A generalization of the logarithmic Gross-Sobolev inequality (Q2279765)

The author presents a new sharp integral inequality implying the Sobolev interpolation inequality. Let \(k>0\) and let \(\rho> 0 \) be a given number such that \(\displaystyle \rho < \frac{2k}{n-k}\) if \(n-k>0\) and \(\rho\) is arbitrary if \(n-k \leq 0\). Let us denote \(\displaystyle \alpha = \frac{n\rho}{k(\rho+2)}\) and \(\displaystyle \chi = \sqrt{\alpha^{\alpha}(1-\alpha)^{1-\alpha}}\). Furthermore, let us denote \[ K_g(\alpha)=\frac{1}{\chi}\left[ \frac{\sigma_n}{k} B\left( \frac nk, \frac{n(1-\alpha)}{k\alpha} \right)\right]^{\frac{\alpha k}{2n}}, \] \[K_B(p)=\left[ \left( \frac{p}{2\pi}\right)^{1/p} \left( \frac{q}{2\pi}\right)^{-1/q}\right] ^{n/2}, \] where \(p\in [1,2]\), \(\displaystyle \frac 1p + \frac 1q =1\), \(\displaystyle \sigma_n=\frac{2\pi^{n/2}}{\Gamma (\frac n2)}\), \(\Gamma\) and \(B\) are the Euler gamma and the Euler beta function, respectively. Let \(\hat{U}\) be the Fourier transform of a function \(U\), i.e., \[ \hat{U}(\xi )=\frac{1}{(2\pi)^{n/2} } \int_{\mathbb{R}^n} e^{i(x,\xi)} U(x) dx, \quad \xi \in \mathbb{R}^n. \] The main result is the following theorem. Theorem. Let \(k, \rho\) and \(\alpha\) be numbers defined above, \(U(x) \in L_2(\mathbb{R}^n)\), \(r^{k/2} \hat{U}(\xi) \in L_2(\mathbb{R}^n)\), \(r=|\xi|\). Then the following multiplicative Sobolev inequality holds \[ \| U\|_{\rho + 2} \leq K_g(\alpha) K_B\left(\frac{\rho+2}{\rho+1} \right) \| r^{k/2} \hat{U}(\xi) \|^{\alpha} \| U \|^{1-\alpha}. \] Putting \(k=2\) and \(k=4\) in the above-mentioned theorem the Gagliardo-Nirenberg-Sobolev interpolation inequality and the Sobolev interpolation inequality appear. The second section is devoted to the logarithmic Gross-Sobolev inequality. Namely, the following theorem is proved. Theorem. Let \(k\) be a positive number, \(U(x) \in L_2(\mathbb{R}^n)\), \(r^{k/2} \hat{U}(\xi) \in L_2(\mathbb{R}^n)\), \(r=|\xi|\). Then, the following logarithmic Gross-Sobolev inequality holds \[ \int_{\mathbb{R}^n} \frac{|U|^2}{\| U \|^2} \mathrm{\ln} \left( \frac{|U|^2}{\| U \|^2} \right) dx \leq \frac nk \mathrm{ln} \left[ \frac{ \displaystyle k \left( \frac{\sigma_n}{k} \Gamma \left(\frac nk\right)\right)^{k/n} \| r^{k/2} \hat{U}(\xi) \|^2 }{n\pi^k e^{k-1} \| U \|^2}\right]. \]