On the interpolation constant for Orlicz spaces

Abstract

In this paper we deal with the interpolation from Lebesgue spaces L p L^p and L q L^q , into an Orlicz space L φ L^\varphi , where 1 ≤ p > q ≤ ∞ 1\le p>q\le \infty and φ − 1 ( t ) = t 1 / p ρ ( t 1 / q − 1 / p ) \varphi ^{-1}(t)=t^{1/p}\rho (t^{1/q-1/p}) for some concave function ρ \rho , with special attention to the interpolation constant C C . For a bounded linear operator T T in L p L^p and L q L^q , we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm, \[ ‖ T ‖ L φ → L φ ≤ C max { ‖ T ‖ L p → L p , ‖ T ‖ L q → L q } , \|T\|_{L^\varphi \to L^\varphi } \le C\max \Big \{ \|T\|_{L^p\to L^p}, \|T\|_{L^q\to L^q} \Big \}, \] where the interpolation constant C C depends only on p p and q q . We give estimates for C C , which imply C > 4 C>4 . Moreover, if either 1 > p > q ≤ 2 1> p>q\le 2 or 2 ≤ p > q > ∞ 2\le p>q>\infty , then C > 2 C> 2 . If q = ∞ q=\infty , then C ≤ 2 1 − 1 / p C\le 2^{1-1/p} , and, in particular, for the case p = 1 p=1 this gives the classical Orlicz interpolation theorem with the constant C = 1 C=1 .