Weighted Hardy spaces associated with elliptic operators. Part I: Weighted norm inequalities for conical square functions

Martell, José María; Prisuelos-Arribas, Cruz

doi:10.1090/tran/6768

Cited by 34 publications

(46 citation statements)

References 37 publications

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“…This will be crucial in the proof of Theorem 3.1. When w ≡ 1 this was proved in [28,Proposition 3.34] for a general p 0 (see also [11,Theorem 3] for the case p 0 = 2 and w, v ≡ 1).…”

Section: 3mentioning

confidence: 84%

Conical square functions for degenerate elliptic operators

Chen

Martell

Prisuelos-Arribas

2017

Advances in Calculus of Variations

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Abstract. The aim of this paper is to study the boundedness of different conical square functions that arise naturally from second order divergence form degenerate elliptic operators. More precisely, let L w = −w −1 div(w A ∇) where w ∈ A 2 and A is an n × n bounded, complex-valued, uniformly elliptic matrix. D. Cruz-Uribe and C. Rios solved the L 2 (w)-Kato square root problem obtaining that √ L w is equivalent to the gradient on L 2 (w). The same authors in collaboration with the second named author of this paper studied the L p (w)-boundedness of operators that are naturally associated with L w , such as the functional calculus, Riesz transforms, or vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in L p (vdw) for v ∈ A ∞ (w)), and in particular a class of "degeneracy" weights w was found in such a way that the classical L 2 -Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on L p (w) and on L p (vdw), with v ∈ A ∞ (w), of the conical square functions that one can construct using the heat or Poisson semigroup associated with L w . As a consequence of our methods, we find a class of degeneracy weights w for which L 2 -estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with L w .

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“…This will be crucial in the proof of Theorem 3.1. When w ≡ 1 this was proved in [28,Proposition 3.34] for a general p 0 (see also [11,Theorem 3] for the case p 0 = 2 and w, v ≡ 1).…”

Section: 3mentioning

confidence: 84%

Conical square functions for degenerate elliptic operators

Chen

Martell

Prisuelos-Arribas

2017

Advances in Calculus of Variations

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“…recall the definition of G 2 j+3 m−1,H in (2.13) and (2.15). Then, for every 0 < p < ∞ and w ∈ A ∞ , taking the L p (w) norm in both sides of the previous inequality and applying change of angles (see [23,Proposition 3.2]), we conclude that…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…Some of the ingredients that are crucial in the present work are taken from the first part of this series of papers [23] (see also [4]), where we already obtained optimal ranges for the weighted norm inequalities satisfied by the heat and Poisson conical square functions associated with the elliptic operator. Here, we need to obtain analogous results for the non-tangential maximal functions associated with the heat and Poisson semigroups (see Section 7).…”

Section: Introductionmentioning

confidence: 99%

Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H^1_L (w)$

Martell¹,

Prisuelos-Arribas²

2018

Publicacions Matemàtiques

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Given a Muckenhoupt weight w and a second order divergence form elliptic operator L, we consider different versions of the weighted Hardy space H 1 L (w) defined by conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by L. We show that all of them are isomorphic and also that H 1 L (w) admits a molecular characterization. One of the advantages of our methods is that our assumptions extend naturally the unweighted theory developed by S. Hofmann and S. Mayboroda in [19] and we can immediately recover the unweighted case. Some of our tools consist in establishing weighted norm inequalities for the non-tangential maximal functions, as well as comparing them with some conical square functions in weighted Lebesgue spaces.

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“…In order to handle such beyond Calderón-Zygmund operators, many researchers study Hardy spaces that are adapted to a linear operator L. For some results on the one-parameter Hardy spaces adapted to an operator, we refer the reader to [3], [20], [7], [25], [19], [26], [24], [9], [16], [38], [28] and their references. For the theory of weighted Hardy spaces associated with an operator, we refer the reader to a series of papers [1], [39], [31], [33], [34].…”

Section: Introductionmentioning

confidence: 99%