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17
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Riesz transform, Gaussian bounds and the method of wave equation
 Math. Z
"... Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We al ..."
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Cited by 20 (1 self)
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Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the LaplaceBeltrami operator on Riemannian manifolds for p> 2. 1.
Plancherel type estimates and sharp spectral multipliers
 J. FUNCT. ANAL
, 2002
"... We study general spectral multiplier theorems for selfadjoint positive definite operators on L²(X, µ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmandertype spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of ..."
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Cited by 7 (0 self)
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We study general spectral multiplier theorems for selfadjoint positive definite operators on L²(X, µ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmandertype spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmandertype spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R³ and new spectral multiplier theorems for the Laguerre and Hermite expansions.
Maximal operator for multilinear singular integrals with nonsmooth kernels
 Indiana Univ. Math. J
"... Abstract. In this article we prove Cotlar’s inequality for the maximal singular integrals associated with operators whose kernels satisfy regularity conditions weaker than those of the standard mlinear CalderónZygmund kernels. The present study is motivated by the fundamental example of the maxima ..."
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Cited by 1 (1 self)
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Abstract. In this article we prove Cotlar’s inequality for the maximal singular integrals associated with operators whose kernels satisfy regularity conditions weaker than those of the standard mlinear CalderónZygmund kernels. The present study is motivated by the fundamental example of the maximal mth order Calderón commutators whose kernels are not regular enough to fall under the scope of the mlinear CalderónZygmund theory; the Cotlar inequality is a new result even for these operators.
COMPARISON OF THE CLASSICAL BMO WITH THE BMO SPACES ASSOCIATED WITH OPERATORS AND APPLICATIONS
, 2006
"... Abstract. Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMOL space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMOL spaces in the theory of singular integration such as BMOL est ..."
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Cited by 1 (0 self)
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Abstract. Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMOL space associated with L which was introduced recently by Duong and Yan. We discuss applications of the new BMOL spaces in the theory of singular integration such as BMOL estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space BMOL might coincide with or might be essentially different from the classical BMO space. 1.
Predual Spaces of Banach Completions of OrliczHardy Spaces Associated with Operators
, 906
"... Abstract. Let L be a linear operator in L 2 (R n) and generate an analytic semigroup {e −tL}t≥0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞]. Let ω on (0, ∞) be of upper type 1 and of critical lower type ˜p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(t) = t − ..."
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Cited by 1 (1 self)
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Abstract. Let L be a linear operator in L 2 (R n) and generate an analytic semigroup {e −tL}t≥0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞]. Let ω on (0, ∞) be of upper type 1 and of critical lower type ˜p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(t) = t −1 /ω −1 (t −1) for t ∈ (0, ∞). In this paper, the authors first introduce the VMOtype space VMOρ,L(R n) and the tent space T ∞ ω,v(R n+1 and characterize the space VMOρ,L(Rn) via the space T ∞ ω,v (Rn+1 +). Let ˜ Tω(R n+1 +) be the Banach completion of the tent space Tω(R n+1 +). The authors then prove that ˜ Tω(R n+1). As an application of this, the authors finally show is the dual space of T ∞ ω,v(R n+1 that the dual space of VMOρ,L ∗(Rn) is the space Bω,L(R n), where L ∗ denotes the adjoint operator of L in L 2 (R n) and Bω,L(R n) the Banach completion of the OrliczHardy space Hω,L(R n). These results generalize the known recent results by particularly taking ω(t) = t for t ∈ (0, ∞). 1
Multiple Weighted Norm Inequalities for Maximal Multilinear Singular Integrals with NonSmooth Kernels
"... Abstract Weighted norm inequalities for maximal truncated operators of multilinear singular integrals with nonsmooth kernels in the sense of Duong, Grafakos, and Yan are obtained; this class of operators extends the class of multilinear CalderónZygmund operators introduced by Coifman and Meyer and ..."
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Abstract Weighted norm inequalities for maximal truncated operators of multilinear singular integrals with nonsmooth kernels in the sense of Duong, Grafakos, and Yan are obtained; this class of operators extends the class of multilinear CalderónZygmund operators introduced by Coifman and Meyer and includes the higher order commutators of Calderón. The weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of Lerner, Ombrosi, Pérez, Torres, and TrujilloGonzález. The key ingredient in the proof is the introduction of a new multisublinear maximal operator that plays the role of the HardyLittlewood maximal function in a version of Cotlar’s inequality. As applications of these results, new weighted estimates for the mth order Calderón commutators and their maximal counterparts are deduced. 1
MULTILINEAR HARMONIC ANALYSIS
, 2011
"... This article contains an expanded version of the material covered by ..."
SPECTRAL MULTIPLIERS FOR SCHRÖDINGER OPERATORS WITH PÖSCHLTELLER POTENTIAL
, 2008
"... Abstract. We prove a sharp MihlinHörmander multiplier theorem for Schrödinger operators H on R n. The method, which allows us to deal with general potentials, improves Hebisch’s method relying on heat kernel estimates for positive potentials [22, 12]. Our result applies to, in particular, the negat ..."
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Abstract. We prove a sharp MihlinHörmander multiplier theorem for Schrödinger operators H on R n. The method, which allows us to deal with general potentials, improves Hebisch’s method relying on heat kernel estimates for positive potentials [22, 12]. Our result applies to, in particular, the negative PöschlTeller potential V (x) = −ν(ν + 1)sech 2 x, ν ∈ N, for which H has a resonance at zero. 1.
OLD AND NEW MORREY SPACES VIA HEAT KERNEL BOUNDS
, 2006
"... Abstract. Given p ∈ [1, ∞) and λ ∈ (0, n), we study Morrey space Lp,λ (Rn) of all locally integrable complexvalued functions f on Rn such that for every open Euclidean ball B ⊂ Rn with radius rB there are numbers C = C(f) (depending on f) and c = c(f, B) (relying upon f and B) satisfying r −λ ..."
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Abstract. Given p ∈ [1, ∞) and λ ∈ (0, n), we study Morrey space Lp,λ (Rn) of all locally integrable complexvalued functions f on Rn such that for every open Euclidean ball B ⊂ Rn with radius rB there are numbers C = C(f) (depending on f) and c = c(f, B) (relying upon f and B) satisfying r −λ