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A sharp form of the Marcinkiewicz interpolation theorem for Orlicz spaces

Volume 255 / 2020

Ron Kerman, Rama Rawat, Rajesh K. Singh Studia Mathematica 255 (2020), 109-158 MSC: Primary 46E30; Secondary 46M35. DOI: 10.4064/sm180111-28-10 Published online: 13 May 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.

Authors

  • Ron KermanDepartment of Mathematics
    Brock University
    St. Catharines, Ontario, L2S 3A1, Canada
    e-mail
  • Rama RawatDepartment of Mathematics and Statistics
    Indian Institute of Technology
    Kanpur 208016, India
    e-mail
  • Rajesh K. SinghDepartment of Mathematics and Statistics
    Indian Institute of Technology
    Kanpur 208016, India
    e-mail

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