Weak-L ∞ and BMO

Bennett, Colin; DeVore, Ronald A.; Sharpley, Robert

doi:10.2307/2006999

Cited by 176 publications

(199 citation statements)

References 6 publications

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“…It is known that by [17], [1,3], L(∞, q) are defined as the class of all measurable functions f for which f * (t) < ∞ for all t > 0 and for which f * * (t) − f * (t) is a bounded function of t such that…”

Section: T X M W F (T) = E 2πiw(t−x) F (T − X) or M W T X F (T) = E 2mentioning

confidence: 99%

On Lorentz mixed normed modulation spaces

Sandıkçı

2012

J. Pseudo-Differ. Oper. Appl.

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This paper is a study on a new kind modulation spaces M(P, Q) (R d ) and A(P, Q, r )(R d ) for indices in the range 1 < P < ∞, 1 ≤ Q < ∞ and 1 ≤ r < ∞, modelled on Lorentz mixed norm spaces instead of mixed norm L P spaces as the spaces 2006). First, we prove the main properties of these spaces. Later, we describe the dual spaces and determine the multiplier spaces for both of them. Moreover, we investigate the boundedness of Weyl operators and localization operators on M(P, Q)(R d ). Finally, we give an interpolation theorem for M(P, Q)(R d ).Keywords Gabor transform · Lorentz mixed norm space · modulation space · Weyl operator · Multiplier IntroductionIn this paper we will work on R d with Lebesgue measure dx. We denote by C c (R d ) and S(R d ) the spaces of complex-valued continuous functions with compact support and the space of complex-valued continuous functions on R d rapidly decreasing at infinity, respectively. Let f be a complex valued measurable function on R d . The operators T x f (t) = f (t − x) and M w f (t) = e 2πiwt f (t) are called translation and A. Sandıkçı (B)

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Section: T X M W F (T) = E 2πiw(t−x) F (T − X) or M W T X F (T) = E 2mentioning

confidence: 99%

On Lorentz mixed normed modulation spaces

Sandıkçı

2012

J. Pseudo-Differ. Oper. Appl.

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“…That (6) => (9) is obtained by getting, via Cotlar's inequality, a good-A inequality between the "singular integral" and the "maximal function" (see [9,Chapters 2 and 4]). Then, trivially (9) => (4) and (4) =» (2). That way we have that (1), (2), (3), (4), (5), (6) and (9) are equivalent.…”

Section: Proof the Detailed Proof Is Given In [8]mentioning

confidence: 81%

“…Then, trivially (9) => (4) and (4) =» (2). That way we have that (1), (2), (3), (4), (5), (6) and (9) are equivalent. Then (10) follows from (2) in a rather straightforward way.…”

Section: Proof the Detailed Proof Is Given In [8]mentioning

confidence: 81%

Maximal operators and B.M.O. for Banach lattices

Garcı́a-Cuerva¹,

Macías²,

Torrea³

1998

Proceedings of the Edinburgh Mathematical Society

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We investigate the behaviour of the classical (non-smooth) Hardy-Littlewood maximal operator in the context of Banach lattices. We are mainly concerned with end-point results for p = oo. Naturally, the main role is played by the space BMO. We analyze the range of the maximal operator in BMO X . This turns out to depend strongly on the convexity of the Banach lattice X. We apply these results to study the behaviour of the commutators associated to the maximal operator. We also consider the parallel results for the maximal fractional integral operator.1991 Mathematics subject classification: 42A28, 46B20, 43A25. IntroductionIn our paper [8] we have introduced the Hardy-Littlewood property (sometimes abbreviated as H.L. property) of a Banach lattice and have characterized it by several boundedness properties of the lattice version M of the Hardy-Littlewood maximal operator or its smooth variants. We refer the reader to this paper and also to [12] for the basic notation and terminology concerning Banach lattices and function spaces.As we explain in [8], the Hardy-Littlewood property is intimately connected to the U.M.D. property, which plays a central role in the development of Vector-Valued Fourier Analysis and, in spite of having been extensively studied (see [6] and [3]), still has some aspects that need to be completely understood. By studying systematically the Hardy-Littlewood property, we hoped to cast some light on the behaviour of Banach lattices regarding the U.M.D. property.The key idea in [8] is that the maximal operator M has a smooth version, obtained by replacing the averages over balls by the convolutions with an appropriate approximate identity. This smooth version can be viewed as a vector-valued singular integral, and its boundedness properties are a consequence of the general theory which was originally set up in [1] and then fully developed in [14]. This latter paper will also be a basic reference for the present work. In particular, the notation for all our function spaces will be taken from it.When one is interested in estimates on the Bochner-Lebesgue spaces L X (K"), only the size matters, and it is irrelevant whether we look at M or to its smooth version

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“…This can be considered as the probabilistic version of the result for Hardy-Littlewood maximal operator established in [1]. We give an example of a function/ whose maximal is in BMO but/ is not in BMO (see Example 3.6), that, in particular, shows that the BMO norms of/ and/* are not equivalent.…”

Section: Introductionmentioning

confidence: 89%

“…A Banach space B is said to be of martingale cotype q if 5,/ = ( £~ , \\dj ||«) 1 only depending on q. This property was introduced by Pisier, see [15] and [16].…”

Section: Introductionmentioning

confidence: 99%

Boundedness of vector-valued martingale transforms on extreme points and applciations

Martínez¹,

Torrea²

2004

J. Aust. Math. Soc.

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Let Bi, B2 be a pair of Banach spaces and T be a vector valued martingale transform (with respect to general filtration) which maps Bi-valued martingales into B 2 -valued martingales. Then, the following statements are equivalent: T is bounded from L p Ki into L^ for some p (or equivalently for every p) in the range 1 < p < 00; T is bounded from Ljjfj into BMO Kl ; T is bounded from BMO Bl into BMOn 2 ; T is bounded from H^ into H^. Applications to UMD and martingale cotype properties are given. We also prove that the Hardy space H^ defined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if B has the Radon-Nikodym property.2000 Mathematics subject classification: primary 60G42; secondary 60B11,46B20.

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Weak-L ∞ and BMO

Cited by 176 publications

References 6 publications

On Lorentz mixed normed modulation spaces

On Lorentz mixed normed modulation spaces

Maximal operators and B.M.O. for Banach lattices

Boundedness of vector-valued martingale transforms on extreme points and applciations

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