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Title: On pointwise interpolation inequalities for derivatives (English)
Author: Maz'ya, Vladimir
Author: Shaposhnikova, Tatyana
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 124
Issue: 2
Year: 1999
Pages: 131-148
Summary lang: English
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Category: math
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Summary: Pointwise interpolation inequalities, in particular, \left\vert\nabla_ku(x)\right\vert\leq c\left({\cal M}u(x)\right) ^{1-k/m} \left({\cal M}\nabla_mu(x)\right)^{k/m}, k<m, and |I_zf(x)|\leq c ({\cal M}I_{\zeta}f(x))^{\mathop Re z/\mathop Re \zeta}({\cal M}f(x))^{1-\mathop Re z/\mathop Re \zeta}, 0<\mathop Re z<\mathop Re\zeta<n, where $\nabla_k$ is the gradient of order $k$, ${\cal M}$ is the Hardy-Littlewood maximal operator, and $I_z$ is the Riesz potential of order $z$, are proved. Applications to the theory of multipliers in pairs of Sobolev spaces are given. In particular, the maximal algebra in the multiplier space $M(W_p^m({\Bbb R}^n)\to W_p^l({\Bbb R}^n))$ is described. (English)
Keyword: Landau inequality
Keyword: interpolation inequalities
Keyword: Hardy-Littlewood maximal operator
Keyword: Gagliardo-Nirenberg inequality
Keyword: Sobolev multipliers
MSC: 26D10
MSC: 42B25
MSC: 46E25
MSC: 46E35
idZBL: Zbl 0936.26008
idMR: MR1780687
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Date available: 2009-09-24T21:36:21Z
Last updated: 2015-09-15
Stable URL: http://hdl.handle.net/10338.dmlcz/126252
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Reference: [8] E. Gagliardo: Ulteriori propietà di alcune classi di funzioni on più variabli.Ric. Mat. 8 (1) (1959), 24-51. MR 0109295
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Reference: [11] V. Maz'ya T. Shaposhnikova: Theory of multipliers in spaces of differentiable functions.Pitman, London, 1985.
Reference: [12] V. Maz'ya I. Verbitsky: Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers.Ark. Mat. 33 (1995), 81-115. MR 1340271, 10.1007/BF02559606
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