  In this paper, we study solutions to the linearized vacuum Einstein equations
centered at higher-dimensional Schwarzschild met- rics. We employ Hodge
decomposition to split solutions into scalar, co-vector, and two-tensor pieces;
the first two portions respectively cor- respond to the closed and co-closed,
or polar and axial, solutions in the case of four spacetime dimensions, while
the two-tensor portion is a new feature in the higher-dimensional setting.
Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge
decomposition, we produce decoupled gauge-invariant master quantities
satisfying Regge- Wheeler type wave equations in each of the three portions.
The scalar and co-vector quantities respectively generalize the
Moncrief-Zerilli and Regge-Wheeler quantities found in the setting of four
spacetime dimen- sions; beyond these quantities, we further discover a
higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the
co-vector por- tion. In the analysis of the master quantities, we strengthen
the mode stability result of Kodama-Ishibashi to a uniform boundedness estimate
in all dimensions; further, we prove decay estimates in the case of six or
fewer spacetime dimensions. Finally, we provide a rigorous argument that
linearized solutions of low angular frequency are decomposable as a sum of pure
gauge solution and linearized Myers-Perry solution, the lat- ter solutions
generalizing the linearized Kerr solutions in four spacetime dimensions.
