### BMO and smooth truncation in Sobolev spaces

David Adams, Michael Frazier (1988)

Studia Mathematica

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David Adams, Michael Frazier (1988)

Studia Mathematica

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Loukas Grafakos (1992)

Revista Matemática Iberoamericana

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We continue the study of multilinear operators given by products of finite vectors of Calderón-Zygmund operators. We determine the set of all r ≤ 1 for which these operators map products of Lebesgue spaces L(R) into the Hardy spaces H(R). At the endpoint case r = n/(n + m + 1), where m is the highest vanishing moment of the multilinear operator, we prove a weak type result.

Jun Tateoka (1994)

Studia Mathematica

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C. Watari [12] obtained a simple characterization of Lipschitz classes $Li{p}^{\left(p\right)}\alpha \left(W\right)(1\ge p\ge \infty ,\alpha >0)$ on the dyadic group using the ${L}^{p}$-modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces ${B}_{p,q}^{\alpha}$ on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces ${B}_{p,q}^{\alpha}$ by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality...

Shan Lu, Da Yang (1995)

Studia Mathematica

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Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces $HK{\u0307}_{q}^{\alpha ,p}\left({\mathbb{R}}^{n}\right)$ which are the local versions of ${H}^{p}\left({\mathbb{R}}^{n}\right)$ spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth’s sense. We also prove an interpolation theorem for operators on $HK{\u0307}_{q}^{\alpha ,p}\left({\mathbb{R}}^{n}\right)$ and discuss the $HK{\u0307}_{q}^{\alpha ,p}\left({\mathbb{R}}^{n}\right)$-boundedness of Calderón-Zygmund operators. Similar results can also be obtained...

M. Mateljević, M. Pavlović (1984)

Studia Mathematica

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Chang-Pao Chen, Dah-Chin Luor (2000)

Studia Mathematica

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Let s* denote the maximal function associated with the rectangular partial sums ${s}_{mn}(x,y)$ of a given double function series with coefficients ${c}_{jk}$. The following generalized Hardy-Littlewood inequality is investigated: ${\left|\right|s*\left|\right|}_{p,\mu}\le {C}_{p,\alpha ,\beta}{{\Sigma}_{j=0}^{\infty}{\Sigma}_{k=0}^{\infty}{\left(j\u0305\right)}^{p-\alpha -2}{\left(k\u0305\right)}^{p-\beta -2}{\left|{c}_{jk}\right|}^{p}}^{1/p}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on ${c}_{jk}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property...

Akihito Uchiyama (1985)

Studia Mathematica

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Der-Chen Chang, Song-Ying Li (1999)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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R. Paley (1931)

Studia Mathematica

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G. Sampson (1993)

Studia Mathematica

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We consider operators of the form $\left(\Omega f\right)\left(y\right)={\u0283}_{-\infty}^{\infty}\Omega (y,u)f\left(u\right)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h\in {L}^{\infty}$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space ${\u1e02}_{1}^{0,1}$ (= B) into itself. In particular, all operators with $h\left(y\right)={e}^{{i\left|y\right|}^{a}}$, a > 0, a ≠ 1, map B into itself.