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31
New maximal functions and multiple weights for the multilinear CalderónZygmund theory
 MATH
, 2010
"... A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller that the mfold product of the HardyLittlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of CalderónZygmund type and to ..."
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Cited by 53 (5 self)
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A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller that the mfold product of the HardyLittlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of CalderónZygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear CalderónZygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp endpoint estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.
On multilinear singular integrals of CalderónZygmund type
, 2011
"... A variety of results regarding multilinear CalderónZygmund singular integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discret ..."
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Cited by 19 (4 self)
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A variety of results regarding multilinear CalderónZygmund singular integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur’s test, and a multilinear version of the T1 Theorem suitable for the study of multilinear pseudodifferential and translation invariant operators. A maximal operator associated with multilinear singular integrals is also introduced and employed to obtain weighted norm inequalities.
Multilinear CalderonZygmund operators on Hardy spaces
 Collect. Math
"... Abstract. It is shown that multilinear CalderónZygmund operators are bounded on products of Hardy spaces. 1. ..."
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Cited by 14 (2 self)
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Abstract. It is shown that multilinear CalderónZygmund operators are bounded on products of Hardy spaces. 1.
THE MULTILINEAR STRONG MAXIMAL FUNCTION
"... Abstract. A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of ..."
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Cited by 12 (4 self)
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Abstract. A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Bilinear interpolation results between distributional estimates, such as those satisfied by the multivariable strong maximal function, are also proved. 1.
Endpoints estimates for iterated commutators of multilinear singular integrals
 Bull. London Math. Soc
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Use of abstract Hardy spaces, Real interpolation and Applications to bilinear operators.
, 2008
"... This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H 1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarif ..."
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Cited by 11 (8 self)
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This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H 1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarify the use of Hardy spaces. Then with the help of the bilinear interpolation theory, we then give applications to study bilinear operators on Lebesgue spaces. These ideas permit us to study singular operators with singularities similar to those of bilinear CalderónZygmund operators in a far more abstract framework as in the euclidean case.
ON THE BOUNDEDNESS OF BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES
"... Abstract. We prove mapping properties of the form T: B ˙ α1,q1 p1 × L p2 → B ˙ α2,q2 p3 and T: B ˙ α1,q1 p1 × ˙ B α2,q2 p2 → Lp3, for certain related indices p1, p2, p3, q1, q2, α1, α2 ∈ R, where T is a bilinear HörmanderMihlin multiplier or a molecular paraproduct. Applications to bilinear Little ..."
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Cited by 8 (3 self)
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Abstract. We prove mapping properties of the form T: B ˙ α1,q1 p1 × L p2 → B ˙ α2,q2 p3 and T: B ˙ α1,q1 p1 × ˙ B α2,q2 p2 → Lp3, for certain related indices p1, p2, p3, q1, q2, α1, α2 ∈ R, where T is a bilinear HörmanderMihlin multiplier or a molecular paraproduct. Applications to bilinear LittlewoodPaley theory are discussed. 1.
HIGHER ORDER ENTROPIES FOR COMPRESSIBLE FLUID MODELS
, 2007
"... We investigate higher order entropies for compressible fluid models and related a priori estimates. Higher order entropies are kinetic entropy estimators suggested by Enskog expansion of Boltzmann entropy. These quantities are quadratic in the density ρ, velocity v, and temperature T renormalized de ..."
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Cited by 5 (5 self)
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We investigate higher order entropies for compressible fluid models and related a priori estimates. Higher order entropies are kinetic entropy estimators suggested by Enskog expansion of Boltzmann entropy. These quantities are quadratic in the density ρ, velocity v, and temperature T renormalized derivatives. We investigate governing equations of higher order entropy correctors and related differential inequalities in the natural situation where the volume viscosity, the shear viscosity, and the thermal conductivity depend on temperature, essentially in the form T κ. Entropic inequalities are established when ‖log ρ‖BMO, ‖v / √ T ‖L∞, ‖ log T ‖BMO, ‖h∂xρ/ρ‖L∞, ‖h∂xv / √ T ‖L∞, ‖h∂xT/T ‖L∞, and ‖h2 ∂ 2 xT/T ‖L ∞ are small enough, where h = 1/ρT 1 2 −κ is a weight associated with the dependence of the local mean free path on density and temperature. As an example of application, we investigate global existence of solutions when the initial values log(ρ0/ρ∞), v0 / √ T0, and log(T0/T∞) are small enough in appropriate spaces.
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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Cited by 4 (3 self)
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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