Let $p\in [1,\infty]$, $\Omega$ an open bounded domain with (smooth, if necessary) boundary $\partial \Omega$.

Is there a subspace $X\subset L^p(\Omega)$ - a *simply describable* space, ideally a Sobolev or Besov or Triebel-Lisorkin or somebody else's space - such that the trace operator is surjective from $\{f\in X:\Delta f\in L^p(\Omega)\}$ onto $L^p(\partial\Omega)$?

The answer is affirmative if $p=2$: One can take $X=H^\frac{1}{2}(\Omega)$. That the trace operator is surjective from
$$
\{f\in H^\frac{1}{2}(\Omega):\Delta f\in L^2(\Omega)\}\quad \hbox{onto}\quad L^2(\partial\Omega)
$$
has been checked in [DOI 10.1007/s00020-002-1163-2], Lemma 3.1: The idea is to use a certain isomorphism that allows to construct a *harmonic* $H^\frac{1}{2}(\Omega)$-function with given initial data, but the proof is essentially an application of Lions-Magenes theory, which is restricted to the Hilbert case. (For me, the obvious place to look for a Banach space extension is the book of Grisvard, but I could not find anything there, either).

EDIT: My conjecture is of course that, more generally, $X=W^{\frac{1}{p},p}(\Omega)$ can be taken. This would yield the space $W^{\frac{1}{p},p}(\Omega)\cap D(\Delta;L^p(\Omega))$ with the notation of Grisvard. Such spaces do not seem to have been introduced in the books of Adams-Fourier or Demengel-Demengel.

EDIT #2: Perhaps it is convenient to explain my interest in this question. The point is that I would like to define not only a right inverse of the trace operator, as customary; but rather an operator that maps a given function on the boundary into the (unique) solution of an eigenvalue equation for $\Delta$ whose boundary values are prescribed by the given function - something that was considered e.g. in many articles of the Japanese school active on Dirichlet forms in the 1960s (Fukushima, Sato etc.). Now, I am willing to consider very weak notions of solution, but at the very least I want to make sure that both sides of the the eigenvalue equation are in $L^p$.