That (2) implies (1) is contained in the proof of Theorem 1 of [4].
Clearly, $A_{\infty}$ weights are sharp weights. That there are no others is the main result of Section 2.
The degree of $P$ equals that of $Q$.
The continuity of $f$ implies that of $g$.
The diameter of $F$ is about twice that of $G$.
It is this point of view which is close to that used in $C^*$-algebras.
Define $f(z)$ to be that $y$ for which ......
Where there is a choice of several acceptable forms, that form is selected which ......
Associated with each Steiner system is its automorphism group, that is, the set of all ......
The usefulness and interest of this correspondence will of course be enhanced if there is a way of returning from the transforms to the functions, that is to say, if there is an inversion formula.
We now state a result that will be of use later.
A principal ideal is one that is generated by a single element.
Let $I$ be the family of all subalgebras that contain $F$. [Or: which contain $F$; you can use either that or which in defining clauses.]
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