Results 1  10
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14
Variations on the Theme of Journe’s Lemma
, 2004
"... Journé’s Lemma [10] is a critical component of many questions related to the product BMO theory of S.Y. Chang and R. Fefferman. This article presents several different variants of the Lemma, some known, some implicit in the literature, and some new. ..."
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Journé’s Lemma [10] is a critical component of many questions related to the product BMO theory of S.Y. Chang and R. Fefferman. This article presents several different variants of the Lemma, some known, some implicit in the literature, and some new.
Multiparameter Riesz Commutators
"... It is shown that product BMO of S.Y. A. Chang and R. Fefferman, defined on the space R d1 ⊗ · · · ⊗ R dt, can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical oneparameter result of R. Coifman, R. Rochberg, and G. Weiss [8], and at the same tim ..."
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It is shown that product BMO of S.Y. A. Chang and R. Fefferman, defined on the space R d1 ⊗ · · · ⊗ R dt, can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical oneparameter result of R. Coifman, R. Rochberg, and G. Weiss [8], and at the same time extends the work of M. Lacey and S. Ferguson [12] and M. Lacey and E. Terwilleger [19], on multiparameter commutators with Hilbert transforms.
A SHORT PROOF OF THE COIFMANMEYER MULTILINEAR THEOREM
"... Abstract. We give a short proof of the well known CoifmanMeyer theorem on multilinear operators. 1. ..."
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Abstract. We give a short proof of the well known CoifmanMeyer theorem on multilinear operators. 1.
WEIGHTED NORM INEQUALITIES FOR PARAPRODUCTS AND BILINEAR PSEUDODIFFERENTIAL OPERATORS WITH MILD REGULARITY
, 2008
"... Abstract. We establish boundedness properties on products of weighted Lebesgue, Hardy, and amalgam spaces of certain paraproducts and bilinear pseudodifferential operators with mild regularity. We do so by showing that these operators can be realized as generalized bilinear CalderónZygmund operator ..."
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Abstract. We establish boundedness properties on products of weighted Lebesgue, Hardy, and amalgam spaces of certain paraproducts and bilinear pseudodifferential operators with mild regularity. We do so by showing that these operators can be realized as generalized bilinear CalderónZygmund operators. 1. Bilinear pseudodifferential operators with mild regularity Let us motivate our main result on bilinear pseudodifferential operators (ΨDOs) by revisiting some facts from the linear theory. A sufficiently regular function σ(x, ξ) defined on Rn × Rn has an associated ΨDO Tσ defined by Tσ(f)(x) = σ(x, ξ) ˆ f(ξ)e ix·ξ dξ x ∈ R n, f ∈ S(R n). R n Here S(Rn) is the Schwartz class and ˆ f denotes the Fourier transform of f, ˆf(ξ) = e −ix·ξ f(x) dx, ξ ∈ R n. R n For m ∈ R, 0 ≤ δ, ρ ≤ 1, the symbol σ(x, ξ) belongs to Hörmander’s class S m ρ,δ if (1.1)  ∂ α x ∂ β ξ σ(x, ξ)  ≤ Cα,β(1 + ξ) m+δα−ρβ  , x, ξ ∈ R n, where α, β ∈ Zn and α, β  depend on the context. The exploration of classes of smooth symbols, in particular the classes Sm ρ,δ, appears to be predominant in the ΨDO literature. However, as diverse problems in Analysis and PDEs demand, the case in which the symbol has mild or no regularity in x has received
Hörmander type theorems for multilinear and multiparameter Fourier multiplier operators with limited smoothness
 Nonlinear Anal
"... MSC: 42B15 42B25 42B20 Keywords: Multiparameter and multilinear multiplier CoifmanMeyer theorem Hörmander multiplier Minimal smoothness condition LittlewoodPaley's inequality A p weights a b s t r a c t The main purpose of this paper is threefold. First of all, we are concerned with the l ..."
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MSC: 42B15 42B25 42B20 Keywords: Multiparameter and multilinear multiplier CoifmanMeyer theorem Hörmander multiplier Minimal smoothness condition LittlewoodPaley's inequality A p weights a b s t r a c t The main purpose of this paper is threefold. First of all, we are concerned with the limited smoothness conditions in the spirit of Hörmander on the multilinear and multiparameter CoifmanMeyer type Fourier multipliers studied by C. Muscalu, J. Pipher, T. Tao, C. Thiele (2004, 2006) where they established the L r estimates for the multiplier operators under the assumption that the multiplier has smoothness of sufficiently large order. Under our limited smoothness assumption, we will prove the L for 1 < p 1 , . . . , p n < ∞ and 0 < r < ∞. Second, our proof of L r estimates also offers a different and more direct approach than the one given in
Endpoint Estimates and Multiparameter Paraproducts on Higher Dimensional Tori
"... Analogues of multiparameter multiplier operators on R d are defined on the torus T d. It is shown that these operators satisfy the classical CoifmanMeyer theorem. In addition, L logL and L(log L) n endpoint estimates are proved. ..."
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Analogues of multiparameter multiplier operators on R d are defined on the torus T d. It is shown that these operators satisfy the classical CoifmanMeyer theorem. In addition, L logL and L(log L) n endpoint estimates are proved.