论文标题
关于卷积不平等$ f \ geq f \ star f $
On the convolution inequality $f \geq f\star f$
论文作者
论文摘要
我们考虑不平等$ f \ geqslant f \ star f $用于$ d $ d $ dimensional Euclidean Space上的真实集成功能，其中$ f \ star f $表示与自身的卷积。我们表明，所有这些功能$ f $都是非负的，对于任何$ 1 <p \ leqslant 2 $，在$ l^p $中的同一不平等情况并非如此，为此定义了卷积。我们还表明，所有集成解决方案$ f $ assimal $ \ int f（x）{\ rm d} x \ leqslant \ tfrac12 $。此外，如果$ \ int f（x）{\ rm d} x = \ tfrac12 $，则$ f $必须相当缓慢地衰减：$ \ int | x | f（x）{\ rm d} x = \ infty $，这很敏锐，因为对于所有$ r <1 $，都有$ \ int f（x）{\ rm d} {\ rm d} x = \ tfrac12 $和$ \ int | x | x | x | x |^r f（x）{\ rm f（x）{\ rm d} x <\ rm d} x <\ infty $。但是，如果$ \ int f（x）{\ rm d} x =：a <\ tfrac12 $，无穷大的衰减可能要快得多：我们表明，对于所有$ a <\ tfrac12 $
We consider the inequality $f \geqslant f\star f$ for real integrable functions on $d$ dimensional Euclidean space 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 integrable solutions $f$ satisfy $\int f(x){\rm d}x \leqslant \tfrac12$. Moreover, if $\int f(x){\rm d}x = \tfrac12$, then $f$ must decay fairly slowly: $\int |x| f(x){\rm d}x = \infty$, and this is sharp since for all $r< 1$, there are solutions with $\int f(x){\rm d}x = \tfrac12$ and $\int |x|^r f(x){\rm d}x <\infty$. However, if $\int f(x){\rm d}x = : a < \tfrac12$, the decay at infinity can be much more rapid: we show that for all $a<\tfrac12$, there are solutions such that for some $ε>0$, $\int e^{ε|x|}f(x){\rm d}x < \infty$.
