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**1 - 8**of**8**### unknown title

, 903

"... Abstract. Let X be a space of homogeneous type in the sense of Coifman and Weiss and D a collection of balls in X. The authors introduce the localized atomic Hardy p, q space HD (X) with p ∈ (0, 1] and q ∈ [1, ∞] ∩ (p, ∞], the localized Morrey-Campanato α, p space ED (X) and the localized Morrey-Ca ..."

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Abstract. Let X be a space of homogeneous type in the sense of Coifman and Weiss and D a collection of balls in X. The authors introduce the localized atomic Hardy p, q space HD (X) with p ∈ (0, 1] and q ∈ [1, ∞] ∩ (p, ∞], the localized Morrey-Campanato α, p space ED (X) and the localized Morrey-Campanato-BLO space ˜α, p

### Localized BMO and BLO Spaces on RD-Spaces and Applications to Schrödinger Operators

, 903

"... Abstract. An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in X. Let ρ be an admissible function on RD-space X. The authors first introduce the localized spaces BMOρ(X) and BLOρ(X) and establish their ..."

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Abstract. An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in X. Let ρ be an admissible function on RD-space X. The authors first introduce the localized spaces BMOρ(X) and BLOρ(X) and establish their basic properties, including the John-Nirenberg inequality for BMOρ(X), several equivalent characterizations for BLOρ(X), and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal functions and their localized versions associated to ρ, and the Littlewood-Paley g-function associated to ρ, where the Littlewood-Paley g-function and some of the radial maximal functions are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on R d, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups. 1

### Functions on Localized BMO Spaces over Doubling Metric Measure Spaces Boundedness of Lusin-area and g ∗ λ

, 2009

"... Abstract. Let X be a doubling metric measure space. In this paper, the authors introduce the notions of Property ( ˜ M) and Property (P) of X, prove that Property ( ˜ M) implies Property (P) and give some equivalent characterizations of Property ( ˜ M) and Property (P). If X has Property (P), the ..."

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Abstract. Let X be a doubling metric measure space. In this paper, the authors introduce the notions of Property ( ˜ M) and Property (P) of X, prove that Property ( ˜ M) implies Property (P) and give some equivalent characterizations of Property ( ˜ M) and Property (P). If X has Property (P), the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, from localized spaces BMOρ(X) to BLOρ(X) without invoking any regularity of considered kernels. The same is true for the g ∗ λ function and, moreover, unlike the Lusin-area function, in this case, X is not necessary to have Property (P). These results are new even on Rd with the Lebesgue measure and the Heisenberg group, and apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on Rd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups. Moreover, via some results on pointwise multipliers of bmo(R), the authors construct a counterexample to show that there exists a nonnegative function which is in bmo(R), but not in blo(R). 1

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"... The multilinear Calderón–Zygmund theory is developed in the setting of RD-spaces, namely, spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón–Zygmund theory in this context is also developed in this work. The bilinear T ..."

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The multilinear Calderón–Zygmund theory is developed in the setting of RD-spaces, namely, spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón–Zygmund theory in this context is also developed in this work. The bilinear T 1-theorems for Besov and Triebel–Lizorkin spaces in the full range of exponents are among the main results obtained. Multilinear vector-valued T 1 type theorems on Lebesgue spaces, Besov spaces, and Triebel–Lizorkin spaces are also proved. Applications include the boundedness of paraproducts and bilinear multiplier operators on products of Besov and Triebel–Lizorkin spaces.

### unknown title

"... 2. Real analysis on spaces of homogeneous type........................................... 8 2.1. Spaces of homogeneous type and RD-spaces....................................... 8 2.2. Dyadic cubes, covering lemmas, and the Calderón–Zygmund decomposition........ 10 ..."

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2. Real analysis on spaces of homogeneous type........................................... 8 2.1. Spaces of homogeneous type and RD-spaces....................................... 8 2.2. Dyadic cubes, covering lemmas, and the Calderón–Zygmund decomposition........ 10

### HARDY SPACES, COMMUTATORS OF SINGULAR INTEGRAL OPERATORS RELATED TO SCHRÖDINGER OPERATORS AND APPLICATIONS

, 2012

"... Abstract. Let L = −∆+V be a Schrödinger operator on R d, d ≥ 3, where V is a nonnegative function, V = 0, and belongs to the reverse Hölder class RH d/2. The purpose of this paper is three-fold. First, we prove a version of the classical theorem of Jones and Journé on weak ∗-convergence in H 1 L (R ..."

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Abstract. Let L = −∆+V be a Schrödinger operator on R d, d ≥ 3, where V is a nonnegative function, V = 0, and belongs to the reverse Hölder class RH d/2. The purpose of this paper is three-fold. First, we prove a version of the classical theorem of Jones and Journé on weak ∗-convergence in H 1 L (Rd). Secondly, we give a bilinear decomposition for the product space H 1 L (Rd)×BMOL(R d). Finally, we study the commutators [b,T] for T belongs to a class KL of sublinear operators containing almost all fundamental operators in harmonic analysis related to L. More precisely, when T ∈ KL, we prove that there exists a bounded subbilinear operator R = RT: H 1 L (Rd)×BMO(R d) → L 1 (R d) such that (1) |T(S(f,b))|−R(f,b) ≤ |[b,T](f) | ≤ R(f,b)+|T(S(f,b))|, where S is a bounded bilinear operator from H 1 L (Rd) × BMO(R d) into L 1 (R d) which does not depend on T. In the particular case of the Riesz transforms Rj = ∂xjL −1/2, j = 1,...,d, and b ∈ BMO(R d), we prove that the commutators [b,Rj] are bounded on H 1 L (Rd) iff b ∈ BMO log L (Rd) – a new space of BMO type, which coincides with the space LMO(Rd) when L = −∆+1. Furthermore, d∑

### ON WEAK ∗-CONVERGENCE IN H 1 L (Rd)

"... Abstract. LetL = −∆+V beaSchrödingeroperatoronR d, d ≥ 3, where V is a nonnegative function, V = 0, and belongs to the reverse Hölder class RH d/2. In this paper, we prove a version of the classical theorem of Jones and Journé on weak ∗-convergence in the Hardy space H 1 L (Rd). hal-00696432, versi ..."

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Abstract. LetL = −∆+V beaSchrödingeroperatoronR d, d ≥ 3, where V is a nonnegative function, V = 0, and belongs to the reverse Hölder class RH d/2. In this paper, we prove a version of the classical theorem of Jones and Journé on weak ∗-convergence in the Hardy space H 1 L (Rd). hal-00696432, version 1- 11 May 2012 1.