Hausdorff operators on modulation and Wiener amalgam spaces

HAUSDORFF OPERATORS ON MODULATION AND WIENER AMALGAM SPACES

arXiv:1709.08282v1 [math.CA] 24 Sep 2017

GUOPING ZHAO,1 DASHAN FAN,2 and WEICHAO GUO3∗ Abstract. We give the sharp conditions for boundedness of Hausdorff operators on certain modulation and Wiener amalgam spaces.

1. Introduction and Preliminary Hausdorff operator, originated from some classical summation methods, has a long history in the study of real and complex analysis. We refer the reader to [1] and [12] for a survey of some historic background and recent developments about Hausdorff operator. For a suitable function Φ, one of the corresponding Hausdorff operator HΦ can be defined by   Z x HΦ f (x) = Φ(y)f dy. (1.1) |y| Rn Although there is a general definition where f (A(y)x), with matrix A, stays in place of f (x/|y|) in (1.1), we only consider the special case in this paper. However we do not exclude that the general case will prove to be of interest as well and we are keep interested in the general case. There are many known results about the boundedness of Hausdorff operators on various function spaces, such as [10, 11, 13, 14]. Unfortunately, the sharp conditions on boundedness of Hausdorff operator can be characterized in only few cases. One can see [21] for the sharp characterization for the boundedness of Hausdorff operators on Lp , and see [3, 17] for the sharp characterization for the boundedness of Hausdorff operators on Hardy spaces H 1 and h1 . We observe that the characterizations of boundedness of Hausdorff operator were also established in some other function spaces (see [1, 6]). However, we find that these spaces have some similar properties as the Lp spaces. Let us briefly describe this fact in the following. In order to prove the necessity of boundedness of Hausdorff operator on Lp , we must choose a suitable function f and estimate kHΦ f kLp from below by some integral regarding Φ. The space Lp is fit for this lower estimates, since for a function f , the norm kf kLp only depends on the absolute value of f and the Lp norm has the scaling property kf (s·)kLp = s−n/p kf kLp . We observe that the function spaces, for which the characterizations of boundedness of Hausdorff Copyright 2016 by the Tusi Mathematical Research Group. Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 42B35; Secondary 47G10. Key words and phrases. Hausdorff operator, sharp conditions, modulation space, Wiener amalgam space. 1

2

G. ZHAO, D. FAN, and W. GUO

operator are established so far, all have the above two properties as the Lp spaces so that the proof of the necessity follows the same line as that on Lp . However, in the case of frequency decomposition spaces, such as the modulation spaces or Wiener amalgam spaces, the situation becomes quite different and complicated. s The modulation spaces Mp,q were first introduced by Feichtinger [4] in 1983. As function spaces associated with the uniform decomposition (see [19]), modulation spaces have a close relationship to the topic of time-frequency analysis (see [7]), and they have been regarded as appropriate function spaces for the study of partial differential equations (see [20]). We refer the reader to [5] for some motivations and historical remarks. One can also refer our recent paper [8, 9] for the properties of modulation spaces and Wiener amalgam spaces. As a frequency decomposition space, the norm of f in a modulation space can not be completely determined by the absolute value of the function. On the other hand, the scaling property of modulation spaces is not as simple as that of Lp (see [15]). Thus, it is interesting to find out the sharp conditions for the boundedness of Hausdorff operator on modulation spaces, since in this case the method used in the Lp case is not adoptable. We also consider the boundedness of Hausdorff operator on Wiener amalgam s spaces Wp,q . In general, a Wiener amalgam space can be represented by W (B, C), where B and C are served as the local and global component respectively. In this paper, we consider a special case W (F −1Lsq , Lp ), which is closely related to s modulation spaces. For simplicity in the notation, we also use Wp,q to denote this function space. Before stating the main theorems, we make some preparations as follows. We need to add some suitable assumptions on Φ. Firstly, in order to establish the sharp conditions for the boundedness of Hausdorff operator, we assume Φ > 0. In the proof of the necessity part, we must make some (pointwise) estimates from below. That is why the assumption Φ > 0 is necessary in most of the known characterizations for the boundedness of Hausdorff operator on function spaces(see [3, 17, 21]). Secondly, we make another assumption for Φ as following: Z

B(0,1)

n

|y| Φ(y)dy < ∞, and

Z

Φ(y)dy < ∞.

(1.2)

B(0,1)c

We would like to give following remarks not only for explaining the reasonability of the assumption (1.2), but also to give some important properties of Hausdorff operator under the assumption (1.2). Remark 1.1 (Assumption (1.2) is weakest). In fact, (1.2) is the weakest assumption to ensure that the Schwartz function can be mapped into tempered distribution by Hausdorff operator HΦ .

HAUSDORFF OPERATORS ON MODULATION AND WIENER AMALGAM SPACES

3

On one hand, if HΦ f ∈ S ′ , it must be locally integrable, and since Φ > 0, we have Z Z Z ∞> |HΦ f (x)|dx = Φ(y)f (x/|y|)dydx B(0,1) B(0,1) Rn Z Z Z Z n = Φ(y) f (x/|y|)dxdy = Φ(y)|y| f (x)dxdy =

Z

Rn

B(0,1)

Φ(y)|y|

n

B(0,1)

Z

Rn

f (x)dxdy + 1 B(0, |y| )

1 B(0, |y| )

Z

Φ(y)|y|

n

B c (0,1)

Z

f (x)dxdy. 1 B(0, |y| )

On the other hand, for any nonnegative Schwartz function f satisfying f = 1 on B(0, 1), we have Z Z Z Z Z n n f (x)dxdy + Φ(y)|y| |HΦ f (x)|dx > Φ(y)|y| dxdy B(0,1)

∼

Z

n

Φ(y)|y| dy +

B(0,1)

This implies that

Z

B c (0,1)

B(0,1)

B(0,1)

n

Φ(y)|y| dy +

B(0,1)

Z

Z

1 B(0, |y| )

Φ(y)dy. B c (0,1)

Φ(y)dy < ∞. B c (0,1)

Remark 1.2 (HΦ f is well defined as a tempered distribution). If Φ satisfies (1.2), HΦ f makes sense for all f ∈ S (Rn ) for the reason that for x 6= 0, Z  Z |HΦ f (x)| 6 + Φ(y)f (x/|y|)dy B(0,1) B c (0,1) Z Z n n −n |y| Φ(y)(|x/|y|| f (x/|y|)) + kf kL∞ 6|x| Φ(y)dy B(0,1) B c (0,1) Z  Z −n n .Cf (1 + |x| ) |y| Φ(y)dy + Φ(y)dy < ∞, B c (0,1)

B(0,1)

and Z

Z

Z

|HΦ f (x)|dx 6 Φ(y)|f (x/|y|)|dydx B(0,1) Rn Z Z Z Z = Φ(y)|f (x/|y|)|dydx + Φ(y)|f (x/|y|)|dydx B(0,1) B(0,1) B(0,1) B c (0,1) Z Z Z ∞ Φ(y)dy 6 Φ(y) |f (x/|y|)|dxdy + |B(0, 1)| · kf kL · B c (0,1) B(0,1) Rn Z Z n Φ(y)dy < ∞. 6 |y| Φ(y)dykf kL1 + |B(0, 1)| · kf kL∞ · B(0,1)

B c (0,1)

B(0,1)

Thus, for f ∈ S (Rn ), HΦ f is a locally integrable function, which has polynomial growth at infinity. It implies that HΦ f is a tempered distribution for f ∈ S (Rn ). Write Z hHΦ f, gi = HΦ f (x)g(x)dx, Rn

4

G. ZHAO, D. FAN, and W. GUO

where hu, f i means the action of a tempered distribution u on a Schwartz function f. Remark 1.3 (HΦ : S → S ′ is continuous). For f, g ∈ S (Rn ), we have that Z

|f (x/|y|)g(x)|dx 6 kf kL∞ kgkL1

Rn

and Z

|f (x/|y|)g(x)|dx 6 kgkL∞ kf (·/|y|)kL1 6 |y|nkgkL∞ kf kL1 . Rn

It follows that Z

|Φ(y)|

Rn

Z

|f (x/|y|)g(x)|dxdy

Rn

.(kf kL1 + kf kL∞ )(kgkL1 + kgkL∞ )

Z

|Φ(y)| min{1, |y|n}dy.

Rn

Thus, Z

|hHΦ f, gi| =| HΦ f (x)g(x)dx| Rn Z Z 6 Φ(y)|f (x/|y|)|dyg(x)dx Rn Rn Z Z 6 Φ(y) |f (x/|y|)| · |g(x)|dxdy Rn Rn Z ∞ ∞ .(kf kL1 + kf kL )(kgkL1 + kgkL )

|Φ(y)| min{1, |y|n}dy.

Rn

Using the definition of Schwartz function space, we have | < HΦ f, gl > | → 0, for f, gl ∈ S (Rn ) satisfying that gl → 0 as l → ∞ in the topology of S . Remark 1.4 (Fourier transform of HΦ f ). Define fΦ f (x) = H

Z

Φ(y)|y|nf (|y|x)dy.

Rn

fΦ f is a tempered distribuBy a similar method used before, we can verify that H ′ fΦ : S → S is continuous. tion and that the map H Moreover, we have d fb H Φ f = HΦ f

in the distribution sense.

HAUSDORFF OPERATORS ON MODULATION AND WIENER AMALGAM SPACES

Indeed, for f, g ∈ S (Rn ), we have

Z

5

Z

d hH gi = Φ(y)f (x/|y|)dyb g(x)dx Φ f , gi =hHΦ f, b n n Z Z R R = Φ(y) f (x/|y|)b g (x)dxdy Rn Rn Z Z f\ (·/|y|)(x)g(x)dxdy = Φ(y) n n R R Z Z = Φ(y) |y|nfb(|y|x)g(x)dxdy n n R ZR Z fΦ fb, gi. = |y|nΦ(y)fb(|y|x)dyg(x)dx = hH Rn

Rn

Remark 1.5 (Adjoint operator of HΦ f ). We define the complex inner product Z hf | gi = f (x)g(x)dx. Rn

The adjoint operator of HΦ f is defined by

hHΦ f | gi = hf | HΦ∗ gi for f, g ∈ S (Rn ). By a direct calculation, we have Z Z Z Φ(y)f (x/|y|) dyg(x)dx hHΦ f | gi = HΦ f (x)g(x)dx = Rn Rn Rn Z Z Z Z n |y| Φ(y) f (x)g(|y|x)dxdy = Φ(y) f (x/|y|) g(x)dxdy = Rn Rn Rn Rn Z Z fΦ gi. = f (x) Φ(y)|y|ng(|y|x)dydx = hf | H Rn

Rn

It follows that

fΦ g HΦ∗ g = H

in the distribution sense.

We turn to give definitions of modulation and Wiener amalgam spaces. Let S := S (Rn ) be the Schwartz space and S ′ := S ′ (Rn ) be the space of tempered distributions. We define the Fourier transform F f and the inverse Fourier transform F −1 f of f ∈ S (Rn ) by Z Z −2πix·ξ −1 ∨ ˆ F f (ξ) = f (ξ) = f (x)e dx, F f (x) = f (x) = f (ξ)e2πix·ξ dξ. Rn

Rn

The translation operator is defined as Tx0 f (x) = f (x − x0 ) and the modulation operator is defined as Mξ f (x) = e2πiξ·x f (x), for x, x0 , ξ ∈ Rn . Fixed a nonzero function φ ∈ S , the short-time Fourier transform of f ∈ S ′ with respect to the window φ is given by Vφ f (x, ξ) = hf, Mξ Tx φi, and that can be written as Z Vφ f (x, ξ) = f (y)φ(y − x)e−2πiy·ξ dy Rn

6

G. ZHAO, D. FAN, and W. GUO

if f ∈ S . We give the (continuous) definition of modulation space Msp,q as follows. Definition 1.6. Let s ∈ R, 0 < p, q 6 ∞. The (weighted) modulation space Msp,q consists of all f ∈ S ′ (Rn ) such that the (weighted) modulation space norm

kf kMs = kVφ f (x, ξ)kLx,p s Lξ,q

p,q

=

Z

Rn

Z

Rn

!1/q q/p |Vφ f (x, ξ)|pdx hξisq dx

is finite, with the usual modifications when p = ∞ or q = ∞. This definition is independent of the choice of the window φ ∈ S . Applying the frequency-uniform localization techniques, one can give an alternative definition of modulation spaces (see [19] for details). We denote by Qk the unit cube with the center at k. Then the family {Qk }k∈Zn constitutes a decomposition of Rn . Let η ∈ S (Rn ), η : Rn → [0, 1] be a smooth function satisfying that η(ξ) = 1 for |ξ|∞ 6 1/2 and η(ξ) = 0 for |ξ| > 3/4. Let ηk (ξ) = η(ξ − k), k ∈ Zn be a translation of η. Since ηk (ξ) = 1 in Qk , we have that all ξ ∈ Rn . Denote !−1 X ηl (ξ) , k ∈ Zn . σk (ξ) = ηk (ξ)

P

k∈Zn

ηk (ξ) > 1 for

l∈Zn

It is easy to know that {σk }k∈Zn constitutes a smooth partition of the unity, and σk (ξ) = σ(ξ − k). The frequency-uniform decomposition operators can be defined by k := F −1 σk F s for k ∈ Zn . Now, we give the (discrete) definition of modulation space Mp,q . s Definition 1.7. Let s ∈ R, 0 < p, q 6 ∞. The modulation space Mp,q consists ′ of all f ∈ S such that the (quasi-)norm !1/q X s hkisq kk f kqp kf kMp,q := k∈Zn

0 is finite. We write Mp,q := Mp,q for short. We also recall that this definition is s are independent of the choice of {σk }k∈Zn and the definitions of Msp,q and Mp,q equivalent [20].

Definition 1.8. Let 0 < p, q 6 ∞, s ∈ R. Given a window function φ ∈ S \{0}, s the Wiener amalgam space Wp,q consists of all f ∈ S ′ (Rn ) such that the norm

s kf kWp,q = kVφ f (x, ξ)kLsξ,q Lx,p !1/p  Z Z p/q

|Vφ f (x, ξ)|q hξisq dξ

=

Rn

Rn

dx

HAUSDORFF OPERATORS ON MODULATION AND WIENER AMALGAM SPACES

7

is finite, with the usual modifications when p = ∞ or q = ∞. We write Wp,q := 0 Wp,q for short. Now, we state our main results as follows. Theorem 1.9. Let 1 6 p, q 6 ∞, (1/p −1/2)(1/q −1/p) > 0, Φ be a nonnegative function satisfying the basic assumption (1.2). Then HΦ is bounded on Mp,q if and only if Z   n/p n/q ′ |y| + |y| Φ(y)dy < ∞. Rn

Theorem 1.10. Let 1 6 p, q 6 ∞, (1/q−1/2)(1/q−1/p) 6 0, Φ be a nonnegative function satisfying the basic assumption (1.2). Then HΦ is bounded on Wp,q if and only if Z   ′ |y|n/p + |y|n/q Φ(y)dy < ∞. Rn

Our paper is organized as follows. In Section 2, we collect some basic properties of modulation and Wiener amalgam spaces and give the proof of Theorem 1.9 and 1.10. Throughout this paper, we will adopt the following notations. We use X . Y to denote the statement that X 6 CY , with a positive constant C that may depend on n, p, but it might be different from line to line. The notation X ∼ Y means the statement X . Y . X. We use X .λ Y to denote X 6 Cλ Y , meaning that the implied constant Cλ depends on the parameter λ. For a multi-index k = (k1 , k2 , ..., kn ) ∈ Zn , we denote |k|∞ := max |ki | and hki := (1 + |k|2)1/2 . i=1,2,...,n

2. Proof of main theorem First, we list some basic properties about modulation spaces as follows. Lemma 2.1 (Symmetry of time and frequency). kF −1 f kMp,q ∼ kf kWq,p ∼ kF f kMp,q . Proof. By the fact that |Vφ f (x, ξ)| = |Vφˆfˆ(ξ, −x)|, the conclusion follows by the definition of modulation and Wiener amalgam space.  Lemma 2.2 ([18] Dilation property of modulation space). Let 1 6 p, q 6 ∞, (1/p − 1/2)(1/q − 1/p) ≥ 0. Set fλ (x) = f (λx). Then ′

kfλ kMp,q . max{λ−n/p , λ−n/q }kf kMp,q . Lemma 2.3 ([16] Embedding relations between modulation and Lebesgue spaces). The following embedding relations are right: (1) Mp,q ֒→ Lp for 1/q > 1/p > 1/2; (2) Lp ֒→ Mp,q for 1/q 6 1/p 6 1/2. Lemma 2.4 ([2] Embedding relations between Wiener amalgam and Lebesgue spaces). The following embedding relations are right:

8

G. ZHAO, D. FAN, and W. GUO

(1) Wp,q ֒→ Lp for 1/p > 1/q > 1/2; (2) Lp ֒→ Wp,q for 1/p 6 1/q 6 1/2. Lemma 2.5. Let 1 6 p, q 6 ∞. We have (1) Z 6 kf kM ′ ′ kgkMp,q f (x)¯ g (x)dx p ,q n R

(2)

Z

Rn

f (x)¯ g (x)dx 6 kf kWq′ ,p′ kgkWq,p .

Proof. ByPLemma 2.1, we only give the proof of the first inequality. Denote ηk∗ = ηk and ∗k = F −1 ηk∗ F . By the definition of modulation spaces l∈Zn :ηk ηl 6=0

and Plancherel’s equality we get that Z Z Z X X ˆ ˆgˆdξ = = σ g ˆ dx σ f · f f (x)¯ g (x)dx l k n n n R k∈Zn R R n l∈Z Z Z X X ∗ ∗ ˆ k f · k gdx σk f · σk gˆdx = = Rn Rn k∈Zn k∈Zn X XZ kk f kLp′ k∗k gkLp |k f · ∗k g| dx 6 6 k∈Zn

6

X

Rn

′ kk f kqLp′

k∈Zn

!1/q′

k∈Zn

X

k∗k gkqLp

k∈Zn

!1/q

6 kf kMp′ ,q′ kgkMp,q ,

where we use H¨older’s inequality in the last two and third inequality and the fact that the definition of modulation space is independent of the decomposition function.  In order to make the proof more clear, we give the following technical proposition. Proposition 2.6 (For technique). Let 1/2 6 1/p 6 1/q 6 1, and Φ be a nonnegative function satisfying the basic assumption (1.2). Then (1) if HΦ : Mp,q → Lp is bounded, we have Z |y|n/pΦ(y)dy < ∞; Rn

(2) if HΦ∗ : Wq,p → Lq is bounded, we have Z ′ |y|n/q Φ(y)dy < ∞; Rn

(3) if HΦ∗ : Mp,q → Lp is bounded, we have Z ′ |y|n/p Φ(y)dy < ∞; Rn

HAUSDORFF OPERATORS ON MODULATION AND WIENER AMALGAM SPACES

9

(4) if HΦ : Wq,p → Lq is bounded, we have Z

|y|n/q Φ(y)dy < ∞. Rn

Proof. We only give the proof of statements (1) and (2), since the other cases can be handled similarly. Suppose HΦ : Mp,q → Lp is bounded. Let ψ : Rn → [0, 1] be a smooth bump function supported in the ball {ξ : |ξ| < 23 } and be equal to 1 on the ball {ξ : |ξ| 6 43 }. Let ρ(ξ) = ψ(ξ) − ψ(2ξ). Then ρ is a positive smooth function supported in the annulus {ξ : 23 < |ξ| < 32 }, satisfying ρ(ξ) = 1 on a smaller annulus {ξ : 43 6 |ξ| 6 43 }. Denote ρj (ξ) := ρ(ξ/2j ). We have suppρj ⊂ {ξ : 2 · 2j 6 |ξ| 6 23 · 2j } and ρj (ξ) = 1 on {ξ : 34 · 2j 6 |ξ| 6 34 · 2j }. Thus, we have 3 N N P P supp ρj (ξ) ⊂ {ξ : 43 6 |ξ| 6 23 · 2N } and ρj (ξ) = 1 on {ξ : 32 6 |ξ| 6 34 · 2N }. j=1

j=1

Take ϕ to be a nonnegative smooth function satisfying that suppϕ b ⊂ B(0, 1/2), ! NP +1 ϕ(0) = 1. Choose fN (x) = ρj (x) · |x|−n/p ∗ ϕ. So we have j=0

suppfc N ⊂ B(0, 1/2) and fN (x) &

N X

ρj (x) · |x|−n/p .

(2.1)

j=1

In above, the previous inclusion relation follows from the support condition of ϕ. b We interpret the latter inequality. We only need to prove it when the right hand is nonzero, i.e. x ∈ { 34 6 |x| 6 32 · 2N }. For the nonnegative function ϕ satisfying ϕ(0) = 1, there exists a positive constant δ < min{4/3, 1/12}, such that ϕ(x) > 1/2 when |x| < δ. By the triangle inequality and the properties of ϕ we have that

fN (x) =

N +1 X

ρj (x) · |x|−n/p

j=0

&

Z

4 ≤|x−y|≤ 23 ·2N 3

|y| Φ(y)|y|n/p · ρj (x/|y|) · |x|−n/p dy

B(0, 2 ·2M )\B(0, 3 ·2−M )

j=1 3 4 Lp

Z



n/p −n/p Φ(y)|y| · χ{2M Φ(y)|y|n/q dy · |x|−n/q χ{2M