Representation and uniqueness for boundary value elliptic problems via first order systems

Auscher, Pascal; Mourgoglou, Mihalis

doi:10.4171/rmi/1054

Cited by 55 publications

(62 citation statements)

References 47 publications

(58 reference statements)

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“…Now we introduce the Orlicz-slice space and the Orlicz-amalgam space. The former is a generalization of the slice spaces introduced in [6,7], and the latter is a generalization of the classical amalgam space (L p , ℓ q ) defined by N. Wiener in 1926, in the formulation of his generalized harmonic analysis.…”

Section: Orlicz-slice Spacesmentioning

confidence: 99%

“…In particular, E q t (R n ) := (E q 2 ) t (R n ) was introduced in [6]. A subspace (C q r ) t (R n ) of (E q r ) t (R n ) was also introduced in [7] in a way similar to T q r (R n+1 + ) (see also Definition 6.5 below).…”

Section: Introductionmentioning

confidence: 99%

“…They are also special cases of the Wiener-amalgam spaces (see [21]) which were first introduced by N. Wiener and further developed in time-frequency analysis and sampling theory. Properties of slice spaces such as the duality, the atomic decomposition and the interpolation were also clarified in [6,7]. Observe that the Hardy type space (C q r ) t (R n ) [and also T q r (R n+1 + )] was introduced in [7] via atoms and no other real-variable characterizations of these Hardy type spaces are known so far.…”

Section: Introductionmentioning

confidence: 99%

“…Let Φ be an Orlicz function on [0, ∞) and q, t ∈ (0, ∞). Motivated by the aforementioned works, in this article, we first introduce a class of Orlicz-slice spaces, (E q Φ ) t (R n ), which generalize the slice spaces [in this case, Φ(τ) := τ r for any τ ∈ [0, ∞) with r ∈ (0, ∞)] recently defined and studied by Auscher and Mourgoglou [6] (the case r = 2) as well as by Auscher and Prisuelos-Arribas [7]. Based on these Orlicz-slice spaces, we then introduce a new kind of Hardy-type spaces, the Orlicz-slice Hardy spaces (HE q Φ ) t (R n ), via the radial maximal functions.…”

Section: Introductionmentioning

confidence: 99%

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Real-variable characterizations of Orlicz-slice Hardy spaces

et al. 2019

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In this article, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by P. Auscher et al. Based on these Orlicz-slice spaces, the authors introduce a new kind of Hardy type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz-Hardy space of A. Bonami and J. Feuto as well as the Hardy-amalgam space of Z. V. de P. Ablé and J. Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood-Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of δ-type Calderón-Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood-Paley function characterizations, the dual spaces and the boundedness of δ-type Calderón-Zygmund operators on these Hardy-type spaces are also new. )(R n ), where Φ(t) := t log(e+t) for any t ∈ [0, ∞) is an Orlicz function, and applied these Hardy-type spaces to study the linear decomposition of the product of the Hardy space H 1 (R n ) and its dual space BMO (R n ) as well as the local Hardy space h 1 (R n ) and its dual space bmo (R n ). Moreover, very recently, Cao et al. [12] applied h Φ * (R n ) to study the bilinear decomposition of the product of the local Hardy space h 1 (R n ) and its dual space bmo (R n ). Recall that both the Hardy type spaces H Φ * (R n ) and h Φ * (R n ) were defined in [8] via the (local) radial maximal functions, while h Φ * (R n ) in [12] was defined via the local grand maximal function. Moreover, no other real-variable characterizations of both the Hardy type spaces H Φ * (R n ) and h Φ * (R n ) are known so far. On the other hand, recently, to study the classification of weak solutions in the natural classes for the boundary value problems of a t-independent elliptic system in the upper plane, Auscher and Mourgoglou [6] introduced the slice spaces E

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Section: Orlicz-slice Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Real-variable characterizations of Orlicz-slice Hardy spaces

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“…The first one uses the extrapolation method pioneered by Blunck and Kunstmann in [20], and developed by Auscher in [3] to show that the relevant L 2 bounds remain valid in certain intervals (p − , p + ) about 2 which depend on the operator involved. This approach has been mostly developed to study second order differential operators, but has also been adapted to first order operators by Ajiev [1] and by Auscher and Stahlhut in [16,17]. The other approach to L p estimates for the holomorphic functional calculus of Hodge-Dirac operators consists in adapting the entire machinery of [18] to L p .…”

Section: Introductionmentioning

confidence: 99%

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in L P

Frey

McIntosh

Portal

2018

JAMA

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Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L 2 spaces, and allowed for an extension of these estimates to other systems with applications to nonsmooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator, to prove that they have a bounded holomorphic functional calculus in those L p spaces. We also obtain functional calculi results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L 2 extends to L p for all p ∈ (1, ∞), while the restrictions in p come from the operator-theoretic part of the L 2 proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces, and about the relationship between conical and vertical square functions. (2010): 47A60, 47F05, 42B30, 42B37 Contents 1 2 ∀u ∈ R(Γ), Mathematics Subject Classificationwhere Π = Γ + Γ * is a first order differential (Hodge-Dirac) operator with constant coefficients, and Π B = Γ + B 1 Γ * B 2 is a perturbation by L ∞ coefficients B 1 , B 2 . See Section 2

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Introduction and Main Results

Auscher¹,

Egert²

2023

Progress in Mathematics

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Representation and uniqueness for boundary value elliptic problems via first order systems

Cited by 55 publications

References 47 publications

Real-variable characterizations of Orlicz-slice Hardy spaces

Real-variable characterizations of Orlicz-slice Hardy spaces

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in L P

Introduction and Main Results

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