Sobolev, Besov and Bessel-potential spaces satisfy two obvious embedding
relations:
- For fixed pp (and arbitrary qq in the case of Besov spaces), the
spaces get larger as ss decreases.
- In the case where ΩΩ a bounded domain, for fixed ss (and fixed qq
in the case
of Besov spaces), the spaces get larger as pp decrease, since
∥f∥Lp1≤C∥f∥Lp2∥f∥Lp1≤C∥f∥Lp2 if p1≤p2p1≤p2.
A less trivial type of embedding is known as Sobolev embedding. In
the case
of Sobolev spaces, it states that the continuous embedding
W
s
1
,
p
1
⊂
W
s
2
,
p
2
if
p
1
≤
p
2
and
s
1
-
s
2
≥
d
(
1
/
p
1
-
1
/
p
2
)
,
W
s
1
,
p
1
⊂
W
s
2
,
p
2
if
p
1
≤
p
2
and
s
1
-
s
2
≥
d
(
1
/
p
1
-
1
/
p
2
)
,
(1)holds except in the case where
p2=+∞p2=+∞ and s2s2 is an integer, for which one needs
to assume
s1-s2>d(1/p1-1/p2)s1-s2>d(1/p1-1/p2).
For example in the univariate case, any H1H1 function has also C1/2C1/2
smoothness.
In the case of Besov spaces the embedding relation are given by
B
p
1
,
p
1
s
1
⊂
B
p
2
,
p
2
s
2
if
p
1
≤
p
2
and
s
1
-
s
2
≥
d
(
1
/
p
1
-
1
/
p
2
)
,
B
p
1
,
p
1
s
1
⊂
B
p
2
,
p
2
s
2
if
p
1
≤
p
2
and
s
1
-
s
2
≥
d
(
1
/
p
1
-
1
/
p
2
)
,
(2)with no other restrictions on the indices s1,s2≥0s1,s2≥0.
In the case where ΩΩ is a bounded domain, these embedding
are compact if and only if the strict inequality s1-s2>d(1/p1-1/p2)s1-s2>d(1/p1-1/p2) holds.
The proof of these embeddings can be found in [1] for
Sobolev spaces and [6] for Besov spaces.
As an exercise, let us see how these embeddings can be used to derive the range
of rr such that B2,qr([0,1])B2,qr([0,1]) can contain discontinuous functions. If
r>1/2r>1/2, then there exists ε>0ε>0 such that r-2ε>1/2r-2ε>1/2; We remark
that B2,qr⊂B2,qr-ε⊂B∞,∞ε=CεB2,qr⊂B2,qr-ε⊂B∞,∞ε=Cε, so all functions in B2,qrB2,qr are
continuous.
Therefore only B2,qrB2,qr with r≤1/2r≤1/2 can contain discontinuous
functions. In the limiting case r=1/2r=1/2, a closer inspection reveals that
the functions in B2,q1/2B2,q1/2 are continuous if q<∞q<∞, while
B2,∞1/2B2,∞1/2 includes discontinuous functions, such as the characteristic
funciton of an interval [0,a][0,a] for 0<a<10<a<1.
-
Adams, R. (1975). Sobolev Spaces. Academic Press.
-
Cohen, A. (2003). Numerical Analysis of Wavelet Methods. Elsevier.
-
Daubechies, Ingrid. (1992). Ten Lectures on Wavelets. [Notes from the 1990 CBMS-NSF Conference on Wavelets and Applications at Lowell, MA]. Philadelphia, PA: SIAM.
-
DeVore, R. (1998). Nonlinear Approximation. Acta Numerica.
-
R. DeVore, B. Jawerth, and V. Popov,. (1992). Compression of wavelet decompositions. American Journal of Math, 114, 737-285.
-
H. Triebel. (1983). Theory of Function Spaces. Birkhauser.
(What does "Endorsed by" mean?)
"This is an introduction to fractional measures of smoothness for functions via Besov spaces that generalize the classical Sobolev smoothness spaces."