When weighted $L^2-$norm are comparable?

Question

Let $f:\mathbb R^2 \to \mathbb C$ be a measurable function. Let $r_1, r_2 \in \mathbb N$ such that $r_1+r_2=r.$

Question:Can we expect $$\int_{\mathbb R^2} |x|^{2r} |f(x)|^2 dx \leq C \int_{\mathbb R^2} x_1^{2r_1}x_2^{2r_2}|f(x)|^2 dx$$ for some constant $C>0$?

Edit: $(x_1, x_2)\in \mathbb R^2, |x|^{2r}= (x_1^2+ x_2^2)^r$

Using the hint given below, I have tried to do this. $\int_{\mathbb R^2}|x|^{2r}|f|^2 \leq C \int_{\mathbb R^2} x_1^{2r_1+2r_2}|f|^2 dx + C \int_{\mathbb R^2} x_2^{2r_1+2r_2}|f|^2 dx $.

From this I do not know how to proceed? (Maybe I have to compare $x_1$ and $x_2$?)

Answer 1

HINT: use the inequality $$ 1\le\frac{(x+y)^p}{x^p+y^p}\le2^{p-1} $$ which holds for all $x,y>0$ in which $p>1$.