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Estimates of Fourier transforms in Sobolev spaces. (English) Zbl 0896.42008

For \(f\) in the anisotropic Sobolev space \(W_1^{r_1, \dots, r_n} (\mathbb{R}^n) \), the author proves that \(\| \widehat f\|_{n/r,1}\) and the quantity \[ \int_{\mathbb{R}^n} \bigl | \widehat f(\xi) \bigr| \left(\sum_{1\leq j\leq n} |\xi_j |^{r_j/r} \right)^{r-n} d\xi, \] where \(r=n (r_1^{-1} +\cdots +r_n^{-1})^{-1}\), are dominated by \(\sum_{1\leq j\leq n}\| D_j^{r_j} f\|_1\). The emphasis is on the absence of mixed derivatives in the latter expression.

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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