Sharp pointwise estimates for functions in the Sobolev spaces Hs (Rn)

arXiv:1602.01902v1 [math.FA] 5 Feb 2016

Sharp pointwise estimates for functions in the Sobolev spaces Hs(Rn) ¨tz 1, Juliana S. Ziebell2, Lineia Schu Jana´ına P. Zingano 1 and Paulo R. Zingano 1 1 Instituto

de Matem´ atica e Estat´ıstica Universidade Federal do Rio Grande do Sul Porto Alegre, RS 91509-900, Brazil 2 Instituto

de Matem´ atica, Estat´ıstica e F´ısica Universidade Federal do Rio Grande Rio Grande, RS 96201-900, Brazil

Abstract We provide the optimal value of the constant K(n, m) in the Gagliardo-Nirenberg 1−

n

n

m u kL2m supnorm inequality k u kL∞ (Rn ) ≤ K(n, m) k u kL2(R2m n) k D 2 (Rn ) , m > n/2, and

its generalizations to the Sobolev spaces H s (Rn ) of arbitrary order s > n/2 as well.

1. Introduction In recent decades there has been a growing interest in determining the sharpest form of many important inequalities in analysis, see e.g. [1, 2, 3, 4, 5, 8, 10] and references therein. A noticeable miss is the fundamental Gagliardo-Nirenberg supnorm inequality n 1 − 2m

k u kL∞ (Rn ) ≤ K(n, m) k u k

L2 (Rn )

n

k D m u k 2m 2

(1.1)

L (Rn )

for functions u ∈ H m (Rn ) when m > n/2 (see [7, 9]), as well as some of its generalizations. Here, as usual, H m (Rn ) is the Sobolev space of functions u ∈ L2 (Rn ) with all derivatives of order up to m in L2 (Rn ), which is a Banach space under its natural norm defined by k u kH m (Rn ) = where k Dm u k

L2 (Rn )

=

n

k u k2 2

 X n n X

L (Rn )

...

n X

+ k Dm u k2 2

im = 1

i1 = 1 i2 = 1

L (Rn )

o1/2 ,

k Di Di ... Di u k 2 2 1

2

m

L (Rn )

(1.2a) 1/2

(1.2b)

(with Di denoting the weak derivative with respect to the variable xi ). In this brief note, we will review some basic results in order to derive the optimal (i.e., minimal) value for the constant K(m, n) in (1.1) above. It will be seen in Section 2 that it turns out to be 1

 − n4 K(n, m) = 4π



−1 n  2 n 2m ) n ) σ( 2m 2m −

1  n − 2  sin σ(

n Γ 2 2

n

n  − 4m

2m 2m − n

21

(1.3)

where σ (r) := rπ and Γ(·) is the Gamma function (for its definition, see e.g. [6], p. 7). For example, we get, with m = 2 and n = 1, 2, 3, the sharp pointwise estimates k u kL∞ (R) kuk

√ 4 1 3 2 4 4 2 k D u k , ≤ √ k u k 8 L2 (R) L2 (R) 27 1 1 1 k u k 22 2 k D2 u k 22 2 , L (R ) L (R ) 2 √ 8 1 3 12 k u k 42 3 k D2 u k 42 3 , ≤ √ L (R ) L (R ) 6π

L∞ (R2 )

kuk

L∞ (R3 )

L∞ (R3 )

which is also shown to be optimal.) Also, setting

H

(1.4b)

≤

and so forth. (In [10], it is obtained that k u k

k u k ˙s

(1.4a)

(Rn )

:=

Z

≤ √

2s

2

1 1 1 k D u k 22 3 k D2 u k 22 3 , L (R ) L (R ) 2π

| ξ | | uˆ(ξ) | dξ

Rn

(1.4c)

1 2

(1.5)

for real s > 0,1 where u ˆ(·) denotes the Fourier transform of u(·), that is, u ˆ(ξ) = 

and letting H s(Rn ) =

2π

Z
− n2

−ix·ξ

e

ξ ∈ Rn,

u(x) dx,

Rn

u ∈ L2 (Rn ) : k u k ˙ s H

(Rn )

n/2, then n 1 − 2s k L2 (Rn )

k u kL∞ (Rn ) ≤ K(n, s) k u k

n

u k 2s ˙s

(1.7a)

H (Rn )

for all u ∈ H s(Rn ), with the optimal value of the constant K(n, s) being given by K(n, s) =



4π

− n4



n −1 − 12   − 4s 21 n  2 sin σ( 2s ) n n n 2s Γ n 2 2 σ( 2s 2s − n 2s − n )

(1.7b)

for any s > n/2, where, as before, σ (r) := r π. The proof of (1.1), (1.3), (1.7) and of the sharpness of the values for K given in (1.3), (1.7b) above is provided in the sequel; in addition, a second classical estimate for k u kL∞ (Rn ) when u ∈ H s(Rn ), s > n/2, is also reexamined here (see (2.4) below), so as to be similarly presented in its sharpest form. 1

Note that (1.5) corresponds to (1.2b) when s = m (m integral), that is: k u k ˙ m H

2

(Rn )

= k Dm u k

L2 (Rn )

.

2. Proof of (1.1), (1.3), (1.7) and other optimal supnorm results in H s(Rn ) To obtain the results stated in Section 1, we first review the following basic lemma. We recall that u ˆ denotes the Fourier transform of u, cf. (1.6) above. Lemma 2.1. Let u ∈ L2 (Rn ). If u ˆ ∈ L1 (Rn ), then u ∈ L∞ (Rn ) and k u kL∞ (Rn ) ≤

2π


− n/2

ku ˆk

L1 (Rn )

.

(2.1)

Moreover, equality holds in (2.1) if u ˆ is real-valued and of constant sign (say, nonnegative). Proof: Clearly, (2.1) is valid if u ∈ S(Rn ), where S(Rn ) denotes the Schwartz class of smooth,

rapidly decreasing functions at infinity ([6], p. 4), since we have, in this case, the representation (see e.g. [6], p. 16) Z
− n2 ix·ξ e uˆm (ξ) dξ, ∀ x ∈ Rn. (2.2) um (x) = 2π Rn

For general u ∈ L2 (Rn ) with u ˆ ∈ L1 (Rn ), let { u ˆm } be a sequence of Schwartz approximants to 1 n 2 n u ˆ ∈ L (R ) ∩ L (R ) such that k u ˆm − u ˆ kL1 (Rn ) → 0 and k u ˆm − u ˆ kL2 (Rn ) → 0 as m → ∞, and let um ∈ S(Rn ) be the Fourier inverse of u ˆm , for each m. Applying (2.1) to { um }, we see that { um } is Cauchy in L∞ (Rn ), so that k um − v kL∞ (Rn ) → 0 for some v ∈ L∞ (Rn ) ∩ C 0 (Rn ). Since we have k um − u kL2 (Rn ) → 0, it follows that u = v. This shows that u ∈ L∞ (Rn ) ∩ C 0 (Rn ) and, letting m → ∞ in (2.2), we also have u(x) =

2π

Z
− n2

R

e

i x·ξ

∀ x ∈ Rn,

u ˆ(ξ) dξ,

n

(2.2′ )

from which (2.1) immediately follows. In particular, in the event that u ˆ(ξ) ≥ 0 for all ξ, we get

from (2.2′ ) that u(0) = (2π)− n/2 k uˆ kL1 (Rn ) , so that (2.1) becomes an identity in this case.



We observe that the hypotheses of Lemma 2.1 are satisfied for u ∈ H s(Rn ) if s > n/2. An important consequence of this fact is the fundamental embedding property revisited next, where the norm in H s(Rn ) is set to be (in accordance with (1.2a) above): k u kHs(Rn ) =

n

ku

k2 2 n L (R )

+ ku

k 2˙ s n H (R )

o1/2

=

Z

Rn

2s

1 + |ξ |




1/2 |u ˆ(ξ) | dξ . (2.3) 2

Theorem 2.1. Let s > n/2. If u ∈ H s(Rn ), then u ∈ L∞ (Rn ) ∩ C 0 (Rn ) and k u kL∞ (Rn ) ≤



4π

− n4



1

1

− − 2  n  2 ) sin σ( 2s n n · k u kHs(Rn ), Γ n ) 2 2 σ( 2s

(2.4)

where σ(r) = rπ. Moreover, equality holds in (2.4) when u ˆ(ξ) = c / ( 1 + | ξ | 2s ) ∀ ξ ∈ Rn for some constant c ≥ 0, so that the constant given in (2.4) above is optimal. 3

Proof: Let u ∈ H s(Rn ), with s > n/2. By (2.1) and Cauchy-Schwarz’s inequality, we have k u kL∞ (Rn ) ≤

2π

≤

2π

=

Z
− n/2

1 + | ξ |2s

Rn




− n/2

4π

− n/4

Z

Rn




− 1/2

1 + | ξ |2s

1 + | ξ |2s


− 1

dξ


1/2

1/2  Z

Rn

|u ˆ(ξ) | dξ

(2.5a)


1 + | ξ |2s | uˆ(ξ) |2 dξ

1/2

(2.5b)

− 1/2 − 1/2  n  sin σ( 2s ) n n Γ · k u kHs(Rn ) n 2 2 σ( 2s )


− 1 by (2.3), since, using polar coordinates and the change of variable t = 1 + r2s , we obtain

Z

Rn

1 + |ξ |


2s − 1

Z

dξ = ωn

∞

1+r

0


2s − 1 n−1 r

ωn dr = 2s

Z

1 − n 2s

t

0

(1 − t)

n 2s − 1


n ωn σ 2s
dt = n n sin σ 2s

where ωn = 2 π n/2/ Γ(n/2) is the surface area of the unit ball in Rn (see [6], p. 8), and σ(r) = rπ. This shows (2.4). Finally, if u ˆ(ξ) = c/(1 + | ξ |2s ) for all ξ, for some c ≥ 0 constant, then equality

holds in both steps (2.5a) and (2.5b) above, so that (2.4) is an identity in this case, as claimed. 

We are now in very good standing to obtain (1.1), (1.7a) with their sharpest constants. Theorem 2.2. Let s > n/2. If u ∈ H s(Rn ), then n 1 − 2s

n

k u kL∞ (Rn ) ≤ K(n, s) k u k

L2 (Rn )

k u k 2s ˙s

(2.6)

H (Rn )

with K(n, s) defined in (1.7b). Moreover, equality holds in (2.6) if u ˆ(ξ) = c / ( 1 + | ξ | 2s ) n ∀ ξ ∈ R , c ≥ 0 constant, so that the numerical value of K given in (1.7b) is optimal. Proof: Let s > n/2, u ∈ H s (Rn ) fixed. Given λ > 0, setting uλ ∈ H s(Rn ) by uλ (x) := u(λx), we have, by (2.4), Theorem 2.1, k u kL∞ (Rn ) = k uλ kL∞ (Rn ) ≤ =



4π

− n4





4π

− n4



1

1

− − 2  n  2 sin σ( 2s ) n n · k uλ kHs(Rn ) Γ n ) 2 2 σ( 2s

−1 − 12  21 n  2  ) sin σ( 2s 2s − n 2 −n 2 n n Γ k u k ˙s n · λ kuk 2 n + λ n L (R ) H (R ) ) 2 2 σ( 2s

for λ > 0 arbitrary. Choosing λ that minimizes the last term on the right of the above expression gives us (2.6), with K(n, s) defined by (1.7b), as claimed. Now, to show that the value provided in (1.7b) is the best possible, we proceed as follows. First, we observe that, by Young’s inequality, 

n 2s − n

n  − 4s

2s 2s − n

21

kuk

n 1 − 2s L2 (Rn )

ku

n k 2s ˙ s(Rn ) H

≤

n

kuk

2 L2 (Rn )

2

+ k u k ˙s

H (Rn )

o12

= k u kHs(Rn ) (2.7)

4

for all u ∈ H s(Rn ). Therefore, taking w ∈ H s(Rn ) defined by w b (ξ) = c/(1 + | ξ |2s ), c ≥ 0, we get,

with K(n, s) given in (1.7b):

k w kL∞ (Rn ) ≤ K(n, s) k w k ≤



4π

− n4



n 1 − 2s L2 (Rn )

n

k w k 2s ˙s

[ by (2.6) ]

H (Rn )

−1 − 12  n  2 sin σ( 2s ) n n · k w kHs(Rn ) [ by (1.7b), (2.7) ] Γ n ) 2 2 σ( 2s

= k w kL∞ (Rn ) ,

[ by Theorem 2.1 ]

thus showing that we have k w kL∞ (Rn ) = K(n, s) k w k

n 1 − 2s 2

L (R ) n

n

k w k 2s ˙s

, as claimed.

H (Rn )



References [1] M. Agueh, Gagliardo-Nirenberg inequalities involving the gradient L2-norm, C. R. Math. Acad. Sci. Paris, 346 (2008), 757-762. [2] W. Beckner, Inequalities in Fourier analysis, Ann. Math, 102 (1975), 159-182. [3] E. A. Carlen and M. Loss, Sharp constant in Nash’s inequality, Intern. Math. Research Notices, 7 (1993), 213-215. [4] D. Cordero-Erausquin, B. Nazaret and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Advances in Math., 182 (2004), 307-332. [5] M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appliqu´ees, 81 (2002), 847-875. [6] G. B. Folland, An Introduction to Partial Differential Equations (2nd ed.), Princeton University Press, Princeton, 1995. [7] E. Gagliardo, Propriet` a di alcune classi di funzioni in pi` u variabili, Ricerche Mat., 7 (1958), 102-137. [8] E. H. Lieb, Inequalities (Selecta), Springer, New York, 2003. [9] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. [10] W. Xie, A sharp pointwise bound for functions with L2-Laplacians and zero boundary values of arbitrary three-dimensional domains, Indiana Univ. Math. Journal, 40 (1991), 1185-1192.

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