Pitt's inequality and the fractional Laplacian: Sharp error estimates (Q2882752)

The author proves that for \(f\in {\mathcal{S}}(\mathbb{R}^n)\) and \(0<\beta<2\), \(\beta\leq\alpha<n\), NEWLINE\[NEWLINE\begin{multlined} C_\alpha\int_{\mathbb{R}^n}|\xi|^\alpha |\hat{f}(\xi)|^2d\xi \\ \geq \int_{\mathbb{R}^n}|x|^{-\alpha}|f(x)|^2dx +\frac{C_\alpha}{D_\beta}\int_{\mathbb{R}^n\times\mathbb{R}^n}\left(\frac{|g(x)-g(y)|^2}{|x-y|^{n+\beta}} |x|^{-(n-\beta)/2}|y|^{-(n-\beta)/2}\right)dxdy,\end{multlined}NEWLINE\]NEWLINE where NEWLINE\[NEWLINEg(x)=|x|^{(n-\beta)/2}(-\Delta/4\pi^2)^{(\alpha-\beta)/4}f(x),NEWLINE\]NEWLINE and NEWLINE\[NEWLINEC_\alpha=\pi^\alpha\left[\frac{\Gamma(\frac{n-\alpha}4)}{\Gamma(\frac{n+\alpha}4)}\right]^2,\qquad D_\beta=\frac 4{\beta}\pi^{n/2+\beta}\frac{\Gamma(1-\beta/2)}{\Gamma(\frac{n+\beta}2)}.NEWLINE\]NEWLINE This result gives a specific improvement by a Besov norm for the Pitt inequality and the proof depends on the spectral representation with weights for the fractional Laplacian due to \textit{R. L. Frank, E. H. Lieb} and \textit{R. Seiringer} [J. Am. Math. Soc. 21, No. 4, 925--950 (2008; Zbl 1202.35146)] and the sharp Stein-Weiss inequality. And this result leads to an improved Stein-Weiss inequality accompanied by an intriguing monotonicity property at the spectral level. Moreover, a new Stein-Weiss lemma is applied to obtain the Frank-Seiringer ``Hardy inequalities'' for both \(\mathbb{R}^n\) and the upper half-space \(\mathbb{R}^n_+\), and the corresponding extensions for the Heisenberg group and product spaces with mixed homogeneity.