Let $f(x,y)$ be in $L^p(\mathbb{R^2})$. Suppose I define a new function $\mathring{f}$ by $$ \mathring{f}(x) = f(x,\phi(x)) $$ for some suitable real valued smooth function $\phi$. What can I say about the integrability of $\mathring{f}$?
Are there any suitable conditions that one can impose on $\phi$ so that $\mathring{f}$ is in $L^q(\mathbb{R})$ for any $q$ ?