Let $R>0$, and $(B_k)_{k \in \Bbb{N}}$ a collection of disjoint balls of radius R and let $f$ be a measurable function on $\mathbb{R}^n$ of the form
$$f = \sum_{k=1}^{\infty} a_k X_{B_k}$$for some sequence of coefficients $a_n$ that depend on $R$.
For $1 \leq p \leq q \leq \infty$ prove that $$||f||_q \leq C_n R^{n(1/p-1/q)}||f||_p$$
Where $C_n>0$ is a constant which depends only on the dimension n (and, in particular, isindependent of $R$).
We can easily see that $||f||_q^q=\sum_k|a_k|^qR^n$
I solved the exercises for the case $R \geq1$. How can i do this for $R<1$?
Thank you in advance.