Best possible sufficient conditions for the Fourier transform to satisfy the Lipschitz or Zygmund condition
Volume 199 / 2010
Studia Mathematica 199 (2010), 199-205
MSC: Primary 42A38; Secondary 26A16.
DOI: 10.4064/sm199-2-5
Abstract
We consider complex-valued functions $f\in L^1 (\mathbb R)$, and prove sufficient conditions in terms of $f$ to ensure that the Fourier transform $\hat f$ belongs to one of the Lipschitz classes ${\rm Lip} (\alpha)$ and ${\rm lip} (\alpha)$ for some $0< \alpha\le 1$, or to one of the Zygmund classes $\mathop{\rm zyg} (\alpha)$ and ${\rm zyg}(\alpha)$ for some $0<\alpha\le 2$. These sufficient conditions are best possible in the sense that they are also necessary in the case of real-valued functions $f$ for which either $x f(x) \ge 0$ or $f(x) \ge 0$ almost everywhere.
