A new class of function spaces on domains of R^d and its relations to classical function spaces

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ISSN: 2008-6822 (electronic)

http://www.ijnaa.semnan.ac.ir

A New Class of Function Spaces on Domains of

_R

d

and Its Relations to Classical Function Spaces

G. Narimania,∗

a_{Department of Mathematics and Applications, Faculty of Basic Sciences, University of Mohaghegh Ardabili, P.O. Box}

179, Ardabil,Iran.

(Communicated by A. Najati)

Abstract

A new class of function spaces on domains (i.e., open and connected subsets) of_Rd, by means of the asymptotic behavior of modulations of functions and distributions, is defined. This class contains the classes of Lebesgue spaces and modulation spaces. Main properties of this class are studied, its applications in the study of function spaces and its relations to classical function spaces are discussed.

Keywords: Modulation Spaces, Bessel Potential Spaces, Function Spaces on Domains.

2000 MSC: Primary 42B35; Secondary 46E35.

1. Introduction and preliminaries

The main aim of this paper is to introduce a new class of function spaces on domain Ω⊆ _Rd _which

contain the class of Lebesgue spaces and modulation spaces. Considering the fact that in general it

is not possible to define modulation spaces on manifolds (see [7]), this new class of function spaces

is interesting as being an appropriate substitution for modulation spaces in such cases.

LetD(Ω) denote the space of compactly supported infinitely differentiable functions with its usual

topology by a family of seminorms and D0_{(Ω) denote its dual, the space of generalized functions.}

Also, Let S(_Rd_{) and} _S0₍

Rd) denote the Schwartz space and its dual space, the space of tempered

distributions, respectively. Let φ∈ D(_Rd_{) be a smooth radial bump function such that}

φ:_Rd−→[0,1]

∗_{Corresponding author}

Email address: [email protected](G. Narimani)

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Suppφ⊆B(0, √

2d) (1.1)

φ(ξ) = 1,for ξ ∈B(0,pd/2).

Letφk(ξ) =φ(ξ−k),k ∈ Zd and ξ ∈Rd. Since φk(ξ) = 1 in the unit closed cube Qk with centerk

and {Qk}k∈Zd is a covering of R

d_{, clearly} P

k∈Zdφk(ξ)≥1 for all ξ∈R

d_{. We write}

ϕk(ξ) =

φk(ξ)

P

l∈Zdφl(ξ)

, k ∈_Zd.

Consequently, there existc, cm >0 such that



   

   

|ϕk(ξ)| ≥c, ∀ξ ∈Qk

Suppϕk⊆B(k, √

2d),

P

l∈Zdφl(ξ) = 1,

|ϕ(m)_k (ξ)| ≤cm, ∀ξ ∈Rd.

(1.2)

Let

Φ ={{ϕk}k∈Zd : {ϕk}k∈Zd satisfies (1.2)}.

By the construction above Φ 6= ∅. Let {ϕk}k∈Zd ∈ Φ be an arbitrary sequence. For r ∈ R, we

consider the weight functionwr(ξ) = (1 +|ξ|2)r/2 on Rd and define the modulation spaces Mp,qr (Rd),

0< p, q≤ ∞, by

M_p,qr (_Rd) = {f ∈ S0(_Rd) :kf :M_p,qr k:= X

k∈Zd

wr(k)kF−1ϕkFfkp

q !1/q

<∞}. (1.3)

The class of modulation spaces is defined by Hans G. Feichtinger in [3]. For a recent treatment of

modulation spaces we refer to [4] and [5]. For r ∈_R and 0 < p, q ≤ ∞, Mr

p,q(Rd) is a quasi-Banach

space (a Banach space ifp, q ≥1). Now we give definition of Bessel potential spaces. Let s∈_R and

1≤p≤ ∞. The Bessel potential space H_ps(_Rd) is defined by

H_ps(_Rd) = {f ∈ S0(_Rd) :kf :H_psk:=kF−1_w

sF fkp <∞}. (1.4)

For s∈_Rand 1≤p≤ ∞, H_ps(_Rd) is a Banach space. If we use the notation Js_(f_{) =}_F−1_w_sF_f

fors∈_R, thenHs

p(Rd) =J

−s_L

p(Rd). When 1< p <∞and n∈N, Hpn(Rd) is equal to the classical

Sobolev space Wn

p(Rd), defined by

W_pn(_Rd) = {f ∈Lp(Rd) :kf :Wpnk:=

X

|α|≤n

kDαfkp <∞}. (1.5)

Note that H0

p = Wp0 = Lp. We refer to [1], [2], [8] and [9] for the theory of Sobolev and Bessel

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2. Definition and Properties of N_p,qr,s(Ω)

Let Ω ⊆_Rd _{be a domain, that is, an open bounded set. The Sobolev space on domain Ω,}_Wn p(Ω),

may be defined as

W_pn(Ω) ={f ∈Lp(Ω) :kf :Wpn(Ω)k:=

X

|α|≤n

kDαfkLp_(Ω) <∞}. (2.1)

Then we have the following restriction property

W_pn(Ω) ={f ∈Lp(Ω) :∃g ∈Wpn(Rd) such that f =g|Ω} (2.2)

and kf : W_pn(Ω)k0 _{:= inf}_{k_g _: _Wn

p(Rd)k : g ∈Wpn(Rd), f =g|Ω} defines an equivalent norm on

kf : W_pn(Ω)k. For a general theory of function spaces on _Rd as well as on domains we refer to the

references [1], [8] and [9]. The described procedure is not applicable to define Bessel potential spaces

and modulation spaces on Ω. Therefore we define Bessel potential spaces and modulation spaces on

Ω as follows:

Definition 2.1. Let Ω⊆_Rd _{be a domain,} ₁_≤_{p, q} _{≤ ∞} _and _{r, s}_∈

R. Then we define

H_ps(Ω) ={f ∈ D0(Ω) : ∃g ∈H_ps(_Rd) such that g|Ω =f}

with norm kf :H_ps(Ω)k= inf{kg :H_ps(_Rd)k: g ∈H_ps(_Rd), f =g|Ω}, and also

M_p,qr (Ω) ={f ∈ D0(Ω) : ∃g ∈M_p,qr (_Rd) such that g|Ω =f}

with norm kf :M_p,qr (Ω)k= inf{kg :M_p,qr (_Rd)k: g ∈M_p,qr (_Rd), f =g|Ω}.

The proof of the following theorem is similar to the proof of proposition 3.2.3 in [8], so omitted.

Theorem 2.2. For 1≤p, q ≤ ∞ andr, s∈_R, H_ps(Ω) andM_p,qr (Ω) are Banach spaces of generalized functions.

The operators of modulation and translation are defined as follows

Maf(x) = e2πia.xf(x) and Taf(x) = f(x−a).

Definition 2.3 (Definition of Nr,s

p,q(Ω)). Let r, s ∈ R, 1 ≤ p, q ≤ ∞ and Ω ⊆ Rd be a domain.

For f ∈Hs

p(Ω) we say that f ∈Np,qr,s(Ω) if

kf :N_p,qr,s(Ω)k:= X

k∈_Zd

kMkf :Hps(Ω)kwr(k)

q !1/q

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Theorem 2.4. Let r, s∈_R and 1≤p, q ≤ ∞.

1. Nr,s

p,q(Ω) is a Banach space of distributions on Ω,

2. N_p,qr,s(Ω),→H_ps(Ω)

3. N_p,qr,s(_Rd),→ S0(_Rd)

4. S(_Rd₎_,_→_Nr,s

p,q(Rd) if r+s ≤0.

Proof .

1. Clearly Nr,s

p,q(Ω) is a normed space. Let {fn} be a Cauchy sequence in Np,qr,s(Ω). Then {fn} is

also a Cauchy sequence in Hs

p(Ω) and therefore there exists f ∈ Hps(Ω) such thatfn −→ f in

H_ps(Ω), which in turn implies thatMkfn−→Mkf inHps(Ω), for allk ∈Z. On the other hand,

since {J−s_M

kfn} is Cauchy in `rq(Lp) and `rq(Lp) is complete, there is {hk}k∈_Zd ∈`r_q(L_p) such

that

{J−s_M

kfn} −→ {hk} (in `rq(Lp)), (2.3)

which implies that J−s_M

kfn −→ hk in Lp or Mkfn −→ Jshk in Hps(Ω), for each k ∈ Zd.

Comparing this with the fact that Mkfn −→ Mkf in Hps(Ω) shows that hk =J−sMkf. Now

(2.3) implies thatfn−→f inNp,qr,s(Ω).

2. This follows from Nr,s

p,q(Rd),→Hps(Rd),→ S

0₍ Rd).

3. f ∈ S(_Rd_{). Since}_`r

∞(Zd),→`r −t

1 (Zd), fort > d, we have the embeddingNp,r,s∞ ,→N

r+t,s

p,1 ,→Np,qr,s.

Therefore we need only to show that S(_Rd),→N_p,1r,s. To this end we have

kf :N_p,r,s_∞k = sup

k∈Zd

kMkf :Hpskwr(k)

= sup

k∈_Zd

kF−1wsFMkfkpwr(k)

= sup

k∈Zd

kF−1_w

s(·+k)Ffkpwr(k)

≤ sup

k∈Zd

kwLF−1ws(·+k)Ffk∞wr(k)

= sup

k∈Zd

kF−1₍₁₋_∆)L_w

s(·+k)Ffk∞wr(k)

≤ C sup

k∈_Zd

k(1−∆)Lws(·+k)Ffk1wr(k)

≤ C0 sup

k∈_Zd

k(1−∆)LFfk1wr+s(k),

for sufficiently largeL. Now the result follows fromr+s≤0 and the fact thatS(_Rd₎_,_→_L 1(Rd).

The following theorem shows that this new class contains the classes of modulation spaces and

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we denote the norm of the operator f −→ F−1_ϕ_F_f _from _L

p to Lp bykϕ:MF(Lp)k, whenever it is

bounded.

Theorem 2.5. We have

1. Nr,s

p,q(Ω) =Lp(Ω) if s= 0 and r <−d/q

2. N_p,qr,s(_Rd) = M_p,qr (_Rd) if r <−|s| −d

3. Nr,s

p,q(Ω) ={0} if s≥0 and r >− d q

Proof . The proofs of (1) and (3) are simple. We prove (2). We have

kf :M_p,qr k = k{kF−1_ϕ

kFfk:Lp}k∈_Zd :`r_qk

= k{kF−1TkwsTkw−sϕkFf :Lpk}k∈Zd :`

r qk

= k{kF−1_T

kwsϕkF F−1Tkw−sFfk:Lp}k∈Zd :`r_qk

≤ k{kTkwsϕk :MF(Lp)kkF−1Tkw−sFf :Lpk}k∈Zd :`r_qk

≤ kwsϕ:MF(Lp)kk{kF−1w−sFM−kf :Lpk}k∈_Zd :`r_qk

≤ Ck{kM−kf :Hp−sk}:` r

qk. (2.4)

Forj ∈_Zd _let

j∗ ={i∈_Zd _{: supp(ϕ}

i)∩supp(ϕj)6=∅}.

Since

kM−kf :Hp−rk = kF

−1_w

−rF(M−kf) :Lpk

= kF−1_(T

kw−r)Ff :Lpk

and

F−1_(T

kw−r)Ff =

X

j∈Zd

X

i∈j∗

F−1_(T

kw−r)ϕjϕiFf

= X

j∈_Zd X

i∈j∗

F−1_ϕ

i(Tkw−r)F F−1ϕjFf,

with convergence inS0_{, we have}

kM−kf :Hp−rk ≤

X

j∈_Zd X

i∈j∗

kϕi(Tkw−r) :MF(Lp)kkF−1ϕjFf :Lpk.

Since

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we have

kM−kf :Hp−rk.

X

j∈_Zd X

i∈j∗

w−r(k−i)kF−1ϕjFf :Lpk. (2.5)

Since i ∈j∗, we have w−r(k−i) ∼w−r(k−j), and therefore we may replace i by j in (2.5). After

this substitution, by using the fact |j∗| _∼ 1, we see that the right hand side of (2.5) is actually

a convolution of two sequences {w−r(j)}j∈_Zd and {kF₋₁ϕ_jFf : L_pk}_j_∈

Zd. By invoking Young’s inequality and using the w|s|-moderateness of the weight functionws, i.e.

ws(x+y)≤w|s|(x)ws(y),

we have

k{kM−kf :Hp−rk}k∈_Zd :`s_qk.k{w−r(j)}j∈_Zd :`

|s|

1 kk{kF

−1_ϕ

jFf :Lpk_j_∈

Zd :`

s qk.

Since r >|s|+d, we have k{w−r(j)}j∈Zd :` |s|

1 k<∞, and finally we arrive at

k{kM−kf :Hp−rk}k∈Zd :`

s

qk.kf :M s

p,qk. (2.6)

Now (2.4) and (2.6) completes the proof.

References

[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Volume 140 of Pure and Applied Mathematics, Elsevier, Academic Press, 2nd edition, 2003.

[2] J. Bergh and J. L¨ofstr¨om,Interpolation Spaces, Springer, Berlin, 1976.

[3] H. G. Feichtinger,Modulation spaces on locally compact Abelian groups, in Proc. Internat. Conf. on Wavelets and Applications (Chennai, 2002), R. Radha, M. Krishna, S. Thangavelu (eds.), New Delhi Allied Publishers, (2003) 1-56. (Reprint of the 1983 technical report).

[4] H. G. Feichtinger,Modulation Spaces: Looking Back and Ahead, Sampl. Theory Signal Image Process., 5 (2006) 109-140.

[5] K. Gr¨ochenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis. Birkh¨auser Boston, Inc., Boston, MA, 2001.

[6] G. Narimani, Smoothness properties of modulation spaces, Technical report, University of Mohaghegh Ardabili, Ardabil, Iran, 2011.

[7] K. Okoudjou,A Beurling–Helson type theorem for modulation spaces, J. Funct. Spaces Appl., 7 (2009) 33-41. [8] H. Triebel,Theory of function spaces, Monographs in Mathematics, Birkh¨auser Verlag, Basel, 1983.