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Spaces of Bounded Λ-Central Mean Oscillation, Morrey Spaces, and Λ-Central Carleson Measures

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Spaces of Bounded Λ-Central Mean Oscillation, Morrey Spaces, and Λ-Central Carleson Measures Collect. Math. 51, 1 (2000), 1–47 c 2000 Universitat de Barcelona Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures Josefina Alvarez and Joseph Lakey Department of Mathematics, New Mexico State University, Las Cruces, NM 88003, USA E-mail: [email protected] [email protected] Martha Guzman-Partida´ ∗ Instituto de Matematicas,´ Universidad Nacional Autonoma´ de Mexico,´ Ciudad Universitaria, Mexico,´ D.F., 04510, Mexico´ E-mail: [email protected] Received August 23, 1998. Revised March 5, 1999 Abstract We prove that the Morrey space is contained in the space CMOq of functions with bounded central mean oscillation, explaining in this way the dependence of CMOq on q. We also define a natural extension of CMO2, to prove the existence of a continuous bijection with central Carleson measures of order λ. This connection is further studied to extend and refine duality results involving tent spaces and Hardy spaces associated with Herz-type spaces. Finally, we prove continuity results on these spaces for general non-convolution singular integral operators, including pseudo-differential operators with symbols in the m H¨ormander class Sρ,δ , ρ<1, and linear commutators. ∗ This work was done while the third named author was a post-doctoral fellow at New Mexico State University supported by UNAM, M´exico. 1 2 Alvarez, Guzman-Partida´ and Lakey 1. Introduction In his monumental papers, “Generalized Harmonic Analysis” [31, pp. 148, 159], and “Tauberian Theorems” [32, pp. 80, 91], Norbert Wiener looked for ways other than the O and o symbols to describe the behavior of a function at infinity. To this effect, he considered several alternatives, for instance 1 T |f(x)|2 dx is bounded for large T, T 0 1 T lim |f(x)|2 dx =0, T →∞ T − T T 1 | | ∈ 1−α f(x) dx is bounded for some α (0, 1) and every T>0 . T 0 Wiener also observed that these conditions are related to appropriate weighted Lq spaces. A. Beurling [5] extended these ideas looking for a more general setting in which the Wiener-L´evy Theorem and the Wiener Approximation Theorem still hold. So, Beurling defined a pair of dual Banach spaces, Aq and Bq ,1/q +1/q =1. More precisely, Aq is a Banach algebra with respect to the convolution, expressible as a union of weighted Lq spaces. The space Bq is expressible as the intersection of weighted Lq spaces, or equivalently, it is the space of functions bounded in the central mean of order q. C. Herz [18] further generalized the space Aq, into the space Ap,q, depending on a second parameter p. H. Feichtinger [13] observed that the space Bq can be described by the condition −kn/q ∞ f Bq = sup 2 fχk q < (1.1) k≥0 { ∈ Rn | |≤ } where χ0 is the characteristic function of the unit ball x : x 1 , χk is the characteristic function of the annulus x ∈ Rn :2k−1 < |x|≤2k , k =1, 2, 3, ... and · q q is the norm in L . Actually, this observation is a special case of much earlier results of J. Gilbert [16]. By duality, the space Aq, appropriately called now the Beurling algebra, can be described by the condition ∞ kn/q ∞ f Aq = 2 fχk q < .(1.2) k=0 Spaces of bounded λ-central mean oscillation 3 Chen and Lau [9] (see also Garc´ıa-Cuerva [14]) introduced an atomic space HAq associated with the Beurling algebra Aq, and identified its dual as the space CMOq defined by the condition 1/q 1 | − |q ∞ sup | | f fR dx < (1.3) R≥1 B(0,R) B(0,R) { ∈ Rn | | } | | where B (0,R)= x : x <R , B(0,R) is the measure of B(0,R) and fR = 1 q q |B(0,R)| B(0,R) f(x)dx. It is clear that CMO contains both BMO and B . Chen and Lau [9] showed by means of a counterexample that CMOq depends on q. This observation establishes a dramatic difference between CMOq and BMO. On the other hand, both spaces are duals of atomic spaces. Furthermore, Chen and Lau [9] and Lu and Yang [25] showed that functions in CMO2 are related to a notion of central Carleson measure in the same way that functions in BMO are related to classical Carleson measures. So, CMOq exhibits both similarities and differences with respect to the space BMO. In recent years, several authors have extended and studied these and other related spaces and have considered the action of various operators on them. We mention, for instance, the work of J. Garc´ıa-Cuerva [14], J. Garc´ıa-Cuerva and M. J. Herrero [15], S. Lu and D. Yang [24], [25], [26], [27], X. Li and D. Yang [23], J. Lakey [22], E. Hern´andez, G. Weiss and D. Yang [21], L. Grafakos, X. Li, and D. Yang [17]. The main purpose of our paper is to make precise and extend several properties pertaining to the spaces Bq, CMOq, and HAq. In Section 3 we prove that the Morrey space is contained in Bq. This obser- vation explains why CMOq depends on q [9]. In fact, it was proved in [1] that the distribution function of functions in the Morrey space, in general, decays at infinity not better than t−α for some appropriate exponent α>0. Thus, from this point of view, we can say that Bq and BMO are roughly the bad part and the good part of CMOq. In Section 4 we investigate duality results at two levels. Downstairs, that is Rn p,α to say in , we consider Herz-type spaces Aq and their associated Hardy spaces p,α HAq . In this context, we need to define the following extension of (1.3), 1/2 1 sup inf |f(x) − c|2 dx < ∞ (1.4) ∈C 1+2λ R≥1 c |B(0,R)| B(0,R) for 0 <λ<1/n. When λ ≥ 1/n we will subtract higher degree polynomials, and when λ<0, we do not need any polynomial correction. We prove that (1.4) 4 Alvarez, Guzman-Partida´ and Lakey 2,λ p,α ≤ − describes the dual CMO of HA2 for 0 <p 1, λ = α 1/2, and analogously for the homogeneous version using supR>0. The case p = 1 provides the Banach space end point. Rn+1 Upstairs, that is to say, in + , we introduce central Carleson measures of type λ in the sense of Amar and Bonami [4]. We consider both the continuous and the discrete case and use classical techniques and wavelet decomposition techniques, respectively, to obtain in each case the correspondence between the measures and functions in CMO2,λ. Our proofs apply in particular to obtain the classical corre- spondence between Carleson measures and BMO functions. We remark that the results in Section 4 apply to both the homogeneous and inhomogeneous cases of the spaces we consider. These results are related to the work of Hern´andez, Weiss, and Yang [21]. The connections will be made precise in Section 4. In Section 5we study the action of non-convolution singular integral operators on inhomogeneous spaces. The main observation is that the inhomogeneous nature of the space, allows for the continuous action of operators associated to kernels more singular than standard kernels in the sense of Coifman and Meyer [8]. The results we obtain extend and refine previous work of Garc´ıa-Cuerva [14] and Lu and Yang [24]. In Section 2, we collect preliminary definitions and results. The paper ends with a list of references. The notation used in this paper is standard. It may be useful to point out that given 1 ≤ q ≤∞, q will denote the conjugate exponent, 1/q +1/q = 1. The ∞ S D q q p symbols C0 , , , L , Lloc, l , etc. will indicate the usual spaces of distributions, Rn sequences, or functions defined on , with complex values. Moreover, f q will denote the Lq norm of the function f. D Rn × Rn q Rn+1 When a particular setting is needed, we will write ( ), Lloc + , q | | L (µ), f q Rn+1 , as needed. With A we will denote the Lebesgue measure of L ( + ) a measurable set A, and B(z,R) will be a ball centered at z with radius R. With χB we will denote the characteristic function of a set B. As usual, the letter C will indicate an absolute constant, probably different at different occurrences. Other notations will be introduced at the appropriate time. 2. Preliminary definitions and results Definition 2.1. Given λ<1/n,1<q<∞ we define the space CMOq,λ by the condition 1/q 1 | − |q ∞ f CMOq,λ = sup 1+λq f(x) fR dx < (2.1) R≥1 |B(0,R)| B(0,R) 1 where fR = |B(0,R)| B(0,R) f(x)dx. Spaces of bounded λ-central mean oscillation 5 CMOq,λ becomes a Banach space if we identify functions that differ in a con- stant. It is easy to see that (2.1) is equivalent to the condition 1/q 1 sup inf |f(x) − c|q dx < ∞ . ∈C 1+λq R≥1 c |B(0,R)| B(0,R) Definition 2.2. Given λ ∈ R,1<q<∞, we define the space Bq,λ by the condition 1/q 1 | |q ∞ f Bq,λ = sup 1+λq f(x) dx < . (2.2) R≥1 |B(0,R)| B(0,R) Bq,λ is a Banach space continuously included in CMOq,λ. When λ = 0 we obtain the spaces CMOq and Bq defined in the introduction.
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