In this paper we derive some Sobolev inequalities for radially symmetric functions in ˙H s with 1 2 < s < n 2 . We show the end point case s = 1 2 on the homogeneous Besov space ˙B 12 2;1. These results are extensions of the well-known Strauss’ inequality [11]. Also non-radial extensions are given, which show that compact embeddings are possible in some Lp spaces if a suitable angular regularity is imposed

Cho, Yonggeun

Ozawa, Tohru

Hokkaido University Collection of Scholarly and Academic Papers

English

Sobolev inequalities with symmetry

EPrint Series of Department of Mathematics, Hokkaido University

SOBOLEV INEQUALITIES WITH SYMMETRY
YONGGEUN CHO AND TOHRU OZAWA
Abstract. In this paper we derive some Sobolev inequalities for radially symmetric
functions in H˙s with 1
2
< s < n
2
. We show the end point case s = 1
2
on the homogeneous
Besov space B˙
1
2
2,1. These results are extensions of the well-known Strauss’ inequality [11].
Also non-radial extensions are given, which show that compact embeddings are possible
in some Lp spaces if a suitable angular regularity is imposed.
1. Introduction
In this paper we derive Sobolev inequalities with symmetry. We first consider several
Sobolev inequalities for radially symmetric functions in H˙s(Rn) with 12 < s <
n
2 . There is
a sharp result by Sickel and Skrzypczak [8], although the argument below is much simpler
and direct and a constant in the inequality in Proposition 1 below is given explicitly in
terms of s.
Definition 1.
H˙srad = {u ∈ H˙s(Rn) : u is radially symmetric }, s ≥ 0.
B˙sp,q,rad = {u ∈ B˙sp,q(Rn) : u is radially symmetric}, s ≥ 0, 1 ≤ p, q ≤ ∞.
The inhomogeneous spaces of radially symmetric functions are defined by the same way
with spaces Hs and Bsp,q.
Proposition 1. Let n ≥ 2 and let s satisfy 1/2 < s < n/2.
Then
sup
x∈Rn\{0}
|x|n/2−s|u(x)| ≤ C(n, s)‖(−∆)s/2u‖L2 (1)
for all u ∈ H˙srad, where
C(n, s) =
(
Γ(2s− 1)Γ(n2 − s)Γ(n2 )
22spin/2Γ(s)2Γ(n2 − 1 + s)
)1/2
and Γ is the gamma function.
Remark 1. For s = 1 with n ≥ 3, the inequality (1) reduces to Ni’s inequality [6, 7].
Remark 2. The restriction 1/2 < s < n/2 is necessary for C(n, s) to be finite.
2000 Mathematics Subject Classification. Primary 46E35; Secondary 42C15.
Keywords and phrases. Sobolev inequality, function space with radial symmetry, angular regularity.
1
2 Y. CHO AND T. OZAWA
Remark 3. The inequality (1) fails for s = n/2. Indeed, u(x) = F−1
(
1
(1+|ξ|)n(1+log(1+|ξ|))
)
satisfies u ∈ Hn/2rad , and u 6∈ L∞ where F is the Fourier transform [12] and F−1 is its inverse.
Remark 4. The inequality (1) fails if 0 ≤ s < 1/2 and n ≥ 3. Indeed, u =
F−1(Jn
2
−1(|ξ|)|ξ|−n/2) satisfies u ∈ H˙srad and u(x) = ∞ for all x ∈ Sn−1, where we note
that u ∈ H˙srad if and only if 1− n/2 < s < 1/2, since
‖(−∆)s/2u‖2L2 = cn
∫ ∞
0
|Jn
2
−1(ρ)|2ρ2s−1dρ
and that by the asymptotic behavior of Bessel function (10)
u(x) =
∫ ∞
0
|Jn
2
−1(ρ)|2dρ =∞, x ∈ Sn−1.
See also the proof of Proposition 1 below.
In the endpoint case s = 1/2, we have the following propostion.
Proposition 2. Let n ≥ 2. Then there exists a constant C such that
sup
x∈Rn\{0}
|x|(n−1)/2|u(x)| ≤ C‖u‖
B˙
1/2
2,1
(2)
for all u ∈ B˙1/22,1,rad.
Remark 5. The inhomogeneous version of (2) has been given in [8] whose proof is
based on the atomic decomposition.
Proposition 3. Let n ≥ 2 and let s satisfy 1/2 ≤ s < 1. Then there exists C such that
for all u ∈ H1rad
sup
x∈Rn\{0}
|x|n/2−s|u(x)| ≤ C(n, s)‖u‖1−s
L2
‖∇u‖sL2 . (3)
Remark 6. For s = 1/2, the inequality (3) reduces to Strauss’ inequality [11].
Remark 7. For s = 0, the inequality (3) holds for nonincreasing functions in |x| [2].
For s = 1, the inequality (3) holds for n ≥ 3 and fails for n = 2. See Proposition 1 and
Remark 2.
Now we extend the results above on radial functions to the non-radial functions with
additional angular regularity. For details, let us define function spaces Hs,mω and B
s,m
2,1,ω,
s ≥ 0,m ≥ 0 as follows.
SOBOLEV INEQUALITIES 3
Definition 2.
Hs,mω =
{
u ∈ Hs : ‖u‖Hs,mω ≡ ‖(1−∆ω)
m
2 u‖Hs <∞
}
,
Bs,m2,1,ω =
{
u ∈ B
1
2
2,1 : ‖u‖2Bs,m2,1,ω ≡ ‖(1−∆ω)
m
2 u‖Bs2,1 <∞
}
,
where ∆ω is the Laplace-Beltrami operator on Sn−1.
The homogeneous spaces H˙s,mω and B˙
s,m
2,1,ω is similarly defined by the definition of H˙
s
and B˙s2,1. Then we have the following.
Proposition 4. (1) If 1/2 < s < n/2 and m > n− 1− s, then there exists a constant
C such that for any u ∈ Hs,mω
sup
Rn\{0}
|x|n/2−s|u(x)| ≤ C‖u‖H˙s,mω . (4)
(2) If m > n− 32 , then there exists a constant C such that for any u ∈ B
1
2
,m
2,1,ω
sup
Rn\{0}
|x|(n−1)/2|u(x)| ≤ C‖u‖
B˙
1
2 ,m
2,1,ω
. (5)
Remark 8. Hs,mω and B
1
2
,m
2,1,ω are closed subspaces of H
s and Bs2,1, respectively and they
contain Hsrad and B
1
2
2,1,rad naturally, respectively. We can identify the spaces H
s with
(1−∆ω)m/2Hs,mω and also B
1
2
2,1 with (1−∆ω)m/2B
1
2
,m
2,1,ω.
Remark 9. Hs,mω is a Hilbert space with the same inner product as L2 space and its
dual space is given by H−s,−mω .
From the decay at infinity we deduce compact embeddings of Hs,mω and B
1
2
,m
2,1,ω into some
Lp spaces as follows. See [2, 3, 4, 8, 9] for the radial case.
Corollary 1. The embedding Hs,mω ↪→ Lp is compact for 1/2 < s < n/2, m > n− 1− s
and 2 < p < 2n/(n− 2s).
Corollary 2. The embedding B
1
2
,m
2,1,ω ↪→ Lp is compact for m > n − 3/2 and 2 < p <
2n/(n− 1).
2. Proofs
2.1. Proof of Proposition 1. We use the following Fourier representation for radially
symmetric functions as
u(x) = |x|1−n2
∫ ∞
0
Jn
2
−1(|x|ρ)û(ρ)ρ
n
2 dρ, (6)
where Jν is the Bessel function of order ν, û is the Fourier transform normalized as
û(ξ) = (2pi)−n/2
∫
e−ix·ξu(x)dx,
4 Y. CHO AND T. OZAWA
and we have identified radially symmetric functions on Rn with the corresponding functions
on (0,∞).
By the Cauchy-Schwarz inequality and the Plancherel formula, we have
|x|n2−s|u(x)|
≤ |x|1−s
(∫ ∞
0
|Jn
2
−1(|x|ρ)|2ρ1−2sdρ
)1/2(∫ ∞
0
|û(ρ)|2ρ2s+n−1dρ
)1/2
=
(∫ ∞
0
|Jn
2
−1(ρ)|2ρ1−2sdρ
)1/2( Γ(n2 )
2pin/2
∫
|ξ|2s|û(ξ)|2dξ
)1/2
= C(n, s)‖(−∆)s/2u‖L2 ,
(7)
as required.
2.2. Proof of Proposition 2. We use the following estimates on Bessel functions:
sup
r≥0
|Jn
2
−1(r)| ≤ 1. (8)
sup
r≥0
r1/2|Jn
2
−1(r)| ≤ C. (9)
The first inequality (8) follows from the integral representation (see [13])
Jn
2
−1(r)2 =
2
pi
∫ pi/2
0
Jn−2(2r cos θ)dθ,
Jm(t) =
1
2pi
∫ 2pi
0
cos(mθ − t sin θ)dθ, m ∈ Z.
The second inequality (9) follows from the first and the well-known asymptotics (see [10])
Jν(r) ∼
√
2
pir
cos
(
r − (2ν + 1)pi
4
)
as r →∞. (10)
We apply the Littlewood-Paley decomposition {ϕj}j∈Z on Rn \ {0} to (4) to obtain
u(x) = |x|1−n2
∑
j∈Z
∫ ∞
0
Jn
2
−1(|x|ρ)ψj(ρ)ϕj(ρ)û(ρ)ρ
n
2 dρ, (11)
where ψj = ϕj−1 + ϕj + ϕj+1 and supp ϕj ⊂ {2j−1 ≤ ρ ≤ 2j+1}.
As in (7), we have
|x|(n−1)/2|u(x)|
≤ |x|1/2 sup
j∈Z
(∫ ∞
0
|Jn
2
−1(|x|ρ)|2ψj(ρ)2dρ
)1/2
·
∑
j∈Z
(∫ ∞
0
|ϕj(ρ)û(ρ)|2ρndρ
)1/2
.
SOBOLEV INEQUALITIES 5
By (9), we estimate
|x|1/2 sup
j∈Z
(∫ ∞
0
|Jn
2
−1(|x|ρ)|2ψj(ρ)2dρ
)1/2
≤ C sup
j∈Z
(∫ ∞
0
1
ρ
ψj(ρ)2dρ
)1/2
≤ C sup
j∈Z
(∫ 2j+2
2j−2
1
ρ
dρ
)1/2
≤ C.
(12)
This proves (2) since ∑
j∈Z
(∫ ∞
0
|ϕj(ρ)û(ρ)|2ρndρ
)1/2
is equivalent to the seminorm on B˙1/22,1,rad.
2.3. Proof of Proposition 3. If we use Cauchy-Schwartz inequality as in (7), we have
for any M > 0
|x|n/2−s|u(x)|
≤ |x|1−s
(∫ M |x|
0
|Jn
2
−1(|x|ρ)|2ρdρ
)1/2(∫ M |x|
0
|û(ρ)|2ρn−1dρ
)1/2
+|x|1−s
(∫ ∞
M |x|
|Jn
2
−1(|x|ρ)|2ρ−1dρ
)1/2(∫ ∞
M |x|
|û(ρ)|2ρn+1dρ
)1/2
≤ |x|−s
(∫ M
0
|Jn
2
−1(r)|2rdr
)1/2
‖u‖L2
+|x|1−s
(∫ ∞
M
|Jn
2
−1(r)|2r−1dr
)1/2
‖∇u‖L2 .
From (8) and (9) we deduce that supr≥0 |r1−sJn2−1(r)| ≤ C for any
1
2 ≤ s < 1. Hence we
have for any M > 0
|x|n/2−s|u(x)|
≤ |x|−s
(∫ M
0
|Jn
2
−1(r)|2r dr
) 1
2
‖u‖L2
+ |x|1−s
(∫ ∞
M
|Jn
2
−1(r)|2r−1 dr
) 1
2
‖∇u‖L2
≤ C|x|−sM s‖u‖L2 + |x|1−sM−(1−s)‖∇u‖L2 .
The minimization of the RHS of the last inequality with respect to M yields Proposition
3.
6 Y. CHO AND T. OZAWA
2.4. Proofs of Proposition 4 and Corollaries. The proof for (4) follows from the one
of Proposition 1 and the spherical harmonic expansion of functions in Hs,mω [5, 10]. In
fact, if we write u(rω) =
∑
k≥0
∑
1≤l≤d(k) fk,l(r)Yk,l(ω), where d(k) is the dimension of
space of spherical harmonic functions of degree k and
d(k) ≤ Ckn−2 for large k. (13)
Then we have
|x|n2−su(|x|ω) = cn
∑
k,l
|x|1−s
∫ ∞
0
Jν(k)(|x|ρ)ρ
n
2 gk,l(ρ)dρYk,l(ω), (14)
where ω ∈ Sn−1, ν(k) = n+2k−22 and f̂k,lYk,l(ρω) = gk,l(ρ)Yk,l(ω). Here
gk,l(ρ) = cn,k
∫ ∞
0
fk,l(r)rn−1(rρ)−
n−2
2 Jν(k)(rρ) dr.
The absolute value of cn,k is bounded by a constant depending only on n. See [10] for this.
Using the Cauchy-Schwarz inequality as in (7), we have
|x|n2−s|u(|x|ω)|
≤ C
∑
k,l
‖Yk,l‖L∞(Sn−1)
(∫ ∞
0
|Jν(k)(|x|ρ)|2ρ1−2s
) 1
2
(∫ ∞
0
|gk,l(ρ)|2ρ2s+n−1dρ
) 1
2
≤ C
∑
k,l
k
n−2
2
(
Γ(ν(k) + 1− s)
Γ(ν(k) + s)
) 1
2
(∫ ∞
0
|gk,l(ρ)|2ρ2s+n−1dρ
) 1
2
‖Yk,l‖L2(Sn−1).
Here we used the inequality that ‖Yk,l‖L∞ ≤ Ck n−22 ‖Yk,l‖L2 (see for instance [10]). Using
the Stirling’s formula for gamma function that Γ(t) ≈ tt− 12 e−(t−1) for large t (for instance,
see [1]) and the fact −∆ωYk, l = k(k + n− 2)Yk, l, we have from (13)
|x|n2−s|u(|x|ω)|
≤ C
∑
k
k
n−2
2 d(k)
1
2
(
Γ(ν(k) + 1− s)
Γ(ν(k) + s)
) 1
2
·
 ∑
1≤l≤d(k)
∫ ∞
0
|gk,l(ρ)|2ρ2s+n−1dρ‖Yk,l‖2L2(Sn−1)
 12
≤ C
(∑
k
k2(n−
3
2
−s−m)
) 1
2
∑
k,l
k2m
∫ ∞
0
|gk,l(ρ)|2ρ2s+n−1dρ‖Yk,l‖2L2(Sn−1)
 12
≤ C
∑
k,l
k2m
∫ ∞
0
∫
Sn−1
|F(fk,lYk,l)(ρω)|2ρ2s+n−1dρdω
 12
≤ C
∑
k,l
∫ ∞
0
∫
Sn−1
∣∣∣F((1−∆ω)m2 (fk,lYk,l))(ρω)∣∣∣2 ρ2s+n−1dρdω
 12
≤ C‖u‖H˙s,mω ,
SOBOLEV INEQUALITIES 7
where F is the Fourier transform. This proves part (1).
For the part (2), if we use the Littlewood-Paley decomposition {ϕj}j∈Z as in the proof
of Proposition 2, the we have
|x|n−12 u(|x|ω) = cn
∑
j∈Z
∑
k,l
|x| 12
∫ ∞
0
Jν(k)(|x|ρ)ρ
n
2 ψj(ρ)ϕj(ρ)gk,l(ρ)dρYk,l(ω), (15)
Since m > n− 3/2, by (12) we deduce that
|x|n−12 |u(|x|ω)|
≤ C
∑
j∈Z
∑
k,l
(∫ ∞
0
|ϕj(ρ)gk,l(ρ)|2ρn dρ
) 1
2
‖Yk,l‖L∞(Sn−1)
≤ C
∑
j∈Z
(∑
k
k2(n−2−m)
) 1
2
∑
k,l
k2m
∫ ∞
0
|ϕj(ρ)gk,l(ρ)|2ρn dρ‖Yk,l‖2L2(Sn−1)
 12
≤ C
∑
j∈Z
2
j
2 ‖ϕjF((1−∆ω)m2 u)‖L2 = C‖u‖
B˙
1
2
2,1,ω
.
To show Corollary 1 we use the fact that Hs,mω is a Hilbert space. Hence any bounded
sequence {uj} in Hs,mω satisfies uj(x) → 0 as |x| → ∞ uniformly and has a subsequence
converges to u in Hs,mω weakly. Let us denote the subsequence by ujk .
Now choose a smooth function ϕ supported in the ball of radius R + 1 and with value
1 in the ball of radius R. By the standard argument one can easily show that for each R
the mapping u 7→ ϕu is compact from Ht to Ht′ if t′ < t. By the compactness above and
Sobolev embedding we may assume that the sequence ϕujk satisfies that for 2 ≤ q < 2nn−2s
‖ϕujk − ϕu‖Lq → 0 as k →∞. (16)
Thus we have
‖ujk − u‖Lp ≤ ‖ϕ(ujk − u)‖Lp + ‖(1− ϕ)(ujk − u)‖Lp ≡ Ik + IIk
with Ik → 0 as k → ∞ by (16) since 2 < p < 2nn−2s . From the uniform convergence that
|ujk(x)|+ |u(x)| → 0 as |x| → ∞ it follows that
lim sup
k→∞
IIk ≤ sup
k
‖ujk − u‖
p−2
p
L∞(|x|>R) → 0 as R→∞.
This proves the compactness of the embedding Hs,mω ↪→ Lp.
Since B
1
2
,m
2,1,ω ↪→ H
1
2
,m
ω , one can adapt the same arguments (compactness of cut-off map-
ping and uniform convergence at infinity) as above except for weak-∗ convergence of ujk
to u in B
1
2
,m
2,1,ω for the compactness of embedding B
1
2
,m
2,1,ω to L
p. This completes the proof.
8 Y. CHO AND T. OZAWA
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Department of Mathematics, POSTECH, Pohang 790-784, Republic of Korea
E-mail address: changocho@postech.ac.kr
Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan
E-mail address: ozawa@math.sci.hokudai.ac.jp


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Sobolev inequalities with symmetry

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