On pointwise interpolation inequalities for derivatives

Vladimir G. Maz'ya; Tatjana Olegovna Shaposhnikova

Mathematica Bohemica (1999)

  • Volume: 124, Issue: 2-3, page 131-148
  • ISSN: 0862-7959

Abstract

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Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where ∇ k is the gradient of order k , 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 ( ℝ n ) → W p l ( ℝ n ) ) is described.

How to cite

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Maz'ya, Vladimir G., and Shaposhnikova, Tatjana Olegovna. "On pointwise interpolation inequalities for derivatives." Mathematica Bohemica 124.2-3 (1999): 131-148. <http://eudml.org/doc/248463>.

@article{Mazya1999,
abstract = {Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where $\nabla _k$ is the gradient of order $k$, $\{\mathcal \{M\}\}$ 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 $M(W_p^m(\{\mathbb \{R\}\}^n)\rightarrow W_p^l(\{\mathbb \{R\}\}^n))$ is described.},
author = {Maz'ya, Vladimir G., Shaposhnikova, Tatjana Olegovna},
journal = {Mathematica Bohemica},
keywords = {Landau inequality; interpolation inequalities; Hardy-Littlewood maximal operator; Gagliardo-Nirenberg inequality; Sobolev multipliers; Landau inequality; interpolation inequalities; Hardy-Littlewood maximal operator; Gagliardo-Nirenberg inequality; Sobolev multipliers},
language = {eng},
number = {2-3},
pages = {131-148},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On pointwise interpolation inequalities for derivatives},
url = {http://eudml.org/doc/248463},
volume = {124},
year = {1999},
}

TY - JOUR
AU - Maz'ya, Vladimir G.
AU - Shaposhnikova, Tatjana Olegovna
TI - On pointwise interpolation inequalities for derivatives
JO - Mathematica Bohemica
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 124
IS - 2-3
SP - 131
EP - 148
AB - Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where $\nabla _k$ is the gradient of order $k$, ${\mathcal {M}}$ 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 $M(W_p^m({\mathbb {R}}^n)\rightarrow W_p^l({\mathbb {R}}^n))$ is described.
LA - eng
KW - Landau inequality; interpolation inequalities; Hardy-Littlewood maximal operator; Gagliardo-Nirenberg inequality; Sobolev multipliers; Landau inequality; interpolation inequalities; Hardy-Littlewood maximal operator; Gagliardo-Nirenberg inequality; Sobolev multipliers
UR - http://eudml.org/doc/248463
ER -

References

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  1. E. Landau, Einige Ungleichungen für zweimal differenzierbare Funktionen, Proc. London Math. Soc. 13 (1913), 43-49. (1913) 
  2. V. Maz'ya T. Shaposhnikova, Jacques Hadamard, a universal mathematician, American Mathematical Society and London Mathematical Society, Providence, RI, 1998. (1998) 
  3. L. Nirenberg F. Trèves, 10.1002/cpa.3160160308, Comm. Pure Appl. Math. 16 (1963), 331-351. (1963) MR0163045DOI10.1002/cpa.3160160308
  4. P. D. Lax L. Nirenberg, 10.1002/cpa.3160190409, Comm. Pure Appl. Math. 19 (1966), 473-492. (1966) MR0206534DOI10.1002/cpa.3160190409
  5. V. Maz'ya A. Kufner, 10.1007/BF01171035, Manuscripta Math. 56 (1986), 89-104. (1986) MR0846988DOI10.1007/BF01171035
  6. D. R. Adams L. I. Hedberg, Function spaces and potential theory, Springer-Verlag, Berlin, 1996. (1996) MR1411441
  7. V. Maz'ya S. Poborchi, Differentiable functions on bad domains, World Scientific Publishing, Singapore, 1997. (1997) MR1643072
  8. E. Gagliardo, Ulteriori propietà di alcune classi di funzioni on più variabli, Ric. Mat. 8 (1) (1959), 24-51. (1959) MR0109295
  9. L. Nirenberg, On elliptic partial differential equations: Lecture II, Ann. Sc. Norm. Sup. Pisa, Ser. 3 13 (1959), 115-162. (1959) MR0109940
  10. L. I. Hedberg, 10.1090/S0002-9939-1972-0312232-4, Proc. Amer. Math. Soc. 36 (1972), 505-510. (1972) MR0312232DOI10.1090/S0002-9939-1972-0312232-4
  11. V. Maz'ya T. Shaposhnikova, Theory of multipliers in spaces of differentiable functions, Pitman, London, 1985. (1985) 
  12. V. Maz'ya I. Verbitsky, 10.1007/BF02559606, Ark. Mat. 33 (1995), 81-115. (1995) MR1340271DOI10.1007/BF02559606

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