2
$\begingroup$

It is a well known result that a Muckenhoupt weight $w \in A_p$ if and only if $w= w_1^{1-p} w_2$ with $w_1,w_2 \in A_p$. This decomposition was proved by P.W.Jones.

I am only interested in $A_2$ and ask the following question: Suppose I fix $w_1 \in A_1$ and consider all $w \in A_2 : w = w_1^{-1} w_2$ with $w_2 \in A_1$, i.e let $w_1$ be fixed and consider the subclass of $A_2$ generated by taking all $w_2 \in A_1$. Is it true that $$[w]_{A_2} \approx [w_2]_{A_1}$$

  • 0
    Prof. Jose Martell has recently informed me that this is false and provided a very nice counter-example. I refrain from posting the counterexample here without his explicit permission. All one can show is $$1\leq [w]_{A_2} \leq [w_2]_{A_1}$$ – 2017-01-30

0 Answers 0