Lipschitz Spaces and Bernstein's Theorem on Absolutely Convergent Fourier Transforms*

Herz, Carl

doi:10.1512/iumj.1969.18.18024

Cited by 174 publications

(139 citation statements)

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“…Herz [6] has proved a number of such properties for homogeneous Besov spaces. It is not obvious that they carry over to inhomogeneous Besov spaces because of the added global constraint.…”

Section: Translation Invariant Subspaces Of Kpq the Main Gap That Rementioning

confidence: 98%

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Maximal Subspaces of Besov Spaces Invariant Under Multiplication by Characters

Johnson¹

1979

Transactions of the American Mathematical Society

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Abstract.Unlike the familiar Lp spaces, neither the homogeneous Besov spaces nor the Hp spaces, 0 < p < 1, are closed under multiplication by the functions x -* e'. We determine the maximal subspace of these spaces which are closed under multiplication by these functions, which are the characters of R".In this paper we investigate the maximal subspaces of Besov spaces invariant under multiplication by characters. The usefulness of such a study is related to applications to multipliers. Translation invariant operators from LP(R") into Lq(R") are generally characterized by two kinds of properties-smoothness properties and growth restrictions. The smoothness properties are most naturally expressed in terms of the Besov spaces. The convoluteurs from HP(R") into Hq(Rn) for 0 < p < 1 < q < oo satisfy smoothness conditions of positive orders which already imply growth conditions by the Sobolev theorem. In this case one finds Cv(Hp, Hq) = Bqj£/p~X) [10]. The smoothness part of this result remains valid lox p > 1 but it no longer implies a growth condition and we should expect the inclusion Cv(Hp, Hq) c B^/p-V to be a proper inclusion in this case. One approach to imposing growth conditions is to note that while elements of Bqoo may

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“…Herz [6] has proved a number of such properties for homogeneous Besov spaces. It is not obvious that they carry over to inhomogeneous Besov spaces because of the added global constraint.…”

Section: Translation Invariant Subspaces Of Kpq the Main Gap That Rementioning

confidence: 98%

“…Our fourth application is to obtain an improvement of a result proved by Herz [6] and Kellogg [11]. Theorem 3.7.…”

Section: Translation Invariant Subspaces Of Kpq the Main Gap That Rementioning

confidence: 99%

Maximal Subspaces of Besov Spaces Invariant Under Multiplication by Characters

Johnson¹

1979

Transactions of the American Mathematical Society

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“…In 1964, Beurling [22] first introduced some fundamental forms of Herz spaces to study convolution algebras. Later Herz [23] gave versions of the spaces defined below in a slightly different setting. Since then, the theory of Herz spaces has been significantly developed, and these spaces have turned out to be quite useful in harmonic analysis.…”

Section: Preliminaries and Main Lemmasmentioning

confidence: 99%

Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces

Yang

Tao

2017

Journal of Function Spaces

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Let be the singular integral operator with variable kernel defined by ( ) = p.v. ∫ R (Ω( , − )/| − | ) ( )d and let (0 ≤ ≤ 1) be the fractional differentiation operator. Let * and ♯ be the adjoint of and the pseudoadjoint of , respectively. In this paper, the authors prove that − and ( * − ♯ ) are bounded, respectively, from Morrey-Herz spaceṡ, ,1 (R ) to the weak Morrey-Herz spaceṡ, ,1 (R ) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product 1 2 and the pseudoproduct 1 ∘ 2 are also given.

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“…We remark that studies involving these spaces have a long history; and we recommend the references [9,14,17] and [12]. Soria and Weiss [22] give some beautiful generalizations of Stein's results in [23].…”

Section: [ H(x-y)f(y)dymentioning

confidence: 99%