Basic Properties of Sobolev Spaces
1.3 Some Inequalities for Functions of One Variable
1.3.4 Hardy-Type Inequalities with Indefinite Weights
Letp=∞. Then B =
∞
0
x 0
dy
|v(y)|
q−1 ∞
x
w(y)qdy dx
|v(x)| 1/q
=q−1/q ∞
0
w(x)q x 0
dy
|v(y)| q
dx 1/q
.
Hence ∞
0
w(x) x
0
f(t) dt qdx
1/q
≤Bq1/qess sup
0<x<∞|vf|.
To prove the necessity we note thatvdoes not vanish on the set of positive
measure and putf = 1/v. The theorem is proved.
The following more general assertion can be derived from Theorem 1 in the same way as Theorem1.3.2/1 was derived from Theorem 1.3.2/2.
Theorem 2. Let μ and ν be nonnegative Borel measures on (0,∞) and letν∗ be the absolutely continuous part ofν. Inequality (1.3.8) with1 ≤q <
p≤ ∞holds for all Borel functionsf if and only if B=
∞
0
μ
[x,∞) x
0
dν∗ dy
−p
dy
q−1p/(p−q) dν∗
dx −p
dx
(p−q)/pq
<∞.
The best constantC in (1.3.17)is related withB in the same manner as in Theorem1.
The change of variable (0,∞)x→y =x−x−1∈(−∞,+∞) leads to the following necessary and sufficient condition for the validity of (1.3.18):
+∞
−∞
μ
(−∞, x] ∞
x
dν∗ dy
−p
dy
q−1p/(p−q) dν∗
dx −p
dx <∞, where 1≤q < p≤ ∞.
Let us assume thatQis a locally integrable real- or complex-valued func- tion such that
lim
b→+∞
b a
Q(x) dx= ∞
a
Q(x) dx (1.3.26)
exists for everya >0.
Theorem 1.Under the above assumptions on Q, let Γ(x) =
∞
x
Q(t) dt, x >0.
Let 1< p <∞, andp∗= max(p, p). Then (1.3.25)is valid if and only if sup
a>0
ap∗−1 ∞
a
Γ(x)p∗dx <∞. (1.3.27)
It is not difficult to see that (1.3.27) is equivalent to the pair of conditions sup
a>0
ap−1 ∞
a
Γ(x)pdx <∞, sup
a>0
ap−1 ∞
a
Γ(x)pdx <∞. (1.3.28)
Proof.Foru, v∈C0∞(R+), let Qu, v=
∞
0
Q(x)u(x)v(x) dx.
We can extendQu, vby continuity to the case where u(x) =
x 0
f(t) dt, v(x) = x
0
g(τ) dτ, forf, g∈C0∞(R+), by setting
Qu, v= lim
a→+∞
a 0
Q(x)u(x)v(x) dx.
To show that the limit on the right-hand side exists, assume that bothf and g are supported in (δ, b)⊂R+. Then clearly,
a→lim+∞
a 0
Q(x)u(x)v(x) dx= b
δ
Q(x) x
δ
f(t) dt x
δ
g(τ) dτ
dx +
∞
b
Q(x) dx b
δ
f(t) dt b
δ
g(τ) dτ.
Observe that we have to be careful here: In what follows one cannot esti- mate the two terms on the right-hand side of the preceding equation separately because this would lead to the restriction
sup
b>0
b ∞
b
Q(x) dx <∞,
which is not necessary for the boundedness of the bilinear form.
Using Fubini’s theorem, we obtain Qu, v=
b δ
b δ
f(t)g(τ) b
max(t,τ)
Q(x) dxdtdτ +
∞
b
Q(x) dx b
δ
f(t) dt b
δ
g(τ) dτ
= b
δ
b δ
f(t)g(τ) ∞
max(t,τ)
Q(x) dxdtdτ.
By definition
Γ
max{t, τ}
= ∞
max(t,τ)
Q(x) dx.
Thus, (1.3.25) is equivalent to the inequality ∞
0
∞
0
f(t)g(τ)Γ
max{t, τ} dtdτ
≤constfLp(R+)gLp(R+) (1.3.29) for compactly supportedf, g.
Using the reverse H¨older inequality, the preceding estimate can be rewrit- ten in the equivalent form
∞
0
∞
0
Γ
max{t, τ} f(t) dt
pdτ≤cfpLp(R+). (1.3.30) Clearly,
∞
0
Γ
max{t, τ}
f(t) dt=Γ(τ) τ
0
f(t) dt+ ∞
τ
f(t)Γ(t) dt. (1.3.31) Suppose now that (1.3.27), or equivalently, both inequalities in (1.3.28) hold. Then the estimate involving the first term in (1.3.31) is established by means of the weighted Hardy inequality
∞
0
τ 0
f(t) dt
pΓ(τ)pdτ≤CfpLp(R+), (1.3.32) which holds if and only if the first part of condition (1.3.28) is valid (see Theorem1.3.2/1).
The second term in (1.3.31) is estimated by using a similar weighted Hardy inequality
∞
0
∞
τ
f(t)Γ(t) dt
pdτ≤CfpLp(R+),
which, by Theorem 1.3.2/3, is equivalent to the second part of condition (1.3.28). This proves the “if” part of the theorem.
To prove the “only if” part, it suffices to assume that f(x) in (1.3.30) is supported on an interval [δ, a], a >0, and restrict the domain of integration inτ on the left-hand side of (1.3.30) toτ ∈(a,+∞). Taking into account that the second term in (1.3.31) vanishes, we get
∞
a
a δ
Γ
max{t, τ} f(t) dt
pdτ
= a
δ
f(t) dt p ∞
a
Γ(τ)pdτ ≤CfpLp(R+).
Applying the reverse H¨older inequality again, we obtain the first part of (1.3.28)
ap−1 ∞
a
Γ(τ)pdτ ≤C.
Since (1.3.30) is symmetric, a dual estimate in the Lp norm yields the second part of (1.3.28)
ap−1 ∞
a
Γ(τ)pdτ ≤C.
Hence (1.3.27) holds.
Remark 1.Notice that a similar argument works with minor changes if the integration (1.3.25) is performed against real- or complex-valued measure dQ in the place of Q(x) dx. However, the general case whereQ is a distribution requires taking care of some technical problems which are considered in detail by the author and Verbitsky in [592], Sect. 2.
Remark 2.For anyp∈(1,∞), a simple condition sup
a>0
aΓ(a)<∞, (1.3.33)
is sufficient, but generally not necessary for (1.3.25) to hold. However, for nonnegative Q, condition (1.3.33) is equivalent to (1.3.27).
Theorem 1 is easily carried over to the two-weight setting.
Theorem 2.LetW1, W2≥0be locally integrable weight functions onR+
such that, respectively, a
0
W1(x)1−pdx <+∞ and a
0
W2(x)1−pdx <+∞
for everya >0. Then the two-weight bilinear inequality ∞
0
u(x)v(x)Q(x) dx
≤const ∞
0
|u(x)|pW1(x) dx 1/p
× ∞
0
v(x)pW2(x) dx 1/p
(1.3.34) holds for all u, v ∈ C0∞(R+) if and only if the following pair of conditions hold:
sup
a>0
a 0
W1(x)1−pdx
p−1 ∞
a
Γ(x)pW2(x)1−pdx <∞ (1.3.35)
and sup
a>0
a 0
W2(x)1−pdx
p−1 ∞
a
Γ(x)pW1(x)1−pdx <∞, (1.3.36) whereΓ(x) =∞
x Q(t) dt.
For functions defined on the interval (0,1), Theorem 2 can be recast in a similar way.
Theorem 3. The inequality 1
0
u(x)v(x)Q(x) dx
≤constu(x)
Lp(0,1)v(x)
Lp(0,1) (1.3.37) holds for allu, v∈C∞(0,1) such thatu(0) = 0, v(0) = 0if and only ifQcan be represented in the form Q=Γ, where
sup
a>0
ap−1 1
0
Γ(x)p∗dx <∞ (1.3.38) asa→0+. The corresponding compactness criterion holds with the preceding condition replaced by
lim sup
a→0+
ap∗−1 1
0
Γ(x)p∗dx= 0.
For functions with zero boundary values at both endpoints, one only has to add similar conditions at a= 1.
We now state the analog of Theorem 1 on the whole lineRfor the Sobolev space Wp1(R) which consists of absolutely continuous functions u : R → C such that
uWp1(R)=
R
u(x)p+u(x)p dx
1/p
<∞. Theorem 4.Let 1< p <∞, andp∗= max(p, p). The inequality
R
u(x)v(x)Q(x) dx
≤constuWp1(R)vWp1(R) (1.3.39) holds for all u, v ∈ C0∞(R), if and only if Q can be represented in the form Q=Γ+Γ0, whereΓ andΓ0 satisfy the following conditions:
sup
a>0
a+1 0
Γ(x)p∗dx <∞, sup
a>0
a+1 0
Γ0(x)dx <∞. (1.3.40)
The proofs of Theorems 3 and 4 are similar to the proof of Theorem 1.
The usual approach to inequality (1.3.25), in the case where Q is real valued, is to represent it in the form Q = Q+ −Q−, where Q+ and Q− are, respectively, the positive and negative parts of Q, and then treat them separately. However, this procedure ignores a possible cancellation between Q+ andQ− and diminishes the class of admissible potentials Q.
The following examples demonstrate the difference between sharp results which follow from Theorem 1, and the usual approach where Q+ andQ− are treated separately.
Example 1.Let
Q(x) =sinx
x1+, >0.
Then
Γ(x) = +∞
x
sint
t1+dt= cosx x1+ +O
1 x2+
asx→+∞.
As x→0+, clearly,Γ(x) =O(1) for <1,Γ(x) =O(logx) for = 1, and Γ(x) =O(x1−) for >1. From this it is easy to see that (1.3.27) is valid if and only if 0≤≤2, and hence by Theorem 1,L: ˚L1p(R+)→L−p1(R+) is bounded for 1< p <∞. Moreover, the multiplication operatorQ: ˚L1p(R+)→L−p1(R+) is compact if and only if 0< <2.
Note that the same Theorem 1 applied separately to Q+ andQ− gives a satisfactory result only for 1≤≤2.
In the next example,Qis a charge onR+, and the condition imposed on Qdepends explicitly onp.
Example 2.Let
Q=
∞ j=1
cj(δj−δj+1),
whereδa is a unit point mass atx=a. Then clearly Γ(x) =
∞ j=1
cjχ(j,j+1)(x).
It follows that (1.3.25) holds if and only if sup
n≥1
np∗−1
∞ j=n
|cj|p∗ <∞.
In particular, for 1< r≤2, letcj =j−1/rifj = 2m, andcj = 0 otherwise.
ThenL: ˚L1p(R+)→L−p1(R+) if and only if r≤p≤r/(r−1). Note that in this example condition (1.3.27) fails for allr >1.