Theorems about traces and multipliers for functions from Lizorkin--Triebel spaces

Y. V. Netrusov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 24, Tome 200 (1992), pp. 132-138

Résumé

We briefly outline the results. In section 2 we formulate results about multipliers in Besov and Lizorkin–Triebel spaces. In theorems 1 and 2 a description of spaces of multipliers for $FL_{p,\theta}^l$, $l>0$, $0$, and $F_{p,\theta}^l$, $l\in\mathrm{R}$, $0$, $p\leqslant\theta\leqslant\infty$, ig given (recall that the description of spaces of multipliers for space $BL_{1,1}^l=FL_{1,1}^l$ has been obtained by V. G. Mazya (see [2])). In theorem 3 a description of multipliers in space $BL_{p,\infty}^l$, $l>0$, $0$ and some information about multipliers for spaces $B_{p,\infty}^l$, $l\in\mathrm{R}$, $0$, are given. In section 3 we formulate two results about traces of functions from spaces of Lizorkin–Triebel type. In theorem 4 we give a discription of such subsets $A\subset\mathrm{R}^n$ that trace of space $FL_{p,\theta}^l$, $0$, $l>0$, on set $A$ is a quasibanach lattice. In theorem 5 we indicate a class of measures $\nu$ such that trace of space of Lizorkin–Triebel type on measure $\nu$ is the Lebesgue space $L_p(\nu)$, $0$. In particular, it follows from theorem 5 that trace of $W^l_{L_{p,1}}(\mathrm{R}^n)$ (Sobolev space in metric of Lorentz space $L_{p,1}$ on $m$-dimensional plane $\pi$ ($m\in \mathrm{N}$, $l=(n-m)/p\in \mathrm{N}$, $1$) is equal to $L_p(\mu_m)$, where $\mu_m$ is the Lebesgue measure on the plane $\pi$.