# Estimates of Fourier transforms in Sobolev spaces

Studia Mathematica (1997)

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

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topKolyada, 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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