Abstract 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 allmore »
This content will become publicly available on May 20, 2023
Enhanced charge density wave coherence in a lightquenched, hightemperature superconductor
Ultrafast resonant soft xray scattering is used to monitor the dynamics of the charge density wave order in YBa 2 Cu 3 O 6+x .
 Authors:
 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; more »
 Publication Date:
 NSFPAR ID:
 10333214
 Journal Name:
 Science
 Volume:
 376
 Issue:
 6595
 Page Range or eLocationID:
 860 to 864
 ISSN:
 00368075
 Sponsoring Org:
 National Science Foundation
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