Consider the following expression: $$ A^q=\int_{\mathbb{R}^{n}}\left(\int_{\mathbb{R}^{n}}\left|f\left(x+y\right)g\left(x+y+Cy\right)\right|^{p}dy\right)^{q/p}dx, $$ where $1\leq p,q<\infty$, $C$ is a given invertible matrix and $f,g$ are suitable functions - in particular, such that $h(x)=\int_{\mathbb{R}^{n}}f\left(x+y\right)g\left(x+y+Cy\right)dy$ is well-defined. Integration shall be intendend here in Lebesgue sense. Two questions:
- Is it legitimate to perform a simulatneous double change of variables, such as $$\begin{cases} x=u+C^{-1}v\\ y=-C^{-1}v \end{cases}$$ in order to turn the inner integral into a convolution? I don't believe so, but I would like a confirmation before abandoning this way - which would be in fact very useful to prove $A<\infty$.
- Any other general strategy in order to prove that $A<\infty$? I know that without further context this may be a vague question, but I am truly interested in finding general approaches for problems like this.
\begin{align}A &= \left\| \left\| f(x+y)g(x+y+Cy)\right\|_{L^p(dy)} \right\|_{L^q(dx)} \end{align} Not an answer, but the assumptions on $f,g$ need to be stronger to achieve $A<\infty$. That is, the finiteness of the inner norm $\left\| f(x+y)g(x+y+Cy)\right\|_{L^p(dy)}$ does not imply $A<\infty$. To see this, let $n=1$,$C>0$, $f(x) = e^x\mathbb 1_{x\ge 0}$ and $g(x) = e^{-x}$. Then
$$f(x+y)g(x+y+Cy)= e^{-Cy}\mathbb 1_{y \ge -x }, $$
$$ \left\| f(x+y)g(x+y+Cy)\right\|^p_{L^p(dy)} = \int_{-x}^\infty e^{-Cpy}dy = \frac{e^{Cpx}}{Cp}, $$ and yet $\exp(Cx) \notin L^q(\mathbb R)$ for any $q$.
You might want to see what happens in when $p=q$, and also when $q/p\in\mathbb Z$, as then you get repeated integrals in $y$ and a Fubini type argument may work...