Norms of an anisotropically concentrating profile

Statement

Let $V\in L^p(\mathbb R^d)$ , $1\le p\le\infty$ , and set $u_\tau(x)=\tau^{-a}V(D_\tau^{-1}x)$ , where $D_\tau$ has exponents $b_i$ . Then

\|u_\tau\|_{L^p}=\tau^{-a+(b_1+\cdots+b_d)/p}\|V\|_{L^p},

with $1/\infty=0$ . The norm is finite under the stated hypothesis on $V$ .

Proof

For finite $p$ , substitute $x=D_\tau y$ in $\int|u_\tau|^p\,dx$ ; the volume factor is $\tau^{\sum b_i}$ . For $p=\infty$ , an invertible linear change of variables preserves null sets and the essential supremum of the profile.

Energy and peak size

The squared $L^2$ norm scales as $\tau^{-2a+\sum b_i}$ , whereas $\|u_\tau\|_\infty=\tau^{-a}\|V\|_\infty$ . Thus if $0<2a<\sum b_i$ , a nonzero bounded profile has growing peak size but vanishing squared $L^2$ norm. This scaling observation alone proves no PDE solution exists.