A pair of indices for function spaces on the circle

Abstract:

We give here some of the basic properties of the classes $\{ {\Phi _r}\}$, $\{ {\Psi _r}\} , - 1 < r < 1$, of dilation operators acting in rearrangement-invariant spaces $\mathfrak {X}$ on the circle It is shown that to each space $\mathfrak {X}$ there correspond two numbers $\xi ,\eta$, called indices, which satisfy $0 \leq \eta \leq \xi \leq 1$; these numbers represent the rate of growth or decay of $\left \| {{\Psi _r}} \right \|$ as $r \to \pm 1$. By using the operators ${\Psi _r}$ to obtain estimates for certain averaging operators ${A_\gamma }$, we are able to show that the indices $(\xi ,\eta )$ coincide with the Boyd indices $(\alpha ,\beta )$. As a consequence, we obtain a Marcinkiewicz-type interpolation theorem for rearrangement-invariant spaces on the circle.