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Algebras of singular integral operators with PC coefficients in rearrangementinvariant spaces with Muckenhoupt weights
 J. Operator Theory
"... Abstract. We prove necessary conditions for the Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of ..."
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Abstract. We prove necessary conditions for the Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces L p(·) w (weighted Lebesgue spaces with variable exponent). Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever w is a Khvedelidze weight, and the variable exponent p(t) satisfies the estimate p(τ) − p(t)  ≤ A/( − log τ − t). 1.
New Hardy spaces of MusielakOrlicz type and boundedness of sublinear operators
 Integral Equations Operator Theory
"... Abstract. We introduce a new class of Hardy spaces Hϕ(·,·)(Rn), called Hardy spaces of MusielakOrlicz type, which generalize the HardyOrlicz spaces of Janson and the weighted Hardy spaces of GarćıaCuerva, Strömberg, and Torchinsky. Here, ϕ: Rn × [0,∞) → [0,∞) is a function such that ϕ(x, ·) is ..."
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Abstract. We introduce a new class of Hardy spaces Hϕ(·,·)(Rn), called Hardy spaces of MusielakOrlicz type, which generalize the HardyOrlicz spaces of Janson and the weighted Hardy spaces of GarćıaCuerva, Strömberg, and Torchinsky. Here, ϕ: Rn × [0,∞) → [0,∞) is a function such that ϕ(x, ·) is an Orlicz function and ϕ(·, t) is a MuckenhouptA ∞ weight. A function f belongs to Hϕ(·,·)(Rn) if and only if its maximal function f ∗ is so that x 7 → ϕ(x, f∗(x)) is integrable. Such a space arises naturally for instance in the description of the product of functions in H1(Rn) and BMO(Rn) respectively (see [6]). We characterize these spaces via the grand maximal function and establish their atomic decomposition. We characterize also their dual spaces. The class of pointwise multipliers for BMO(Rn) characterized by Nakai and Yabuta can be seen as the dual of L1(Rn)+H log(Rn) where H log(Rn) is the Hardy space of MusielakOrlicz type related to the MusielakOrlicz function θ(x, t) = t log(e+ x) + log(e+ t) Furthermore, under additional assumption on ϕ(·, ·) we prove that if T is a sublinear operator and maps all atoms into uniformly bounded elements of a quasiBanach space B, then T uniquely extends to a bounded sublinear operator from Hϕ(·,·)(Rn) to B. These results are new even for the classical HardyOrlicz spaces on Rn. 1.
Endpoint estimates for commutators of singular integrals related to Schrödinger operators
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Pointwise Multipliers Of Besov Spaces Of Smoothness Zero And Spaces Of Continuous Functions
, 2000
"... . We characterize the set of pointwise multipliers of the Besov spaces B 0 1;1 and B 0 1;1 . These characterizations are used to obtain regularity results for elliptic partial dierential equations. In addition several counterexamples are provided and the relation of various spaces of continuous ..."
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. We characterize the set of pointwise multipliers of the Besov spaces B 0 1;1 and B 0 1;1 . These characterizations are used to obtain regularity results for elliptic partial dierential equations. In addition several counterexamples are provided and the relation of various spaces of continuous functions to these multiplier classes are studied. 1. Introduction The paper is a rst attempt to describe the set of all pointwise multipliers for Besov spaces on the smoothness level 0. We obtain characterizations of multipliers for B 0 1;1 and B 0 1;1 . We call a function f (or distribution) a multiplier for a function space X, denoted by f 2 M(X), if kf jM(X)k = sup h2X;h6=0 kfhjXk khjXk < 1: Since both f and h may be distributions the denition of the product needs some further considerations, which we postpone. We believe that a study of these multipliers is related to interesting and deep questions in analysis. To support this view we apply our results to elliptic equations...
Functional Calculus on BMO and related spaces
, 2001
"... Abstract: Let f be a Borel measurable function of the complex plane to itself. We consider the nonlinear operator Tf defined by Tf [g] = f ◦ g, when g belongs to a certain subspace X of the space BMO(R n) of functions with bounded mean oscillation on the Euclidean space. In particular, we investiga ..."
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Abstract: Let f be a Borel measurable function of the complex plane to itself. We consider the nonlinear operator Tf defined by Tf [g] = f ◦ g, when g belongs to a certain subspace X of the space BMO(R n) of functions with bounded mean oscillation on the Euclidean space. In particular, we investigate the case in which X is the whole of BMO, the case in which X is the space V MO of functions with vanishing mean oscillation, and the case in which X is the closure in BMO of the smooth functions with compact support. We characterize those f’s for which Tf maps X to itself, those f’s for which Tf is continuous from X to itself, and those f’s for which Tf is differentiable in X. 1 Introduction and main results. In this paper, we characterize those Borel measurable functions f of the complex plane C to itself such that the nonlinear superposition operator Tf defined by Tf [g]: = f ◦ g takes BMO(R n) and several spaces related to BMO(R n) to themselves. Also continuity
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 threefold. 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 threefold. 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∑
Van Ngai HUYNH RAPPORTEURS:
, 2013
"... pour obtenir le grade de: Docteur de l’université d’Orleans ..."