Ian Jauslin
2020
We consider the inequality \(f \geqslant f\star f\) for real functions in \(L^1(\mathbb R^d)\) where \(f\star f\) denotes the convolution of \(f\) with itself. We show that all such functions \(f\) are non-negative, which is not the case for the same inequality in \(L^p\) for any \(1 < p \leqslant 2\), for which the convolution is defined. We also show that all solutions in \(L^1(\mathbb R^d)\) satisfy \(\int_{\mathbb R^d}f(x){\rm d}x \leqslant \textstyle\frac12\). Moreover, if \(\int_{\mathbb R^d}f(x){\rm d}x = \textstyle\frac12\), then \(f\) must decay fairly slowly: \(\int_{\mathbb R^d}|x| f(x){\rm d}x = \infty\), and this is sharp since for all \(r< 1\), there are solutions with \(\int_{\mathbb R^d}f(x){\rm d}x = \textstyle\frac12\) and \(\int_{\mathbb R^d}|x|^r f(x){\rm d}x <\infty\). However, if \(\int_{\mathbb R^d}f(x){\rm d}x = : a < \textstyle\frac12\), the decay at infinity can be much more rapid: we show that for all \(a<\textstyle\frac12\), there are solutions such that for some \(\epsilon>0\), \(\int_{\mathbb R^d}e^{\epsilon|x|}f(x){\rm d}x < \infty\).
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