| Title:
|
On pointwise interpolation inequalities for derivatives (English) |
| Author:
|
Maz'ya, Vladimir |
| Author:
|
Shaposhnikova, Tatyana |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
124 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
131-148 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 |
| DOI:
|
10.21136/MB.1999.126252 |
| . |
| Date available:
|
2009-09-24T21:36:21Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126252 |
| . |
| Reference:
|
[1] E. Landau: Einige Ungleichungen für zweimal differenzierbare Funktionen.Proc. London Math. Soc. 13 (1913), 43-49. |
| Reference:
|
[2] V. Maz'ya T. Shaposhnikova: Jacques Hadamard, a universal mathematician.American Mathematical Society and London Mathematical Society, Providence, RI, 1998. |
| Reference:
|
[3] L. Nirenberg F. Trèves: Solvability of a first order linear partial differential equation.Comm. Pure Appl. Math. 16 (1963), 331-351. MR 0163045, 10.1002/cpa.3160160308 |
| Reference:
|
[4] P. D. Lax L. Nirenberg: On solvability for difference schemes, a sharp form of Gårding's inequality.Comm. Pure Appl. Math. 19 (1966), 473-492. MR 0206534, 10.1002/cpa.3160190409 |
| Reference:
|
[5] V. Maz'ya A. Kufner: Variations on the theme of the inequality $(f')^2 \leq 2 f \sup |f''|$.Manuscripta Math. 56 (1986), 89-104. MR 0846988, 10.1007/BF01171035 |
| Reference:
|
[6] D. R. Adams L. I. Hedberg: Function spaces and potential theory.Springer-Verlag, Berlin, 1996. MR 1411441 |
| Reference:
|
[7] V. Maz'ya S. Poborchi: Differentiable functions on bad domains.World Scientific Publishing, Singapore, 1997. MR 1643072 |
| Reference:
|
[8] E. Gagliardo: Ulteriori propietà di alcune classi di funzioni on più variabli.Ric. Mat. 8 (1) (1959), 24-51. MR 0109295 |
| Reference:
|
[9] L. Nirenberg: On elliptic partial differential equations: Lecture II.Ann. Sc. Norm. Sup. Pisa, Ser. 3 13 (1959), 115-162. MR 0109940 |
| Reference:
|
[10] L. I. Hedberg: On certain convolution inequalities.Proc. Amer. Math. Soc. 36 (1972), 505-510. MR 0312232, 10.1090/S0002-9939-1972-0312232-4 |
| 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 |
| . |
