A sharp form of the Marcinkiewicz interpolation theorem for Orlicz spaces
Volume 255 / 2020
Abstract
Let $(X,\mu )$ and $(Y,\nu )$ be $\sigma $-finite measure spaces with $\mu (X)=\nu (Y)=\infty $ and $(Y,\nu )$ nonatomic and separable. Suppose $T$ is a so-called $r$-quasilinear operator mapping the simple functions on $X$ into the measurable functions on $Y$ that satisfies the weak type conditions $$ \lambda \nu ( \lbrace y \in Y : |(Tf)(y)| \gt \lambda \rbrace )^{{1}/{p_{i}}} \leq C_{p_{i},q_{i}} \Big ( \int _{\mathbb {R_+}} \mu ( \lbrace x \in X : |f(x)| \gt t \rbrace )^{{q_{i}}/{p_{i}}} t^{q_{i}-1}\,dt \Big )^{{1}/{q_{i}}},\quad i=0,1, $$ where $1 \lt p_0 \lt p_1 \lt \infty $, $1 \leq q_0, q_1 \lt \infty $ and $C_{p_{i},q_{i}}=C_{p_{i},q_{i}}(T) \gt 0$ is independent of simple $f$ on $X$ and $\lambda \gt 0$.
We give necessary and sufficient conditions on Young functions $\Phi _1$ and $\Phi _2$ in order that any operator $T$ as described above is bounded between the corresponding Orlicz spaces.
