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We have a canonical integration map ∫: C^∞(Dens₁(M))→C. We use this map to define norms on spaces C^∞(Dens_p(M)) for all p∈C such that ℜp>0. First we send f∈C^∞(Dens_p(M)) to |f|=(ff)^1/2∈C^∞(Dens_ℜp(M)). Observe that ff and |f| are positive with respect to the canonical orientations on Dens_2ℜp(M) and Dens_ℜp(M). Then |f|^1/ℜp∈C^∞(Dens₁⁺(M)) and we set ‖f‖:=∫⁠(|f|^1/ℜp). This is a norm for ℜp≤1 and a quasi-norm for ℜp>1.

Thus we defined L_p-spaces of the trivial line bundle on M for an arbitrary p∈C such that ℜp≥0. To extend this definition to an arbitrary hermitian vector bundle V we replace ff by μ(ff) in the above definition of norm. Here μ denotes the hermitan pairing on V.

We have a canonical integration map ∫: C^∞(Dens₁(M))→C. We use this map to define norms on spaces C^∞(Dens_p(M)) for all p∈C such that ℜp>0. First we send f∈C^∞(Dens_p(M)) to |f|=(f*f)^1/2∈C^∞(Dens_ℜp(M)). Observe that f*f and |f| are positive with respect to the canonical orientations on Dens_2ℜp(M) and Dens_ℜp(M). Then |f|^1/ℜp∈C^∞(Dens₁⁺(M)) and we set ‖f‖:=∫⁠(|f|^1/ℜp). This is a norm for ℜp≤1 and a quasi-norm for ℜp>1.

Thus we defined L_p-spaces of the trivial line bundle on M for an arbitrary p∈C such that ℜp≥0. To extend this definition to an arbitrary hermitian vector bundle V we replace f*f by μ(f*f) in the above definition of norm. Here μ denotes the hermitan pairing on V.

For the sake of convenience I denote L_p=L^1/p (see this answer for a motivation), in particular L₀=L^∞ and L_1/2=L². As explained in the link above, p is an arbitrary complex number such that ℜp≥0.

For the sake of convenience I denote L_p=L^1/p (see this answer for a motivation), in particular L₀=L^∞ and L_1/2=L². As explained in the link above, p is an arbitrary complex number such that ℜp≥0.

For the sake of convenience I denote L_p=L^p (see this answer for a motivation), in particular L₀=L^∞ and L_1/2=L². As explained in the link above, p is an arbitrary complex number such that ℜp≥0.

For the sake of convenience I denote L_p=L^1/p (see this answer for a motivation), in particular L₀=L^∞ and L_1/2=L². As explained in the link above, p is an arbitrary complex number such that ℜp≥0.