Estimates of Fourier transforms in Sobolev spaces

V. Kolyada

Studia Mathematica (1997)

  • Volume: 125, Issue: 1, page 67-74
  • ISSN: 0039-3223

Abstract

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We investigate the Fourier transforms of functions in the Sobolev spaces W 1 r 1 , . . . , r n . It is proved that for any function f ∈ W 1 r 1 , . . . , r n the Fourier transform f̂ belongs to the Lorentz space L n / r , 1 , where r = n ( ∑ j = 1 n 1 / r j ) - 1 ≤ n . Furthermore, we derive from this result that for any mixed derivative D s f ( f ∈ C 0 ∞ , s = ( s 1 , . . . , s n ) ) the weighted norm ∥ ( D s f ) ∧ ∥ L 1 ( w ) ( w ( ξ ) = | ξ | - n ) can be estimated by the sum of L 1 -norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

How to cite

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Kolyada, V.. "Estimates of Fourier transforms in Sobolev spaces." Studia Mathematica 125.1 (1997): 67-74. <http://eudml.org/doc/216422>.

@article{Kolyada1997,
abstract = {We investigate the Fourier transforms of functions in the Sobolev spaces $W_1^\{r_1,..., r_n\}$. It is proved that for any function $f ∈ W_1^\{r_1,...,r_n\}$ the Fourier transform f̂ belongs to the Lorentz space $L^\{n/r,1\}$, where $r = n(∑_\{j=1\}^n 1/r_\{j\})^\{-1\} ≤ n$. Furthermore, we derive from this result that for any mixed derivative $D^\{s\}f (f ∈ C_0^∞, s=(s_1,... ,s_n))$ the weighted norm $∥(D^\{s\}f)^∧∥_\{L^1(w)\} (w(ξ) = |ξ|^\{-n\})$ can be estimated by the sum of $L^1$-norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.},
author = {Kolyada, V.},
journal = {Studia Mathematica},
keywords = {Fourier transform; mixed derivative; Hardy inequality; Sobolev space},
language = {eng},
number = {1},
pages = {67-74},
title = {Estimates of Fourier transforms in Sobolev spaces},
url = {http://eudml.org/doc/216422},
volume = {125},
year = {1997},
}

TY - JOUR
AU - Kolyada, V.
TI - Estimates of Fourier transforms in Sobolev spaces
JO - Studia Mathematica
PY - 1997
VL - 125
IS - 1
SP - 67
EP - 74
AB - We investigate the Fourier transforms of functions in the Sobolev spaces $W_1^{r_1,..., r_n}$. It is proved that for any function $f ∈ W_1^{r_1,...,r_n}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n/r,1}$, where $r = n(∑_{j=1}^n 1/r_{j})^{-1} ≤ n$. Furthermore, we derive from this result that for any mixed derivative $D^{s}f (f ∈ C_0^∞, s=(s_1,... ,s_n))$ the weighted norm $∥(D^{s}f)^∧∥_{L^1(w)} (w(ξ) = |ξ|^{-n})$ can be estimated by the sum of $L^1$-norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.
LA - eng
KW - Fourier transform; mixed derivative; Hardy inequality; Sobolev space
UR - http://eudml.org/doc/216422
ER -

References

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  8. [8] V. I. Kolyada, On embedding of Sobolev spaces, Mat. Zametki 54 (3) (1993), 48-71 (in Russian). Zbl0821.46043
  9. [9] J. Peetre, Sur la transformation de Fourier des fonctions à valeurs vectorielles, Rend. Sem. Mat. Padova 42 (1969), 15-26. Zbl0241.46033
  10. [10] A. Pełczyński and M. Wojciechowski, Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm, Studia Math. 107 (1993), 61-100. Zbl0811.46028
  11. [11] S. Poornima, An embedding theorem for the Sobolev space W 1 , 1 , Bull. Sci. Math. 107 (1983), 253-259. Zbl0529.46025
  12. [12] E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482-492. Zbl0072.32402
  13. [13] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971. Zbl0232.42007

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