Mathematical Principles of Theoretical Physics

3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 127

Remark 3.8.WhenMis non-compact, the conclusions (3.2.11) and (3.2.12) are also valid
only forqsatisfying

p≤q≤

np

n−p

ifn>p,

p≤q<∞ ifn=p.

Remark 3.9.By the recurrence relations

Wk,p(M⊗pEN)֒→Wk−^1 ,q(M⊗pEN),

we readily deduce from (3.2.11) that

Wk,p(M⊗pEN)֒→











Lq(M⊗pEN) for 1≤q≤

np

n−kp

ifn>kp,

Lq(M⊗pEN) for 1≤q<∞ ifn=kp,

CM,α(M⊗pEN) form+α=k−n/p ifn<kp.

Based on weakly differentiability properties (3.2.6) and (3.2.7), essence of Theorem3.7
can be seen from embeddings (3.2.11) using the following function:

(3.2.13) u(x) =

|x|α

|ln|x||β

, x∈Rn,

whereβ>1 is given, and the exponentα<1 is to reflect the critical embedding indexq∗in
(3.2.11).
The derivatives ofugiven by (3.2.13) are as follows

(3.2.14)

∇u= (∂ 1 u,···,∂nu),

∂iu=

(

α

|ln|x||β

−

β

|ln|x||β+^1

)

|x|α−^2 xi.

LetBR={x∈Rn| |x|<R}, 0 <R<1. Assume that

u∈W^1 ,p(BR) for some 1≤p<∞.

Then, by (3.2.14) we have

(3.2.15)

∫

BR

|∇u|pdx≤C

∫

BR

|x|(α−^1 )p

|ln|x||βp

dx.

In the spherical coordinate system,

dx=rn−^1 drds,

anddsis the area element of the unit sphere, (3.2.15) becomes

∫

BR

|∇u|pdx≤C

∫R

0

∫

Sn−^1

rn+(α−^1 )p−^1

|lnr|βp

drds≤C

∫R

0

rk

|ln|r||βp

(3.2.16) dr,