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140
On Leray's SelfSimilar Solutions Of The NavierStokes Equations Satisfying Local Energy Estimates
 Arch. Rational Mech. Anal
, 1998
"... . This paper proves that Leray's selfsimilar solutions of the threedimensional NavierStokes equations must be trivial under very general assumptions, for example, if they satisfy local energy estimates. 1. Introduction In 1934 Leray [Le] raised the question of the existence of selfsimilar s ..."
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Cited by 25 (1 self)
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. This paper proves that Leray's selfsimilar solutions of the threedimensional NavierStokes equations must be trivial under very general assumptions, for example, if they satisfy local energy estimates. 1. Introduction In 1934 Leray [Le] raised the question of the existence of selfsimilar solutions of the NavierStokes equations. For a long time, the selfsimilar solutions had appeared to be a good candidate for constructing singular solutions of the NavierStokes equations. Leray's question was unanswered until 1995, when Necas, Ruzicka, and Sver'ak [NRS] showed, among other things, that the only selfsimilar solution satisfying the global energy estimates is zero. Although they answered Leray's original problem, some important questions were left open. For example, can a selfsimilar solution satisfying local energy estimates exist? The goal of this paper is to show that the selfsimilar solutions must be zero under very general assumptions, for example, if they satisfy the local...
The Marcinkiewicz multiplier condition for bilinear operators
 Studia Math. 146 (2001), 115–156. LOUKAS GRAFAKOS
"... Abstract. This article is concerned with the question of whether Marcinkiewicz multipliers on R2n give rise to bilinear multipliers on Rn × Rn.We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions ..."
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Cited by 23 (7 self)
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Abstract. This article is concerned with the question of whether Marcinkiewicz multipliers on R2n give rise to bilinear multipliers on Rn × Rn.We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces. 1.
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 21 (7 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
A new approach to absolute continuity of elliptic measure, with applications to nonsymmetric equations
 Adv. in Math
"... In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc ..."
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Cited by 14 (3 self)
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In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc
Hardy spaces of differential forms on Riemannian manifolds
, 2006
"... Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H pboundedness for Riesz transfo ..."
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Cited by 13 (2 self)
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Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H pboundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ∞ functional calculus and Hodge decomposition, are given.
Hardy spaces and divergence operators on strongly Lipschitz domain
 of R n , J. Funct. Anal
"... Let Ω be a strongly Lipschitz domain of R n. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the nontangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L 1. Under su ..."
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Cited by 10 (2 self)
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Let Ω be a strongly Lipschitz domain of R n. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the nontangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L 1. Under suitable assumptions on L, we identify this maximal Hardy space with atomic Hardy spaces, namely with H 1 (R n) if Ω = R n, H 1 r(Ω) under the Dirichlet boundary condition, and H1 z (Ω) under the Neumann boundary condition. In particular, we obtain a new proof of the atomic decomposition for H1 z (Ω). A version for local Hardy spaces is also given. We also present an overview of the theory of Hardy spaces and BMO spaces on Lipschitz domains with proofs.
L p regularity of averages over curves and bounds for associated maximal operators
 Amer. J. Math
"... Abstract. We prove that for a finite type curve in R3 the maximal operator generated by dilations is bounded on Lp for sufficiently large p. We also show the endpoint Lp → L p regularity result for the averaging operators for large 1/p p. The proofs make use of a deep result of Thomas Wolff about de ..."
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Cited by 10 (8 self)
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Abstract. We prove that for a finite type curve in R3 the maximal operator generated by dilations is bounded on Lp for sufficiently large p. We also show the endpoint Lp → L p regularity result for the averaging operators for large 1/p p. The proofs make use of a deep result of Thomas Wolff about decompositions of cone multipliers. 1.
Hardy space estimates for multilinear operators
 I, Revista Mat. Iberoamericana
, 1992
"... Abstract. We continue the study of multilinear operators given by products of finite vectors of CalderónZygmund operators. We determine the set of all r ≤ 1 for which these operators map products of Lebesgue spaces L p (R n) into the Hardy spaces H r (R n). At the endpoint case r = n/n + m + 1, whe ..."
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Cited by 9 (6 self)
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Abstract. We continue the study of multilinear operators given by products of finite vectors of CalderónZygmund operators. We determine the set of all r ≤ 1 for which these operators map products of Lebesgue spaces L p (R n) into the Hardy spaces H r (R n). At the endpoint case r = n/n + m + 1, where m is the highest vanishing moment of the multilinear operator, we prove a weak type result. A well known by now theorem of P.L. Lions says that the determinant of the Jacobian of a function from Rn → Rn maps the product of Sobolev spaces Ln 1 × · · · × Ln 1 into the Hardy space H1. Coifman, Lions, Meyer and Semmes, [CLMS], went below H1 by showing that for p, q> 1, the Jacobiandeterminant maps L p
Localized Hardy spaces H 1 related to admissible functions on RDspaces and applications to Schrödinger operators
"... Abstract. Let X be an RDspace, which means that X is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X. In this paper, the authors first introduce the notion of admissible functions ρ and then develop a theory of lo ..."
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Cited by 8 (6 self)
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Abstract. Let X be an RDspace, which means that X is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X. In this paper, the authors first introduce the notion of admissible functions ρ and then develop a theory of localized Hardy spaces H1 ρ(X) associated with ρ, which includes several maximal function characterizations of H1 ρ (X), the relations between H1 ρ (X) and the classical Hardy space H1 (X) via constructing a kernel function related to ρ, the atomic decomposition characterization of H1 ρ(X), and (X) via a finite atomic the boundedness of certain localized singular integrals on H1 ρ decomposition characterization of some dense subspace of H1 ρ (X). This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on Rn, or the subLaplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrödinger operators considered here are associated with nonnegative potentials satisfying the reverse Hölder inequality. 1