Mapping properties of Fourier transforms (Q2107772)

The author investigates the mapping properties of the Fourier transform \(\mathcal{F}\) in the framework of the Besov spaces \(B^s_{p,p}(\mathbb{R})\). Necessary and sufficient conditions are found which match except in some limiting cases. More exactly, it is proved the following. Let \(1<p<\infty\), \[ d_p^n:= 2n \Big(\frac 1p -\frac 12\Big)\, , \qquad \tau_p^{n+} := \max \Big(0, d_p^n\Big) \text{ and } \tau_p^{n-} := \min \Big(0, d_p^n\Big) \, . \] Further we put \[ X^s_p (\mathbb{R}^n) := \begin{cases} L_p (\mathbb{R}^n) & \text{if } 2\le p <\infty\, ,\, s=0\, ; \\ B^s_{p,p} (\mathbb{R}^n) & \text{if } 2\le p <\infty\, ,\, s>0\, ; \\ B^s_{p,p} (\mathbb{R}^n) & \text{if } 1< p \le 2\, ,\, s\ge d_p^n \end{cases} \] and \[ Y^s_p (\mathbb{R}^n) := \begin{cases} B^s_{p,p} (\mathbb{R}^n) & \text{if } 2\le p <\infty\, ,\, s\le d_p^n\, ; \\ B^s_{p,p} (\mathbb{R}^n) & \text{if } 1< p \le 2\, ,\, s<0\, ; \\ L_{p} (\mathbb{R}^n) & \text{if } 1< p \le 2\, ,\, s=0\, . \end{cases} \] \begin{itemize} \item[(i)] If \(s_1 \ge \tau_p^{n+}\) and \(s_2 \le \tau_p^{n-}\), then \(\mathcal{F}: X^{s_1}_{p} (\mathbb{R}^n) \to Y^{s_2}_p (\mathbb{R}^n)\) is continuous. \item[(ii)] The mapping \(\mathcal{F}: X^{s_1}_{p} (\mathbb{R}^n) \to Y^{s_2}_p (\mathbb{R}^n)\) is compact if, and only if, both \(s_1 > \tau_p^{n+}\) and \(s_2 < \tau_p^{n-}\). \item[(iii)] Assume that \(\mathcal{F}: X^{s_1}_{p} (\mathbb{R}^n) \to Y^{s_2}_p (\mathbb{R}^n)\) is continuous. Then \(s_1 \ge \tau_p^{n+}\) and \(s_2 \le \tau_p^{n-}\). \end{itemize} In a second part of the paper the degree of compactness of this mapping is studied. Therefore the author is using entropy numbers.