On the interpolation of the spaces and
Eduard Curcă
University “Alexandru Ioan Cuza”, Iasi, Romania
![On the interpolation of the spaces $W^{l,1}( \mathbb{R}^{d})$ and $W^{r,\infty }( \mathbb{R}^{d})$ cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-rmi-volume-40-issue-3.png&w=3840&q=90)
Abstract
We study some properties of spaces obtained by interpolation of the Sobolev spaces and , where and are nonnegative integers, and . We are concerned with the standard real and complex methods of interpolation. In the case of the real method, an old result of De Vore and Scherer (1979) gives that
where and . We complement this result by considering the case . We prove that, when ,
where and , if and only if . Also, we prove a similar fact when is replaced in by a space where is a real number and . Several other problems like the boundedness of the Riesz transforms on interpolation spaces are also considered. In the case of the complex method, it was proved by M. Milman (1983) that, for any ,
where . We show by simple arguments that fails when and , answering a question of P. W. Jones (1984). As an immediate consequence of these arguments, we show that certain closed subspaces of (with ) that are described by Fourier multipliers are not complemented in .
Cite this article
Eduard Curcă, On the interpolation of the spaces and . Rev. Mat. Iberoam. 40 (2024), no. 3, pp. 931–986
DOI 10.4171/RMI/1447