Kolyada, V. I. Estimates of Fourier transforms in Sobolev spaces. (English) Zbl 0896.42008 Stud. Math. 125, No. 1, 67-74 (1997). 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. Reviewer: S.V.Kislyakov (St.Peterburg) Cited in 8 Documents 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 Keywords:Fourier transform; mixed derivative; Hardy inequality; Sobolev space PDFBibTeX XMLCite \textit{V. I. Kolyada}, Stud. Math. 125, No. 1, 67--74 (1997; Zbl 0896.42008) Full Text: DOI EuDML