Scaling property of the Fourier transform

If $f_{\lambda }(x)=f(\lambda x)$ on ${\mathbb{R}}^n$ with $\lambda >0$, thenand consequently

If a nontrivial uniform estimateholds for all integrable $L^p$ functions on ${\mathbb{R}}^n$, dilation by every $\lambda >0$ forcesThus $q$ must be the conjugate exponent of $p$.