Integral bounds for simple partial fractions
A simple partial fraction is defined as \(g_{n}(t)=\sum_{k=1}^{n}\frac{1}{ t-z_{k}}.\) Here it is assumed that \(z_{k}=x_{k}+iy_{k}\in \mathbb{C} \backslash \mathbb{R},\) \(1\leq k\leq n\) and the following notations are used: \(f(t)=\text{Re}\, g_{n}(t)\) and \(\left\| f\right\| _{p}=\left( \int_{\mathbb{R}}\left| f(t)\right| ^{p}dt\right) ^{1/p}.\) The author estimates the \(L_{q}\)-norms of the Fourier transform for the function \(f\): \(\widetilde{f}(\lambda )=\frac{1}{\sqrt{2\pi }}\int_{\mathbb{R} }e^{-i\lambda t}f(t)dt.\) Let \(q\geq 2.\) Then \[ \sqrt{\frac{\pi }{2}}\left( \int_{\mathbb{R}}\left( \sum_{k=1}^{n}e^{-2\left| \lambda \right| \left| y_{k}\right| }\right) ^{q/2}d\lambda \right) ^{1/q}\leq \left\| \widetilde{f}\right\| _{q}\leq \sqrt{\frac{\pi }{2}}\left( \int_{\mathbb{R }}\left( \sum_{k=1}^{n}e^{-2\left| \lambda \right| \left| y_{k}\right| }\right) ^{q}d\lambda \right) ^{1/q}, \] and the upper estimate of the \(L_{q}\)-norm of the Fourier transform is sharp; the equality holds if and only if all \(x_{k}\) coincide. An inequality for \(L_{p}\)-norms of simple partial fractions and a sharp lower estimate for the \(L_{2q}\)-norm of the Fourier transform is also obtained.
scientific article; zbMATH DE number 6108802