Let us fix $1\le p<r<q$; fix an $l>0$ and take an $f\in L^r(\Bbb R^n)$; then we split the function $$ f=\underbrace{f\chi_{\{|f|>l\}}}_{=:f_1}+\underbrace{f\chi_{\{|f|\le l\}}}_{=:f_2} $$ and at this point my teacher wrote that $$ f_1\in L^p(\Bbb R^n)\\ f_2\in L^q(\Bbb R^n) $$
- Why is this true?
- Should I deduce from this that every $f\in L^r(\Bbb R^n)$ is the sum of of an element of $L^p(\Bbb R^n)$ and one of $L^q(\Bbb R^n)$?
For $|f| \geq 1$, $|f|^{p_1} \geq |f|^{p_2}$ iff $p_1 \geq p_2$. (Multiplying by a number bigger than $1$ gives you a bigger number.) Thus $g=f \chi_{|f| \geq 1}$ inherits the integrability of $f$ as well as any lower integrability.
For $|f| \leq 1$, $|f|^{p_1} \geq |f|^{p_2}$ iff $p_1 \leq p_2$. (Multiplying by a number less than $1$ gives you a smaller number.) Thus $g=f \chi_{|f| \leq 1}$ inherits the integrability of $f$ as well as any higher integrability.
You can play around with constants to generalize this to arbitrary $l>0$, but the case $l=1$ is the most intuitive.
Your conclusion from this result is correct (and useful).