For \(\sigma >0\) , let \(B^p_{\sigma }\) be the set of entire functions of exponential type at most \(\sigma \) such that their restriction to the real axis \(\mathbb R\) are in \(L^p(\mathbb R)\) . We investigate in \(B^p_{\sigma }\) a problem of maximal concentration for the fraction of \(L^p\) -norm on a subset of \(\mathbb R\) . To do this, we first prove the following generalization of the classical inequality of Plancherel-Pólya inequality: \(\int _{\mathbb R} |f(x)|^p\,d\Delta (x)\le C(\delta , \sigma , p)\Bigl [\bigl (\sup _{a\in \mathbb R} \Delta [a, a+\delta ]\bigr )/\delta \Bigr ]\cdot \Vert f\Vert ^p_{L^{p}(\mathbb R)}\) for for a given positive sigma-finite measure \(\Delta \) on \(\mathbb R\) , \(\delta >0\) , and \(C(\delta , \sigma , p)=\Vert \cos (\sigma \delta t)\Vert ^{-p}_{L^{p}[-1/2, 1/2]}\) . This inequality shows that if \(\Delta \) has the sparse support in the sense that \(\sup _{a\in \mathbb R}\,\Delta ([a, a+\delta ])\) is much less than \(\delta \) for some fixed \(\delta >0\) , then \(\int |f(x)|^p\,d\Delta (x)\) is also small. In particular, if \(\Delta \) coincides with the restriction of the Lebesgue measure on a sparse subset in \(\mathbb R\) , then only a small portion of \(L^p\) -energy \(\Vert f\Vert _{L^{p}(\mathbb R)}\) for \(f\in B^p_{\sigma }\) can be concentrated on such a set.
