What is the best estimation for the following?

Question

Let $f$ be 1-periodic and $f\in L_{p}[0,1]$ where $p>1.$ Let $D_{n}$ $n=0,1,2,..$ be the dyadic partition of $[0,1].$ Consider $$ F_{n}(x)=\frac{1}{|I^{n}_{j}|}\int_{I^{n}_{j}}f(t)dt, \,\,\,\,x\in I^{n}_{j}\in D_{n}. $$ Q. What is the best estimate (depending on $n$) for the following?
$$\|f-F_{n}\|_{L_{p}[0,1]}\leq?$$