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.