  We develop an $L^p(\mathbb{R}^n)$-functional calculus appropriated for
interpreting "non-classical symbols" of the form $a(-\Delta)$, and for proving
existence in $L^q(\mathbb{R}^n)$, some $q > p$, of solutions to nonlinear
pseudo-differential equations of the form $[1 + a(-\Delta)]^{s/2} (u) =
V(\cdot, u)$. More precisely, we use the theory of Fourier multipliers for
constructing suitable domains on which the formal operator appearing in the
above equation can be rigorously defined, and we prove existence of solutions
belonging to these domains. We also include applications of the theory to
equations of physical interest involving the fractional Laplace operator such
as the Allen-Cahn equation.
