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 nonnegative, 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 |$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$|⁠. Moreover, if |$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$|⁠, then |$f$| must decay fairly slowly: |$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $|⁠, and this is sharp since for all |$r< 1$|⁠, there are solutions with |$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$| and |$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $|⁠. However, if |$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$|⁠, the decay at infinity can be much more rapid: we show that for all |$a<\tfrac 12$|⁠, there are solutions such that for some |$\varepsilon>0$|⁠, |$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $|⁠.

中文翻译：

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关于卷积不等式f≥f⋆f

我们考虑不等式| $ f \ geqslant f \ star f $ | | $ L ^ 1（{\ mathbb {R}} ^ d）$ |中的实函数 其中| $ f \ star f $ | 表示| $ f $ |的卷积 本身。我们显示所有这些功能| $ f $ | 是非负的，| $ L ^ p $ |中的不等式不是这种情况 对于任何| $ 1 <p \ leqslant 2 $ |⁠，为其定义了卷积。我们还显示| $ L ^ 1（{\ mathbb {R}} ^ d）$ |中的所有解。满足| $ \ int _ {{\\ mathbb {R}} ^ {\ textrm {d}}} f（x）\ \ textrm {d} x \ leqslant \ tfrac 12 $ |⁠。此外，如果| $ \ int _ {{\\ mathbb {R}} ^ {\ textrm {d}}} f（x）\ \ textrm {d} x = \ tfrac 12 $ |⁠，则| $ f $ | 必须相当缓慢地衰减：| $ \ int _ {{\\ mathbb {R}} ^ {\ textrm {d}}} | x | f（x）\ \ textrm {d} x = \ infty $ |⁠，这很明显，因为对于所有| $ r <1 $ |⁠，都有| $ \ int _ {{\ mathbb {R} } ^ {\ textrm {d}}} f（x）\ \ textrm {d} x = \ tfrac 12 $ | 和| $ \ int _ {{\\ mathbb {R}} ^ {\ textrm {d}}} | x | ^ rf（x）\ \ textrm {d} x <\ infty $ |⁠。但是，如果| $ \ int _ {{\\ mathbb {R}} ^ {\ textrm {d}}} f（x）\ \ textrm {d} x =：a <\ tfrac 12 $ |⁠，则衰减为无限快得多：我们证明，对于所有| $ a <\ tfrac 12 $ |⁠，存在一些解决方案，例如对于| $ \ varepsilon> 0 $ |⁠，| $ \ int _ {{\ mathbb { R}} ^ {\ textrm {d}}} e ^ {\ varepsilon | x |} f（x）\ \ textrm {d} x <\ infty $ |⁠。