Weak type (p,p) estimates for Riesz transforms

Blunck, Sönke; Kunstmann, Peer Christian

doi:10.1007/s00209-003-0627-7

Cited by 50 publications

(69 citation statements)

References 12 publications

(5 reference statements)

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“…We learned during the preparation of this manuscript that the L q boundedness of the Riesz transforms ∇L − 1 2 on the same range of q has also been obtained independently by S. Blunck and P. C. Kuntsmann [BK1], by essentially the same method as ours. Moreover, they have applied this technique to related matters, including L p estimates for Riesz transforms associated to higher order elliptic operators [BK1], as well as to the existence of an H ∞ functional calculus [BK2] in L p spaces. We are grateful to Pascal Auscher and Alan McIntosh for bringing their work to our attention.…”

Section: Introductionsupporting

confidence: 65%

$L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators

Hofmann¹,

Martell²

2003

Publicacions Matemàtiques

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We consider the Riesz transforms ∇L −1/2 , where L ≡− divA(x)∇, and A is an accretive, n × n matrix with bounded measurable complex entries, defined on R n . We establish boundedness of these operators on L p (R n ), for the range pn < p ≤ 2, where pn = 2n/(n + 2), n ≥ 2, and we obtain a weak-type estimate at the endpoint pn. The case p = 2 was already known: it is equivalent to the solution of the square root problem of T. Kato.

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Section: Introductionsupporting

confidence: 65%

$L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators

Hofmann¹,

Martell²

2003

Publicacions Matemàtiques

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“…Finally, following [AMR], in Section 9 we define H p L for p > 1 (which may be strictly smaller than L p for some p), and we show that these belong to a complex interpolation scale that includes L 2 and H 1 L . We conclude this introduction by remarking that this work, as well as the earlier cited papers [DY1], [DY2], [AMR], and [HM], can in some sense be viewed as a companion to the L p theory developed for general classes of operators in [BK1], [BK2], [HMa] and [Au].…”

mentioning

confidence: 79%

Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

et al. 2011

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Abstract. Let X be a metric space with doubling measure, and L be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characterization, duality of Hardy and BMO spaces. Further specializing to the case that L is a Schrödinger operator on R n with a non-negative, locally integrable potential, we establish addition characterizations of such Hardy space in terms of maximal functions. Finally, we define Hardy spaces H p L (X) for p > 1, which may or may not coincide with the space L p (X), and show that they interpolate with H 1 L (X) spaces by the complex method.

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“…Indeed, we know from [6] that the Riesz transform is bounded in L 2 (R n ). In fact, we also know that there exists p L , 1 ≤ p L < 2n/(n + 2), such that the Riesz transform is bounded in L p (R n ) for p L < p ≤ 2 (see [21], [11], [3], [9], [10] and §2 for more details; see also [12] for related theory). If, in addition, it mapped H 1 (R n ) to L 1 (R n ), then by interpolation ∇L −1/2 : L p (R n ) → L p (R n ) for all 1 < p ≤ 2, which, in general, is not true (i.e., the best possible p L can be strictly greater than 1).…”

mentioning

confidence: 99%