L ) ° = °P K(·, g, Lw (Q0 ), L ) (4.10) ° ° ° ° · · q/p q and since K(s, f, L1w (Q0 ), L∞ )/s is decreasing, we can apply Lemma 2 to obtain °  ° ° K(·, g, L1w (Q0 ), L∞ ) p ° ° ° ° ° · q/p Ãq !p/q °  1 ∞ p ° −1 ° ° p °P K(·, g, Lw (Q0 ), L ) ° ≤ ° ° q · q/p p ° °  p/q 1 ∞ p ° ° p (1−q/p)/(q/p) ° K(·, g, Lw (Q0 ), L ) °. w(Q0 ) + ° ° q · 1 Combining this last inequality, (4.10), and the well-known inequality (x + y) α ≤ x α + y α if 0 < α ≤ 1, we arrive at 571 Gehring’s Lemma for Nondoubling Measures ° ° ° K(·, g, L1w (Q0 ), L∞ ) ° ° ° ° ° · q °  ° ° K(·, g, L1w (Q0 ), L∞ ) p °1/p ° =° ° ° · q/p Ãq !1/q°  ° −1 ° K(·, g, L1w (Q0 ), L∞ ) ° p ° ° ≤ °Pp ° q · q p ° °  1/q 1 ∞ p °1/p ° p (1−q/p)/(q/p) 1/p ° K(·, g, Lw (Q0 ), L ) ° . (w(Q0 ) ) ° + ° q · 1 On the other hand, by the classical Hardy inequality we have à q !1/p° ° °  1 ∞ ° ° ° K(·, h; L1w (Q0 ), L∞ ) ° ° K(·, h; L (Q ), L ) p 0 w ° ≤ ° . ° °Pp ° ° ° ° q · · −1 q q p Collecting these estimates gives à q !1/q ° ° ° K(·, g, L1w (Q0 ), L∞ ) ° p ° ° ° ° q · − 1 q p ° ° ° K(·, g, L1w (Q0 ), L∞ ) ° ° ≤ c° ° ° · q !1/q à ° ° 1 ∞ ° ° 1 (1−q/p)/(q/p) 1/p ° K(·, g, Lw (Q0 ), L ) ° (w(Q0 ) ) ° + q ° · −1 p p à q !1/p° ° ° K(·, h; L1w (Q0 ), L∞ ) ° p ° . ° +c q ° ° · −1 q p We can therefore choose q > p, with q sufficiently close to p, such that à q !1/q p q p −1 − c > 0, and thus we can write ° ° ° ° 1 ∞ ° ° ° K(·, g, L1w (Q0 ), L∞ ) ° ° ≤ C(w(Q0 )(p−q)/q )1/p ° K(·, g, Lw (Q0 ), L ) ° ° ° ° ° ° · · q q ° ° ° K(·, h; L1w (Q0 ), L∞ ) ° ° . + C° ° ° · q Now, using (2.7) (see also [5, Thm. 5.5.1]), (L1w (Q0 ), L∞ )1/q ′,q = Lqw (Q0 ), hence (4.11) can be rewritten as (L1w (Q0 ), L∞ )1/p ′,p = Lpw (Q0 ); (4.11) Joaqu i m M a rt í n & M a r i o M i l m a n 572 ¡
kgkLqw (Q0 ) ≤ C (w(Q0 )(p−q)/q )1/p kgkLpw (Q0 ) + khkLqw (Q0 ) . (4.12) Dividing by w(Q0 )1/q, we thus obtain  1/q 1/p  Z Z 1 1 q p ≤C g(x) w(x) dx g(x) w(x) dx w(Q0 ) Q 0 w(Q0 ) Q 0  1/q Z 1 q +C h(x) w(x) dx . w(Q0 ) Q 0 Obviously this argument can be repeated for every Q ⊂ Q0 in order to obtain inequality (1.2). Remark 1. If we work in R n instead of a cube Q0 and if w(R n ) = ∞, then the term (w(Q0 )(p−q)/q )1/p that appears in (4.12) is equal to 0 and we obtain 1/q Z 1/q Z q q h(x) w(x) dx ≤C g(x) w(x) dx Rn Rn (cf. [13, Thm. 1]). Remark 2. Using inequality (4.8) instead of (4.7) and the fact that (L1w (Q0 ), L∞ )1/p ′,q = Lp,q w (Q0 ), we can show that ¡
+ khkLqw (Q0 ) , kgkLqw (Q0 ) ≤ C (w(Q0 )(p−q)/q )1/p kgkLp,q w (Q0 ) p p,q which implies (4.12) because Lw (Q0 ) ⊂ Lw (Q0 ) if q > p. Remark 3. Our proof of Theorem 1, combined with the argument in [2, Thm. 3.1], can be used to derive—for nondoubling measures—the endpoint version of Gehring’s lemma originally obtained by Fefferman [6]. References [1] I. Asekritova, N. Ya. Krugljak, L. Maligranda, and L. E. Persson, Distribution and rearrangement estimates of the maximal function and interpolation, Studia Math. 124 (1997), 107–132. [2] J. Bastero, M. Milman, and F. Ruiz, Reverse Hölder inequalities and interpolation, Function spaces, interpolation spaces, and related topics, Israel Math. Conf. Proc., 13, pp. 11–23, Bar-Ilan Univ., Ramat Gan, 1999. [3] , On sharp reiteration theorems and weighted norm inequalities, Studia Math. 142 (2000), 7–24. [4] C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Boston, 1988. [5] J. Bergh and J. Löfstrom, Interpolation spaces: An introduction, Grundlehren Math. Wiss., 223, Springer-Verlag, Berlin, 1976. [6] R. Fefferman, A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator, J. Amer. Math. Soc. 2 (1989), 127–135. Gehring’s Lemma for Nondoubling Measures 573 [7] F. W. Gehring, The Lp integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. [8] L. Grafakos and J. Kinnunen, Sharp inequalities for maximal functions associated with general measures, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 717–723. [9] T. Iwaniec, The Gehring lemma, Quasiconformal mappings and analysis (Ann Arbor, MI, 1995) pp. 181–204, Springer, New York, 1998. [10] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. [11] C. E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conf. Ser. in Math., 83, Amer. Math. Soc., Providence, RI, 1994. [12] J. Martin and M. Milman, Reverse Hölder inequalities and approximation spaces, J. Approx. Theory (to appear). [13] M. Mastylo and M. Milman, A new approach to Gehring’s lemma, Indiana J. Math. 49 (2000), 655–679. [14] J. Mateu, P. Mattila, A. Nicolau, and J. Orobitg, BMO for non-doubling measures, Duke Math. J. 102 (2000), 533–565. [15] M. Milman, A note on Gehring’s lemma, Ann. Acad. Sci. Fenn. Ser. A I Math. 21 (1996), 389–398. [16] J. Orobitg and C. Perez, Ap weights for non-doubling measures in R n and applications, preprint. [17] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. [18] , Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993. [19] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1971. [20] X. Tolsa, Cotlar’s inequality without the doubling conditions and existence of principal values for the Cauchy integral of measures, J. Reine Angew. Math. 502 (1998), 199–235. [21] , BMO, H 1, and Calderón–Zygmund operators for non-doubling measures, preprint, 1999. J. Martín Department of Mathematics University of Barcelona Barcelona Spain [email protected] M. Milman Department of Mathematics Florida Atlantic University Boca Raton, FL 33431 [email protected]