Traces of Besov Spaces Revisited

Abstract

For the trace of Besov spaces $B^s_{p,q}$ onto a hyperplane, the borderline case with $s = \frac {n}{p} – (n–1)$ and $0 < p < 1$ is analysed and a new dependence on the sum-exponent $q$ is found. Through examples the restriction operator defined for $s$ down to $\frac {1}{p}$, and valued in $L_p$, is shown to be distinctly different and, moreover, unsuitable for elliptic boundary problems. All boundedness properties (both new and previously known) are found to be easy consequences of a simple mixed-norm estimate, which also yields continuity with respect to the normal coordinate. The surjectivity for the classical borderline $s = \frac {1}{p} (1 ≤ p < \infty)$ is given a simpler proof for all $q \in ]20, 1]$, using only basic functional analysis. The new borderline results are based on corresponding convergence criteria for series with spectral conditions.