A sufficient condition forLp-multipliers

Igari, Satoru; Kuratsubo, Shigehiko

doi:10.2140/pjm.1971.38.85

Cited by 17 publications

(12 citation statements)

References 3 publications

(4 reference statements)

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“…Its L ^-behaviour has been investigated by Igari and Kuratsubo in [6] where they have shown via analytic interpolation between the points ( where each c, > 0 is independent of / E 5. Had we applied the interpolation argument of [6] to the g λ -function defined in (1.…”

Section: =(Jfmentioning

confidence: 99%

A Marcinkiewicz criterion forL^p-multipliers

Dappa¹

1984

Pacific J. Math.

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Section: =(Jfmentioning

confidence: 99%

A Marcinkiewicz criterion forL^p-multipliers

Dappa¹

1984

Pacific J. Math.

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“…For the range p > 2, the condition σ > max{1/2, D(1/2 − 1/p)} − 1 is known to be necessary and sufficient in dimensions D = 1 and 2. In dimensions D 3, there are some partial results, see for instance, for σ > D(1/2 − 1/p) − 1/2 in [14] and [15]. For 0 D(1/p − 1/2) − 1/2, then G σ (∆) is bounded from H p to L p (see [16]).…”

Section: Introductionmentioning

confidence: 99%

“…It is known that the L p boundedness of G σ (∆) for 1 D(1/p−1/2)−1/2 (see [14], [15] and [21]). For the range p > 2, the condition σ > max{1/2, D(1/2 − 1/p)} − 1 is known to be necessary and sufficient in dimensions D = 1 and 2.…”

Section: Introductionmentioning

confidence: 99%

“…For 0 D(1/p − 1/2) − 1/2, then G σ (∆) is bounded from H p to L p (see [16]). Boundedness of the square function G δ (∆) has been studied extensively because of its important role in the Bochner-Riesz analysis and we refer the reader to [5], [14], [15], [16] and [21] and the references therein. Recently, in the abstract framework of a space of homogeneous type (X, d, µ) with dimension n > 0 (see Section 2 below), P. Chen, X. T. Duong and L. X. Yan ( [5]) studied and obtained the L p boundedness of Stein's square function G δ (L) when the semigroup e −tL , generated by −L on L 2 (X), has the kernels p t (x, y) which satisfy the Gaussian upper bounds (see, for example, [18]) for all t > 0 and x, y ∈ X, where C, c are constants.…”

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Boundedness of Stein’s square functions and Bochner-Riesz means associated to operators on hardy spaces

Yan

2015

Czech Math J

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“…IjSa, (f)llp, 1 ½/2, [3] IIVa(f)114 s CllfIl4, a > 1/2, [4] where the constants C are independent of f and m. Inequality 2 follows from the theory of vector-valued singular integrals (2), and inequality 4 follows from proposition 1 in ref. [7]…”

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Radial Fourier multipliers of L ^p (R ² )

Carbery¹,

Gasper²,

Trebels³

1984

Proc. Natl. Acad. Sci. U.S.A.

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We give sufficient conditions (in terms of differentiability and growth properties) for a radial function to be an L!(R2) Fourier multiplier. These conditions are in the nature of best possible.The well-known multiplier theorem of Hormander (ref. 1; see also p. 96 in ref.2) may be stated as follows. Let 4 be a fixed nonnegative smooth bump function supported in [1,2], and let D,, a > 0, denote the fractional derivative defined initially on f(R") (the space of rapidly decreasing C' functions) by (Dng)A(4) = 11agA(e) and then extended to L1(R').For n = 1 we write D' = Di . Suppose m is a bounded function on R' that satisfies SU t2a-n faDn' [4(u) m(1)1 2 = A2 < 00 [1] for some real number a > n/2. Then m is a Fourier multiplier of LP(R") [ Condition 1 for cr > n/2 actually implies that m is bounded, while for a s n/2, the space of functions satisfying 1 contains unbounded functions: in this sense, Hormander's result is sharp; but when additional assumptions about m and p are made, the differentiability conditions required of m may be relaxed (see, for example, refs. 3 and 4).On the other hand, let us consider the Bochner-Riesz multipliers ms(f) = (1 -If12)5 for 8 > 0. Carleson and Sjolin (5) [see also the work of Fefferman (6) and the contributions of C6rdoba (7) 10 equals R(q, a) with equivalent norms when a > -, 1 < q q <0. The case q = 2 and Sobolev-type embeddings give the case q < 2. Interpolation of the case q = 2 with M2(R2) = L'(R2) gives the case q > 2. These matters will be discussed in a forthcoming paper. It should be noted that while the multiplier theorems are most naturally proved in the R(q, a) context, the WBVq,a conditions are easier to verify.(ii) Theorem 2 is best possible in the sense that the "criti- n -i1 await a better understanding of the multipliers m6 for n . 3. However, Dappa (12) has obtained the optimal result in the case q = 1, a > (3n + 1)/(2n + 2) by using the Stein-Tomas restriction theorem (13).(iv) Gasper and Trebels proved the Hankel multiplier version of Theorem 2 in ref. 9 and gave necessary conditions for Hankel multipliers (and hence for radial Fourier multipliers) in terms of WBV-spaces in ref. 14.The proof of Theorem 1 uses square functions in the spirit of the Igari-Kuratsubo multiplier theorem (3) and Stein's proof of Hormander's theorem. Prior to the proof, let us make the following definitions in which all Fourier transforms are in R2, with functions on (0, 00) extended to be radial where appropriate. 3254The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

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A sufficient condition forL^p-multipliers

Cited by 17 publications

References 3 publications

A Marcinkiewicz criterion forL^p-multipliers

A Marcinkiewicz criterion forL^p-multipliers

Boundedness of Stein’s square functions and Bochner-Riesz means associated to operators on hardy spaces

Radial Fourier multipliers of L ^p (R ² )

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A sufficient condition forLp-multipliers

Cited by 17 publications

References 3 publications

A Marcinkiewicz criterion forLp-multipliers

A Marcinkiewicz criterion forLp-multipliers

Boundedness of Stein’s square functions and Bochner-Riesz means associated to operators on hardy spaces

Radial Fourier multipliers of L p (R 2 )

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A sufficient condition forL^p-multipliers

A Marcinkiewicz criterion forL^p-multipliers

A Marcinkiewicz criterion forL^p-multipliers

Radial Fourier multipliers of L ^p (R ² )