Generalized Hardy-Morrey spaces

Generalized Hardy-Morrey spaces Ali Akbulut

arXiv:1511.02020v1 [math.FA] 6 Nov 2015

Department of Mathematics, Ahi Evran University, Kirsehir, Turkey [email protected]

Vagif S. Guliyev Department of Mathematics, Ahi Evran University, Kirsehir, Turkey Institute of Mathematics and Mechanics, Baku, Azerbaijan [email protected]

Takahiro Noi Department of Mathematics and Information Science, Tokyo Metropolitan University [email protected]

Yoshihiro Sawano Department of Mathematics and Information Science, Tokyo Metropolitan University [email protected] Abstract The generalized Morrey space was defined independetly by T. Mizuhara 1991 and E. Nakai in 1994. Generalized Morrey space Mp,φ (Rn ) is equipped with a parameter 0 < p < ∞ and a function φ : Rn × (0, ∞) → (0, ∞). Our experience shows that Mp,φ (Rn ) is easy to handle when 1 < p < ∞. However, when 0 < p ≤ 1, the function space Mp,φ (Rn ) is difficult to handle as many examples show. The aim of this paper is twofold. One of them is to propose a way to deal with Mp,φ (Rn ) for 0 < p ≤ 1. One of them is to propose here a way to consider the decomposition method of generalized Hardy-Morrey spaces. We shall obtain some estimates for these spaces about the Hardy-Littlewood maximal operator. Especially, the vector-valued estimates obtained in the earlier papers are refined. The key tool is the weighted Hardy operator. Much is known about the weighted Hardy operator. Another aim is to propose here a way to consider the decomposition method of generalized Hardy-Morrey spaces. Generalized Hardy-Morrey spaces emerged from generalized Morrey spaces. By means of the grand maximal operator and the norm of generalized Morrey spaces, we can define generalized Hardy-Morrey spaces. With this culmination, we can easily refine the existing results. In particular, our results complement the one the 2014 paper by Iida, the third author and Tanaka; there was a mistake there. As an application, we consider bilinear estimates, which is the “so-called” Olsen inequality.

AMS Mathematics Subject Classification:

42B20, 42B25, 42B35

Key words: generalized Hardy-Morrey spaces, atomic decomposition, maximal operators

1

1

Introduction

In this paper, we are concerned with generalized Hardy-Morrey spaces. The generalized Morrey space Mp,φ (Rn ) is equipped with a function φ and a positive parameter 0 < p < ∞. The generalized Morrey space Mp,φ (Rn ) was defined independetly by T. Mizuhara in 1991 [19] and E. Nakai in 1994 [20]. Although we can disprove Cc∞ (Rn ) is dense in Mp,φ (Rn ), we are still able to develop a theory of the function space HMp,φ (Rn ) called generalized Hardy-Morrey spaces. Denote by Gp the set of all the functions φ : Rn × (0, ∞) → (0, ∞) decreasing in n the second variable such that t ∈ (0, ∞) 7→ t p φ(x, t) ∈ (0, ∞) is almost increasing uniformly over the first variable x, so that there exists a constant C > 0 such that φ(x, r) ≤ φ(x, s),

Cφ(x, r)r n/p ≥ φ(x, s)sn/p

for all x ∈ Rn and 0 < s ≤ r < ∞. All “cubes” in Rn are assumed to have their sides parallel to the coordinate axes. Denote by Q the set of all cubes. For a cube Q ∈ Q, 1 the symbol ℓ(Q) stands for the side-length of the cube Q; ℓ(Q) ≡ |Q| n . We denote by Q(Rn ) the set of all cubes. When we are given a cube Q, we use the following abuse of notations φ(Q) ≡ φ(c(Q), ℓ(Q)), where c(Q) denotes the center of Q. The generalized Morrey space Mp,φ (Rn ) is defined as the set of all measurable functions f for which the norm kf kMp,φ

1 ≡ sup Q∈Q φ(Q)

1 |Q|

Z

p

|f (y)| dy Q

1

p

is finite. n

Observe that, if φ(x, r) = r p , then Mp,φ (Rn ) = Lp (Rn ). In the special case when φ(x, r) ≡ r λ/p−n/p , we write Mp,λ (Rn ) instead of Mp,φ (Rn ). An important observation made by Nakai is that we can assume that φ itself is decreasing and that φ(t)tn/p ≤ φ(T )T n/p for all 0 < t ≤ T < ∞ when φ is independent of x; see [21, p. 446]. Indeed, in the case when 1 ≤ p < ∞, Nakai established that there exists a function ρ such that ρ itself is decreasing, that ρ(t)tn/p ≤ ρ(T )T n/p for all 0 < t ≤ T < ∞ and that Mp,φ (Rn ) = Mp,ρ (Rn ). See [32, (1.2)] for the case when 0 < p ≤ 1. The class Gp is defined to be the set of all φ such that φ itself is decreasing and that φ(t)tn/p ≤ φ(T )T n/p for all 0 < t ≤ T < ∞. This assumption will turn out to be natural even when φ depends on x; see Section 2. One of the primary aims of this paper is to prove the following decomposition result about the functions in generalized Morrey spaces M1,φ (Rn ): Theorem 1.1. Assume that φ ∈ G1 and η ∈ G1 satisfy Z ∞ φ(x, r) φ(x, s) ds ≤ C . η(x, s)s η(x, r) r 2

(1)

n ∞ n ∞ Assume that {Qj }∞ j=1 ⊂ Q(R ), {aj }j=1 ⊂ M1,η (R ) and {λj }j=1 ⊂ [0, ∞) fulfill

X ∞

1 , supp(aj ) ⊂ Qj , λj χQj kaj kM1,η ≤ < ∞.

η(ℓ(Qj ))

j=1

Then f ≡

∞ X

(2)

M1,φ

λj aj converges absolutely in L1loc (Rn ) and satisfies

j=1

kf kM1,φ

∞ X

λj χ Q j ≤C

j=1

.

(3)

M1,φ

The proof of this theorem is not so difficult and it is given in an early stage of the present paper; see Section 2.2, where we do not use the Hardy-Littlewood maximal operator as in [17]. Unlike the case when p > 1, when 0 < p ≤ 1, Mp,φ (Rn ) is a nasty space as the following example shows. Example 1.2. Denote by M the Hardy-Littlewood maximal operator. 1. Let η : (0, ∞) → (0, ∞) be a function which is independent of the position x. One defined ML log L,η by the following norm in [34]; Z |f (x)| |f (x)| 1 inf λ > 0 : log e + dx ≤ |Q| . kf kML log L,η ≡ sup λ λ Q∈Q η(ℓ(Q)) Q In [34, Lemma 3.5], we proved C −1 kf kML log L,η ≤ kM f kM1,η ≤ Ckf kML log L,η for all f ∈ ML log L,η (Rn ). 2. In [34, Lemma 3.4], we proved C −1 kf kM1,η ≤ kM f kMp,η ≤ Ckf kM1,η for all f ∈ M1,η (Rn ). From these examples we see that Mp,φ (Rn ) with p ∈ (0, 1] is difficult to handle. Probably, Theorem 1.1 paves the way to deal with such a nasty space. Another method to handle these nasty spaces to use the grand maximal operator and define generalized Hardy-Morrey spaces. Let t > 0 and f ∈ L1 (Rn ). Then define the heat semigroup by: Z |x − y|2 1 t∆ p exp − f (y)dy (x ∈ Rn ). e f (x) ≡ 4t (4πt)n Rn

In a well-known method using the duality, we naturally extend et∆ f to the case when f ∈ S ′ (Rn ). Let 0 < p ≤ 1 and φ ∈ Gp . The generalized Hardy-Morrey space HMp,φ (Rn ) is the set of all f ∈ S ′ (Rn ) satisfying sup |et∆ f (·)| ∈ Mp,φ (Rn ). We equip HMp,φ (Rn ) with the following norm:

t∆

kf kHMp,φ ≡

sup |e f | t>0

Mp,φ

3

t>0

(f ∈ HMp,φ (Rn )).

(4)

We define dp ≡ n/p − n for 0 < p ≤ 1. In addition to Theorem 1.1, we shall prove the following two theorems in this paper. Theorem 1.3. Let 0 0. Assume in addition that {Qj }∞ j=1 ⊂ Q(R ), {aj }j=1 ⊂ Mq,η (R ) and {λj }∞ j=1 ⊂ [0, ∞) fulfill

 1

p ∞

X

 u such that η(x, s) = s−n/v and that φ(x, s)sn/u ≤ φ(x, r)r n/u for all s and r with s ≥ r, then (48) is satisfied. Theorem 1.3 will refine [18, p. 100 Theorem] in that we can postulate a weaker integrability condition on aj in Theorem 1.3. We shall take its advantage in Section 4. Theorem 1.5. Let L ∈ N ∪ {0}. Let 0 < p ≤ 1 and f ∈ HMp,φ (Rn ). Then under Z ∞ φ(x, r)p φ(x, s)p ds ≤ C η(x, s)p s η(x, r)p r and

∞

ds ≤ Cφ(x, r). s r ∞ n ∞ and φ ∈ G1 , there existsPa triplet {λj }∞ j=1 ⊂ [0, ∞), {Qj }j=1 ⊂ Q(R ) and {aj }j=1 ⊂ ∞ L∞ (Rn ) such that f = j=1 λj aj in S ′ (Rn ) and that, for all v > 0

 1/v

∞

Z

X



v α (λj χQj ) ≤ Cv kf kHMp,φ (9) x aj (x) dx = 0, |aj | ≤ χQj ,

n R

j=1

Z

φ(x, s)

Mp,φ

for all multi-indices α with |α| ≤ L. Here the constant Cv > 0 is independent of f . 4

It is not known that HMp,φ (Rn ) ∩ L1loc (Rn ) is dense in HMp,φ (Rn ). It seems that HMp,φ (Rn ) ∩ L1loc (Rn ) is not dense in HMp,φ (Rn ) as the fact that Mp,φ (Rn ) ∩ n n L∞ comp (R ) is not dense in Mp,2p (R ) implies. Recall that in [40], we resorted to density of H p (Rn ) ∩ L1loc (Rn ) to obtain the atomic decomposition of H p (Rn ). The difficulty willl cause a disability; we can prove Theorem 1.5 only when f ∈ HMp,φ (Rn ). By using a diagonal argument, we circumbent this problem; see (70) and (71). Before we go further, let us recall some special cases related to generalized Morrey spaces. Example 1.6. 1. Generalized Morrey spaces can cover L∞ (Rn ) spaces by letting φ ≡ 1. 2. [35, Theorem 5.1] Let 1 < p < ∞ and 0 < λ < n. Then there exists a positive constant Cp,λ such that Z 1 −1 (10) k(1 − ∆)λ/2p f kLp,λ |f (x)|dx ≤ Cp,λ |B|(1 + |B|) p log e + |B| B holds for all f ∈ Lp,λ (Rn ) with (1 − ∆)λ/2p f ∈ Lp,λ (Rn ) and for all balls B. See [6, Section 5] for more details. In view of the integral kernel of (1 − ∆)−α/2 (see [39]) and the Adams theorem, we have (1 − ∆)−α/2 : Lp,λ (Rn ) → Lq,λ (Rn )

(11)

is bounded as long as the parameters p, q, λ and α satisfy 1 < p, q < ∞, 0 < λ ≤ n,

1 1 α = − . q p λ

However, if α = λp , the number q not being finite, the boundedness assertion (11) is no longer true. Hence (10) can be considered as a substitute of (11). A passage to the Hardy type space from a given function space is not a mere quest to generality. Many people have shown that Hardy spaces H p (Rn ) (0 < p ≤ ∞) can be more informative than Lebesgue spaces Lp (Rn ) when we discuss the boundedness of some operators. For example, the Riesz transform is bounded from H 1 (Rn ) to L1 (Rn ), although they are not bounded on L1 (Rn ). One of the earliest real variable definitions of Hardy spaces were based on the grand maximal operator, which is discussed in [40] and references therein. One can also give an equivalent definition for Hardy spaces by using the atomic decomposition. This definition states that any elements of Hardy spaces can be represented as the series of atoms. An atom is a compactly supported function which enjoys the size condition and the cancellation moment condition. One of the advantages of the atomic decompositions in Hardy spaces is that we can prove the boundedness of some operators can be verified only for the collection of atoms. The concept of the atomic decomposition in Hardy spaces can be developed to other function spaces. Some of these works are the decomposition of Hardy–Morrey spaces 5

[18], the decomposition of Hardy spaces with variable exponent [28], and the atomic decomposition of Morrey spaces [17]. Motivated by these advantages that Hardy spaces enjoy, in our current research, we investigate the atomic decomposition for generalized Hardy-Morrey spaces, where we are based on the definition by means of the grand maximal operator. There are many attempts of obtaining non-smooth atomic decompositions by using the grand maximal operator [8, 17, 22, 23], where the authors handled Morrey spaces, Orlicz spaces and variable exponent Lebesgue spaces. Unlike Orlicz spaces, variable exponent Lebesgue spaces, in general we can take a sequence {fj }∞ j=1 of functions such that f1 ≥ f2 ≥ · · · ≥ fj ≥ fj+1 ≥ · · · → 0, inf kfj kMp,φ > 0. j∈N

For example, when 0 < p < a, the sequence fj (x) ≡ χ(j,∞) (|x|)|x|−n/a does the job. This makes it more difficult to look for a good dense space of Mp,n−a (Rn ). This difficulty prevents us from using (65) directly. We adopt the following notations: 1. N0 ≡ {0, 1, . . .}. 2. Let A, B ≥ 0. Then A . B means that there exists a constant C > 0 such that A ≤ CB and A ≈ B stands for A . B . A, where C depends only on the parameters of importance. 3. By “cube” we mean a compact cube whose edges are parallel to the coordinate axes. The metric ball defined by ℓ∞ is called a cube. If a cube has center x and radius r, we denote it by Q(x, r). From the definition of Q(x, r), its volume is (2r)n . We write Q(r) instead of Q(o, r), where o denotes the origin. Given a cube Q, we denote by c(Q) the center of Q and by ℓ(Q) the sidelength of Q: ℓ(Q) = |Q|1/n , where |Q| denotes the volume of the cube Q. 4. Given a cube Q and k > 0, k Q means the cube concentric to Q with sidelength k ℓ(Q). 5. By a dyadic cube, we mean a set of the form 2−j m + [0, 2−j ]n for some m ∈ Zn and j ∈ Z. The set of all dyadic cubes will be denoted by D. 6. Let Qx (Rn ) be a collection of all cubes that contain x ∈ Rn . 7. In the whole paper, we adopt the following definition of the Hardy-Littlewood maximal operator to estimate some integrals. The Hardy-Littlewood maximal operator M is defined by Z 1 |f (y)|dy, (12) M f (x) ≡ sup Q∈Qx (Rn ) |Q| Q for a locally integrable function f .

6

8. Let 0 < α < n. We define the fractional integral operator Iα by Z f (y) Iα f (x) ≡ dy n−α Rn |x − y| for all suitable functions f on Rn . 9. Let 0 < q ≤ ∞. If F ≡ {fj }∞ j=−∞ is a sequence of complex-valued Lebesgue measurable functions on Rn such that kF kMp,φ (lq ) = kkF klq kMp,φ < ∞, write F ∈ Mp,φ (lq , Rn ). 10. Let 0 < p, q ≤ ∞. If {fj }j∈N0 is a sequence of complex-valued Lebesgue measurable functions on Ω ⊆ Rn , then define;

n

o

≡

{fj }j∈N0 p

kfj kLp (Ω)

lq (L (Ω))

j∈N0 l q

and

{fj }j∈N0

Lp (Ω)(lq )

lq

≡ }

{f j j∈N0

. Lp (Ω)

The space Mp,φ (lq , Rn ) stands for the set of all sequences {fj }j∈N0 of complexvalued Lebesgue measurable functions on Rn for which

< ∞. k{fj }j∈N0 kMp,φ (lq ) ≡ k{fj }j∈N0 klq Mp,φ

Similarly denote by W Mp,φ (lq , Rn ) the set of all sequences {fj }j∈N0 for which

< ∞. k{fj }j∈N0 kW Mp,φ (lq ) ≡ k{fj }j∈N0 klq W Mp,φ

The spaces lq (Mp,φ (Rn )) and lq (W Mp,φ(Rn )) can be also defined similarly by the norms;

n o

0 with kx − yk∞ ≤ r − s such that Mp,φ (Rn ) = Mp,ψ (Rn ) with norm coincidence. Proof. Let us set ψ(x, r) ≡ infn y∈R

inf

v≥r+kx−yk∞

φ(y, v)

v n/p

(x ∈ Rn , r > 0).

r

(14)

Then we have φ(x, r) ≥ ψ(x, r) trivially and hence kf kMp,φ ≤ kf kMp,ψ . Meanwhile, kf kMp,ψ

1 = sup Q∈Q ψ(c(Q), ℓ(Q)) = sup sup

Q∈Q y∈Rn



≤ sup  y∈Rn

1 |Q|

Z

sup v≥ℓ(Q)+kx−yk∞

sup

v≥ℓ(Q)+kx−yk∞

p

|f (y)| dy Q

1 φ(y, v)

1 φ(y, v)

1 vn

1

1 vn

Z

p

Z

|f (y)|p dy Q

Q(y,v)

= kf kMp,φ .

1 ! p

!1  p |f (y)|p dy 

Thus, we have Mp,φ (Rn ) = Mp,ψ (Rn ) with norm coincidence. From the definition of ψ, it is easy to check that we have (13). Lemma 2.2. Let 0 < p < ∞ and let φ : Rn × (0, ∞) → (0, ∞) be a function satisfying (13). Then there exists a function ψ : Rn × (0, ∞) → (0, ∞) satisfying ψ(x, r) ≤ ψ(x, s)

(15)

for all x ∈ Rn and 0 < s ≤ r < ∞ such that Mp,φ (Rn ) = Mp,ψ (Rn ) with norm equivalence. Proof. Let us set ψ(x, r) ≡ inf

00

holds for some C > 0 for all non-negative and non-decreasing g on (0, ∞) if and only if Z ∞ w(s)ds B ≡ sup v2 (t) < ∞. (25) sups 0. Then, for p > 1, M is bounded from Mp,φ1 (Rn ) to Mp,φ2 (Rn ) and, for p = 1, M is bounded from M1,φ1 (Rn ) to W M1,φ2 (Rn ). 13

The following statements, containing Proposition 2.13, was proved by A. Akbulut, V.S. Guliyev and R. Mustafayev [1]; note that (26) is stronger than (27). Proposition 2.14. Let 1 ≤ p < ∞ and suppose the couple (φ1 , φ2 ) satisfies the following condition; n n (27) inf φ1 (x, s)s p τ − p . φ2 (x, t), sup τ >t

τ 1, kM f kMp,φ2 . kf kMp,φ1 for all f ∈ Mp,φ1 (Rn ) and for 1 ≤ p < ∞, kM f kW Mp,φ2 . kf kMp,φ1 for all f ∈ Mp,φ1 (Rn ). From this proposition, when φ1 = φ2 = φ, we have the following boundedness. We know that there is no requirement when we consider the boundedness of the maximal operator [27, Theorem 2.3] when φ is independent of x. Proposition 2.14 naturally extends the assertion above. Corollary 2.15. [20, Theorem 1], [27, Theorem 2.3] Let 1 ≤ p < ∞ and φ ∈ Gp . 1. Let 1 < p < ∞. Then kM f kMp,φ . kf kMp,φ for all f ∈ Mp,φ (Rn ). 2. Let 1 ≤ p < ∞. Then kM f kW Mp,φ . kf kMp,φ for all f ∈ Mp,φ (Rn ).

By using the Planchrel-Pólya Nilokiski’i inequality [25], we have the following estimate of the Peetre maximal operator. Theorem 2.16. Let 0 t

τ 2r − r = r. Next, B(y, t) ∩ (Rn \B(x, 3r)) ⊂ B(x, 2t). Indeed, for z ∈ B(y, t) ∩ (Rn \B(x, 3r)), then we get |x − z| ≤ |y − z| + |x − y| < t + r < 2t. Hence for all j ∈ N0 Z

1

kM F2 (y)klq = sup |fj (z)|dz lq t>0 |B(y, t)| B(y,t)∩(Rn \B(x,2r))

∞ Z

X 1

|f (y)|dy .

j k

|2 Q| 2k Q k=1

.

∞ X k=1

lq

1 |2k Q|

Z

2k Q

kF (y)klq dy.

As a result, we obtain kM F2 (y)klq .

Z

∞ 2r

s−n−1 k kF klq kB(x,s) ds

for all y ∈ B(x, s). Thus, the estimate for M F2 is valid. Then obtain (33) from (36) and (38). 18

(38)

Let p = 1. It is obvious that for any ball B = B(x, r) k kM F klq kW L1 (B(x,r)) ≤ k kM F1 klq kW L1 (B(x,r)) + k kM F2 klq kW L1 (B(x,r)) . By the weak-type Fefferman-Stein maximal inequality (see [7]) we have k kM F1 klq kW L1 (B(x,r)) ≤ k kM F1 klq kW L1 (Rn ) . k kF1 klq kL1 (Rn ) = k kF klq kL1 (B(x,2r)) ,

(39)

where the implicit constant is independent of the vector-valued function F . Then by (39) and (38), we obtain the inequality (34). We transform Lemma 2.22 to an inequality we use to prove Theorem 2.19. Lemma 2.23. Let 1 ≤ p < ∞ and 1 ≤ q ≤ ∞. Then, for any ball B = B(x, r) in Rn , the inequality Z ∞ n − n −1 t p k kF klq kLp (B(x,t)) dt (40) k kM F klq kLp (B(x,r)) . r p 2r

holds for all F = {fj }∞ j=−∞ ⊂

Lploc (Rn ).

Moreover, for any ball B = B(x, r) in Rn , the inequality Z ∞ n k kM F klq kW L1 (B(x,r)) . r t−n−1 k kF klq kL1 (B(x,t)) dt

(41)

2r

1 n holds for all F = {fj }∞ j=−∞ ⊂ Lloc (R ).

Proof. Note that, for all 1 ≤ p < ∞ k kF klq k

Lp (B(x,2r))

≤r

n p

Z

∞

t

−n −1 p

2t

k kF klq kLp (B(x,t)) dt.

(42)

Applying Hölder’s inequality, we get k kM F2 (y)klq kLp (B(x,r)) ≤ 2n vn−1 k1kLp (B(x,r)) ≤ 1

2n vn−1 k1kLp (B(x,r))

Z

∞

Z2r∞ 2r

t−n−1 k kF klq kL1 (B(x,t)) dt t−n/p−1 k kF klq kLp (B(x,t)) dt. (43)

n

Since k1kLp (B(x,r)) = vnp r p , we then obtain (40) from (42) and (43). Let p = 1. The inequality (41) directly follows from (42). With these estimates in mind, let us prove Theorem 2.19.

19

n

Proof of Theorem 2.19. By Lemma 2.23 and Theorem 2.11 with v1 (r) = φ1 (x, r)−1 r − p , n v2 (r) = φ2 (x, r)−1 , g(r) = k kF klq kLp (B(x,r)) and w(r) = r − p −1 , we have kM F kMp,φ

2

(lq )

sup

.

x∈Rn , r>0

sup

.

x∈Rn , r>0

−1

φ2 (x, r)

Z

∞ 2r

n −1 − p

φ1 (x, r)

r

n

t− p −1 k kF klq kLp (B(x,t)) dt

k kF klq kLp (B(x,r))

= kF kMp,φ1 (lq ) , if 1 0

.

sup

x∈Rn , r>0

= kF kM1,φ

φ2 (x, r)

∞

2r −1 −n

φ1 (x, r)

1

Z

r

t−n−1 k kF klq kLp (B(x,t)) dt

k kF klq kLp (B(x,r))

(lq )

if p = 1. As a corollary, we can recover the result in [27]. Corollary 2.24. [27, Theorem 2.5] Let 1 ≤ p < ∞, 1 ≤ q ≤ ∞ and φ ∈ Gp . Assume in addition Z ∞ dt φ(t) . φ(r). t r

Then

kM F kMp,φ (lq ) . kF kMp,φ (lq )

if

p > 1,

(44)

and kM F kW Mp,φ (lq ) . kF kM1,φ (lq ) . We can prove the following estimate, which is a counterpart to Theorem 2.16. Theorem 2.25. Let 0 < p < ∞, 0 < q ≤ ∞, 0 < r < min{p, q} and suppose that the couple (φ1 , φ2 ) satisfies the condition (28). Let {Ωj }j∈N0 be a sequence of compact sets, and let dj be the diameter of Ωj . Then exists a positive constant C such that





n





o

|f (· − y)|

j

,

(45)

sup

≤ C kfj kMp,φ

n/r 1 j∈N0

 y∈Rn 1 + dj |y| 

l q

Mp,φ2 j∈N 0 lq

if we are given a collection of measurable functions {fj }j∈N0 such that supp(Ffj ) ⊂ Ωj .

Proof. We begin with a reduction. Let {fj }j∈N0 ∈ lq (Mp,φ1 ), 0 < q ≤ ∞, η j ∈ Ω and j hj (x) ≡ e−ixη fj (x). We have Fhj (ξ) = Ffj (ξ + η j ). Therefore, supp Fhj ⊂ Ωj − η j .

20

If {fj }j∈N0 ∈ lq (Mp,φ1 ,Ω ), η j ∈ Ω satisfies (1) of Theorem 2.25, then so does {hj }j∈N0 also, where Ω is replaced by {Ωj − η j }j∈N0 , and the converse also holds. Thus, we may suppose 0 ∈ Ωj . It suffices to consider the case Ωj = Dj = B(0, dj ). Second, we have to prove (45), when Ωj = Dj = B(0, dj ) and dj > 0. If {fj }j∈N0 ∈ lq (Mp,φ1 ,Ω ), η j ∈ Ω, then fj ∈ Lp,Ωj . If gj ≡ fj (d−1 ·), then Fgj = dnj (Ffj )(dj ·) and hence supp Fgj ⊂ B(0, 1). From (29), we obtain |fj (x − z)| . [M (|fj |r )]1/r , 1 + |dj z|n/r

for all

x, z ∈ Rn ,

(46)

where the implicit constant is independent of x, z and j. If 0 < q < ∞, then by (46) we have

n

o

|f (· − z)|

j r 1/r

. [M (|fj | )]

sup

n/r

x∈Rn 1 + |dj z|

j∈N 0 lq (Mp,φ2 ) j∈N0 lq (Mp,φ ) 2

1/r

= {M (|fj |r )}j∈N0 . lq/r (Mp/r,φ2 )

Since 0 < r < min{p, q}, we have p/r > 1 and q/r > 1. By Corollary 2.21, we have

1/r

|fj (· − z)|

r . (|f | )}

{M

sup j j∈N 0 n/r

x∈Rn 1 + |dj z| lq/r (Mp/r,φ2 ) j∈N0 lq (Mp,φ ) 2 . k{fj }j∈N0 klq (Mp,φ ) . 1

If q = ∞, by (46), we have sup sup j∈N0

z∈Rn

|fj (· − z)| . sup [M (|fj |r ) (x)]1/r . 1 + |dj z|n/r j∈N0

Thus,

|fj (· − z)|

1/r r

] [M (|f | ) (·)] . sup sup sup

j n/r Mp,φ2 j∈N0 j∈N0 z∈Rn 1 + |dj z| Mp,φ 2

1/r

= sup kM (|fj |r )kMp/r,φ . j∈N0

2

Using Corollary 2.21, we obtain

|f (· − z)| j

sup . sup kfj kMp,φ .

supn 1 + |d z|n/r 1 j∈N0 z∈R j∈N0 j Mp,φ 2

The theorem is therefore proved.

Finally, to conclude this section, we obtain an estimate which is used later. 21

Lemma 2.26. Let {Ek }k∈N be a countable collection of measurable sets in Rn . Let 0 ≤ θ < 1. Suppose in addition that p and κ are positive numbers such that pκ > 1 and that p ≤ 1. Then    1 κ

κ ∞

X

X

X  − nθ

κ   . 2 p χE k

(M χEk )  .

k∈N

Lp ([−1,1]n )

l=1

k∈N

Lp ([−2l ,2l ]n )

Proof. By the scaling, we have

X

(M χEk )κ

k∈N

Lp ([−1,1]n )

θ

 !1

X κ

 κ

=  (M χEk )

k∈N

Lpκ ([−1,1]n )

κ

  .

Recall that (M χ[−1,1]n ) pκ is an A1 -weight of Muckenhoupt and Wheeden. By the weighted vector-valued inequality obtained by Andersen and John [2], we obtain  κ

!1

X

κ

X

θ



 (M χ[−1,1]n ) pκ ≤  (M χEk )κ

(M χEk )κ 

k∈N

pκ n k∈N Lp ([−1,1]n ) L (R ) κ 

1 !

κ X θ   (M χ[−1,1]n ) pκ .  χE k 

pκ n

k∈N L (R ) κ 

!1

∞ κ X

 X 1

χE k .  nθ

pκ

pκ k∈N l=1 2 l l n L ([−2 ,2 ] )    1 κ

κ ∞

X

X  − nθ

  = χE k 2 p  ,

k∈N

l=1

Lp ([−2l ,2l ]n )

as was to be shown.

2.5

Structure of generalized Hardy-Morrey spaces

The grand maximal operator characterizes Hardy-Morrey spaces defined by the norm (4). To formulate the result, we recall the following two fundamental notions. 1. Topologize S(Rn ) by norms {pN }N ∈N given by X pN (ϕ) ≡ sup (1 + |x|)N |∂ α ϕ(x)| |α|≤N

x∈Rn

for each N ∈ N. Define FN ≡ {ϕ ∈ S(Rn ) : pN (ϕ) ≤ 1}. 22

2. Let f ∈ S ′ (Rn ). The grand maximal operator Mf is given by Mf (x) ≡ sup{|t−n ψ(t−1 ·) ∗ f (x)| : t > 0, ψ ∈ FN } (x ∈ Rn ),

(47)

where we choose and fix a large integer N . In analogy to [22, Section 3], we can prove the following proposition. Proposition 2.27. Let 0 < p ≤ 1 and φ ∈ Gp . Then kf kHMp,φ ∼ kMf kMp,φ for all f ∈ S ′ (Rn ). From this proposition, we can use the norm kMf kMp,φ for the space Mp,φ (Rn ). Lemma 2.28. Let 0 0 such that if |y| ≤ 1 and ϕ ∈ FN , then C −1 ϕ(· − y) ∈ FN . Thus, |hf, ϕi| . inf Mf (y) |y|≤1

for all ϕ ∈ FN . This implies |hf, ϕi| . kf kHMp,φ , as was to be shown. Going through the same argument as [22, Theorem 3.4], we obtain the following theorem, whose proof will be omitted. Theorem 2.29. Let N ≫ 1. Then f ∈ HMp,φ (Rn ) if and only if Mf ∈ Mp,φ (Rn ). Remark 2.30. When 1 < p < ∞ and φ ∈ Gp , M is bounded on Mp,φ (Rn ) as Proposition 2.13 implies. Also, similarly to [8], we can show that Mp,φ (Rn ) is realized as a dual space of a Banach space. By combining these facts, we can show the following fact for f ∈ S ′ (Rn ). The distribution f is represented by a function in Mp,φ (Rn ) if and only if Mf ∈ Mp,φ (Rn ).

3

Atomic decomposition

We return to the case where ϕ is independent of x and we prove the remaining theorems.

23

3.1

Proof of Theorem 1.3

Prior to the proof, we remark that (6) implies Z ∞ φ(x, s)p φ(x, r)p ds ≤ C η(x, s)p s η(x, r)p r and

Z

∞

φ(x, s)

r

ds ≤ Cφ(x, r). s

(48)

(49)

For the proof of (48), we refer to [20]. We start with collecting auxiliary estimates. Lemma 3.1. Let p, η, aj , Qj be the same as Theorem 1.3. Then kM aj kMp,η .

1 . η(Qj )

(50)

Proof. When p < 1. we use the boundedness of the Hardy-Littlewood maximal operator M : M1,η (Rn ) → Mp,η (Rn ); [34] for more details. When p = 1, this can be replaced by the Mq,η (Rn )-boundedness of M . Using this boundedness and (5) and (7), we obtain (50). Note that (50) readily yields k(M aj )p kM1,ηp = (kM aj kMp,η )p .

1 . η(Qj )p

(51)

We invoke an estimate from [22]; Maj (x) . χ3Qj (x)M aj (x) +

ℓ(Qj )n+d+1 . ℓ(Qj )n+d+1 + |x − c(Qj )|n+d+1

(52)

For the time being, we assume that there exists N ∈ N such that λj = 0 whenever j ≥ N. Observe first that Mf (x) ≤

∞ X j=1

1  p ∞ X p p  |λj | Maj (x) |λj |Maj (x) ≤ , j=1

since M is sublinear and 0 < p ≤ 1. Set τ≡

n+d+1 ∈ (1, ∞). n

24

Consequently from (52), we obtain 1  p ∞ X p p  |λj | χ3Qj (x)M aj (x) Mf (x) . j=1



+

∞ X j=1

|λj

|p ℓ(Q

j

1

p

)p(n+d+1)

ℓ(Qj )p(n+d+1) + |x − c(Qj )|p(n+d+1)



1   1 p p ∞ ∞ X X p p p pτ    |λj | χ3Qj (x)M aj (x) . |λj | M χ3Qj (x) + . j=1

j=1

Thus, by the quasi-triangle inequality, kMf kMp,φ

 1

p ∞

X

.  (|λj |χ3Qj M aj )p 

j=1



X ∞

 p =  (|λj |χ3Qj M aj )

j=1

 1

p ∞

X

|λj |p (M χ3Qj )pτ  + 

j=1

Mp,φ Mp,φ 

 1 1 p

∞ pτ

 X 

p pτ    + |λ | (M χ )

 j 3Qj 

j=1

p

M1,φ

Mpτ,φτ

τ

  . 

Note that (49) and (48) allows us to use Theorem 1.1 and Corollary 2.20, respectively. Thus, we obtain

 1

p ∞

X

 p (|λj |χQj ) kMf kMp,φ . .

j=1

Mp,φ

Thus, we obtain the desired result.

3.2

Proof of Theorem 1.5

We define the topology on S(Rn ) with the norm {ρN }N ∈N which is given by the following formula: X ρN (ϕ) ≡ sup (1 + |x|)N |∂ α ϕ(x)|. |α|≤N

x∈Rn

We define FN ≡ {ϕ ∈ S(Rn ) : ρN (ϕ) ≤ 1}. Definition 3.2. The grand maximal operator Mf is defined by Mf (x) ≡ sup{|t−n ϕ(t−1 ·) ∗ f (x)| : t > 0, ϕ ∈ FN } for all f ∈ S ′ (Rn ) and x ∈ Rn . 25

(53)

The following proposition can be proved similar to [22, Section 3]. ∞ (Rn ) we denote the set of all compactly We invoke the following lemma. By Ccomp n supported smooth functions in R . We refer to [40] for the proof.

Lemma 3.3. Let f ∈ S ′ (Rn ), d ∈ {0, 1, 2, · · · } and j ∈ Z. Then there exist collections ∞ (Rn ), which are all indexed by a of cubes {Qj,k }k∈Kj and functions {ηj,k }k∈Kj ⊂ Ccomp set Kj for every j, and a decomposition X f = gj + bj , bj = bj,k , k∈Kj

such that: (0) gj , bj , bj,k ∈ S ′ (Rn ). (i) Define Oj ≡ {y ∈ Rn : Mf (y) > 2j } and consider its Whitney decomposition. Then the cubes {Qj,k }k∈Kj have the bounded intersection property, and [ Oj = Qj,k . (54) k∈Kj

(ii) Consider the partition of unity {ηj,k }k∈Kj with respect to {Qj,k }k∈Kj . Then each function ηj,k is supported in Qj,k and X ηj,k = χ{y∈Rn : Mf (y)>2j } , 0 ≤ ηj,k ≤ 1. k∈Kj

(iii) The distribution gj satisfies the inequality: Mgj (x) . Mf (x)χOj c (x) + 2j

X

k∈Kj

ℓj,k n+d+1 (ℓj,k + |x − xj,k |)n+d+1

(55)

for all x ∈ Rn . (iv) Each distribution bj,k is given by bj,k = (f − cj,k )ηj,k with a certain polynomial cj,k ∈ Pd (Rn ) satisfying hf − cj,k , ηj,k · qi = 0 for all q ∈ Pd (Rn ), and Mbj,k (x) . Mf (x)χQj,k (x) + 2j ·

ℓj,k n+d+1 χ n (x) |x − xj,k |n+d+1 R \Qj,k

(56)

for all x ∈ Rn . In the above, xj,k and ℓj,k denote the center and the side-length of Qj,k , respectively, and the implicit constants are dependent only on n. If we assume f ∈ L1loc (Rn ) in addition, then we have kgj kL∞ . 2−j . (57) 26

Lemma 3.4. Let ϕ ∈ S(Rn ) and θ ∈ (0, 1). Keep to the same notation as Lemma 3.3. Then we have ! ∞ 1 κ X

κ − nθ 2 p Mf · χOj Lp ([−2l ,2l ]n ) |hbj , ϕi| ≤ Cϕ,θ (58) l=1

and

∞ 1 X

nθ κ 2− p 2j χOj Lp ([−2l ,2l ]n )

|hgj , ϕi| ≤ Cϕ,θ

l=1

!κ

+Cϕ,θ kMf ·χOjc kLp ([−1,1]n ) , (59)

where the constants Cϕ,θ in (58) and (59) depend on ϕ and θ but not on j or k. Proof. For some large constant M ≡ Mϕ , we have ψx ≡ M −1 ϕ(x − ·) ∈ FN for all x ∈ [−1, 1]n , so that |hbj , ϕi| = |bj ∗ ψx (z)|z=x ≤ M

inf

x∈[−1,1]n

Mbj (x).

Thus, we have |hbj , ϕi| .

inf

x∈[−1,1]n

Mbj (x) .

inf

x∈[−1,1]n

X

Mbj,k (x).

k∈Kj

Observe also that M χQ (x) &

|Q| |Q| ≥ χ n (x) n |Q| + |x − xQ | |x − xQ |n R \Q

(x ∈ Rn ),

if Q is a cube centered at xQ . It follows from (56) that ! ℓj,k n+d+1 χ n (x) Mbj,k (x) . Mf (x)χQj,k (x) + 2 · |x − xj,k |n+d+1 R \Qj,k k∈Kj k∈Kj X n+d+1 (60) M χQj,k (x) n . . Mf (x)χOj (x) + 2j X

X

j

k∈Kj

n+d+1 . From (60), we deduce n

X

κ j

kMbj kLp ([−1,1]n ) . Mf · χOj + 2 (M χQj,k )

p

k∈Kj L ([−1,1]n )

j X

κ

. Mf · χOj Lp ([−1,1]n ) + 2 (M χQj,k )

k∈Kj

We abbreviate κ ≡

. Lp ([−1,1]n )

By using (2.26), we obtain

kMbj kLp ([−1,1]n )

. Mf · χOj Lp ([−1,1]n ) +

∞ X l=1

27

− nθ p

2

1

j

κ

2 χO p j L ([−2l ,2l ]n )

!κ

.

So, the estimate for the first term is valid. In the same way we can prove (59). Indeed, by using the Fefferman-Stein inequality for A1 -weighted Lebesgue spaces [2], we obtain kMgj kLp ([−1,1]n )

j · ℓ n+d+1

X 2 j,k

. Mf · χOj c Lp ([−1,1]n ) +

n+d+1 (ℓ + |x − x |) j,k j,k

p

k∈Kj L ([−1,1]n )

X

n+d+1 j

n c . Mf · χOj Lp ([−1,1]n ) + 2 (M χQj,k )

p

k∈Kj L ([−1,1]n ) ! ∞ 1 κ X

nθ κ − j . Mf · χO c p 2 p 2 χO p . n + l l n j

j

L ([−1,1] )

L ([−2 ,2 ] )

l=1

Thus, (59) is proved.

The key observation is the following. Lemma 3.5. Assume (49). In the notation of Lemma 3.3, in the topology of S ′ (Rn ), we have gj → 0 as j → −∞ and bj → 0 as j → ∞. In particular, f=

∞ X

(gj+1 − gj )

j=−∞

in the topology of S ′ (Rn ). Proof. Let us show that bj → 0 as j → ∞ in S ′ (Rn ). Once this is proved, then we have f = limj→∞ gj in S ′ (Rn ). Let us choose a test function ϕ ∈ S(Rn ). Then we have |hbj , ϕi| .

inf

x∈[−1,1]n

Mbj (x) . kMbj kLp ([−1,1]n ) ,

where the implicit constant does depend on ϕ. Assume (49) and choose θ > 0 so that τ < θ < 1. Note that

kf kHMp,φ ≥ Mf · χOj Lp ([−2l ,2l ]n ) → 0 (j → ∞)

and that

∞ 1 X

nθ κ 2− p Mf · χOj Lp ([−2l ,2l ]n ) l=1

!κ

≤

∞ X

ϕ(2l )2

l=1

= C0 kf kHMp,φ .

28

n(1−θ) p

kf kHMp,φ

1 !κ κ

Hence it follows from (58) that hbj , ϕi → 0 as j → ∞. Likewise by using (59), we obtain 1 !κ ∞

X κ

− nθ j |hgj , ϕi| . . 2 p 2 · χOj + Mf · χ(Oj )c p l l n L ([−2 ,2 ] )

l=1

Hence, gj → 0 as j → −∞ by the Lebesgue convergence theorem. Consequently, it follows that j X (gl+1 − gl ) f = lim gj = lim j→∞

in

j,k→∞

l=−k

S ′ (Rn ).

We prove Theorem 1.5 when f ∈ L1loc (Rn ). Proof of Theorem 1.5 when f ∈ L1loc (Rn ). For each j ∈ Z, consider the level set Oj ≡ {x ∈ Rn : Mf (x) > 2j }.

(61)

Then it follows immediately from the definition that Oj+1 ⊂ Oj .

(62)

If we invoke Lemma 3.3, then f can be decomposed; X f = gj + bj , bj = bj,k , bj,k = (f − cj,k )ηj,k , k

where each bj,k is supported in a cube Qj,k as is described in Lemma 3.3. We know that f=

∞ X

(gj+1 − gj ),

(63)

j=−∞

with the sum converging in the sense of distributions from Lemma 3.5. Here, going through the same argument as the one in [40, pp. 108–109], we have an expression; X X f= Aj,k , gj+1 − gj = Aj,k (j ∈ Z) (64) j,k

k

in the sense of distributions, where each Aj,k , supported in Qj,k , satisfies the pointwise estimate |Aj,k (x)| ≤ C0 2j for some universal constant C0 and the moment condition Z Rn

Aj,k (x)q(x) dx = 0 for every q ∈ Pd (Rn ). With these observations in mind, let us

set

Aj,k , κj,k ≡ C0 2j . C0 2j Then we automatically obtain that each aj,k satisfies Z xα aj,k (x) dx = 0 (|α| ≤ L) |aj,k | ≤ χQj,k , aj,k ≡

Rn

29

and that f =

X

κj,k aj,k in the topology of HMp,φ (Rn ), once we prove the estimate

j,k

of coefficients. Rearrange {aj,k } and so on to obtain {aj } and so on. To establish (9) we need to estimate

 1/v

∞

X

|λj χQj |v  α ≡ 

j=−∞

.

Mp,φ

Since {(κj,k ; Qj,k )}j,k = {(λj ; Qj )}j as a set, we have

 1/v

∞

X

X



α= |κj,k χQj,k |v 

j=−∞ k∈Kj

.

Mp,φ

If we insert the definition of κj , then we have



 1/v 1/v

∞ ∞

X

X



 X jv X

j v  α = C0 2 |2 χQj,k | = C χ

0 Qj,k

k∈Kj

j=−∞ k∈Kj

j=−∞

Mp,φ

.

Mp,φ

Observe that (54) together with the bounded overlapping property yields X χOj (x) ≤ χQj,k (x) . χOj (x) (x ∈ Rn ). k∈Kj

Thus, we have

 1/v

∞

X

v

2j χOj  α . 

j=−∞

.

Mp,φ

Recall that Oj ⊃ Oj+1 for each j ∈ Z. Consequently we have v v   ∞ ∞ ∞ X X X v 2j χOj \Oj+1 (x) 2j χOj (x) ∼  2j χOj (x) ∼  j=−∞

(x ∈ Rn ).

j=−∞

j=−∞

Thus, we obtain

X

∞ j

α. 2 χOj \Oj+1

j=−∞

.

Mp,φ

It follows from the definition of Oj that we have 2j < Mf (x) for all x ∈ Oj . Hence, we have

X

∞

α. χOj \Oj+1 Mf = kMf kMp,φ = kf kHMp,φ .

j=−∞

Mp,φ

This is the desired result.

30

Proof of Theorem 1.5 for general cases. According to (55), gj is a locally integrable function and it satisfies kgj kHMp,φ . kf kHMp,φ . Therefore, applying the above paragraph, we see that each gj has a decomposition; there exist a collection {Ql,j }∞ l=1 of ∞ n ∞ ∞ cubes, {al,j }l=1 ⊂ L (R ), and {λl,j }l=1 ⊂ [0, ∞) such that gj =

∞ X

λl,j al,j

(65)

l=1

unconditionally in S ′ (Rn ), that |al,j | ≤ χQl,j , that Z aQ,j (x)xα dx = 0 Rn

for all |α| ≤ L and that

!1/v

∞ X

v

(λj,l χQj,l )

l=1

≤ Cv kf kHMp,φ .

(66)

Mp,φ

We may assume that each Qj,l is realized as 3Q for some dyadic cube Q. Since v ≤ 1, by using av + bv ≥ (a + b)v for a, b ≥ 0 and taking into account the case when Qj,l = Qj ′ ,l′ for some (j.l) 6= (j ′ , l′ ), we have a decomposition there exist a collection {QQ,j }Q∈D of cubes, {aQ,j }Q∈D ⊂ L∞ (Rn ), and {λQ,j }Q∈D ⊂ [0, ∞) such that X gj = λQ,j aQ,j (67) Q∈D

in S ′ (Rn ), that |aQ,j | ≤ χ3Q , that

Z

Rn

(68)

aQ,j (x)xα dx = 0

for all |α| ≤ L and that

 1/v

X



v (λQ,l χQ )

Q∈D

≤ Cv kf kHMp,φ .

(69)

Mp,φ

∞ n Fix Q ∈ D. Since {aQ,j }∞ j=1 is a bounded sequence in L (R ) from (68), and {λQ,j }∞ j=1 ⊂ [0, ∞) is a bounded sequence in R from (69), we can choose subsequences ∞ ∞ ∞ {aQ,jk }∞ k=1 and {λQ,jk }k=1 ⊂ [0, ∞) so that {aQ,jk }k=1 and {λQ,jk }k=1 ⊂ [0, ∞) are convergent to aQ and λQ respectively, where the convergence of {aQ,jk }∞ k=1 takes place ∞ n in the weak-* topology of L (R ).

Let us set g≡

X

Q∈D

31

λQ aQ .

(70)

Then according to Theorem 1.3, we have g ∈ HMp,φ (Rn ). By the Fatou lemma, we can conclude the proof once we show that f = g.

(71)

To this end, we take a test function ϕ ∈ S(Rn ). If we insert (65) to gJ and use (58), we obtain X hf, ϕi = lim hgJ , ϕi = lim λQ,J haQ,J , ϕi. J→∞

J→∞

If we can change the order of lim and J→∞

X

Q∈D

in the most right-hand side of the above

Q∈D

formula, we have X X X hf, ϕi = lim λQ,J haQ,J , ϕi = lim λQ,J haQ,J , ϕi = λQ haQ , ϕi = hg, ϕi, J→∞

Q∈D

Q∈D

J→∞

Q∈D

showing f = X g. Thus, we are left with the task of justifying the change of the order of lim and . Let ϕ† ∈ Cc∞ (Rn ) satisfy χB(1) ≤ ϕ† ≤ χB(2) . Since ϕ ∈ S(Rn ), by J→∞

Q∈D

decomposing ϕ = ϕϕ† (R−1 ·) + ϕ(1 − ϕ† (R−1 ·)), and using the fact that HMp,φ (Rn ) (defined via the grand maximal operator) is continuously embedded in S ′ (Rn ) as well as Theorem 1.3, we see that the contribution of the function ϕ(1 − ϕ† (R−1 ·)) can be made as small as we wish. In fact, X |λQ,J haQ,J , ϕ(1 − ϕ† (R−1 ·))i| = O(R−1 ), Q∈D

where the implicit constant do not depend on J. This implies that we can and do assume that ϕ is supported in a compact set K. Suppose that K is contained in Q(2N ) for some N > 0. Let us set X |λQ,J haQ,J , ϕi| I ≡ sup J

Q∈D,Q∩K6=∅,ℓ(Q)≤2−A

X

II ≡ sup J

|λQ,J haQ,J − aQ , ϕi|

Q∈D,Q∩K6=∅,2−A2A X

. φ(2A )kf kHMp,φ .

Mp,φ

In view of (73) and (74), we see that I and III contribute little to the sum (72). With this in mind we use the weak-* convergence to II to see (71). Finally, we state a corollary to conclude this section. Corollary 3.6. If φ ∈ Gp satisfies (49), then HM1,φ (Rn ) is embedded into M1,φ (Rn ). Proof. Under the notation of Theorem 1.5, we have ∞ X

λj |aj | ≤

∞ X

λj χ Q j .

j=1

j=1

(9) with v = 1 guarantees that the right-hand side belongs to M1,φ (Rn ). Hence f (x) = ∞ X λj aj (x) converges for almost all x ∈ Rn . Observe also j=1

Z

Rn



|κ(x)| 

∞ X j=1



λj |aj (x)| dx

. sup (1 + |x|)2n/p+1 |κ(x)| x∈Rn

2n/p+1

. sup (1 + |x|) x∈Rn

|κ(x)|

Z

Rn

∞ X



(1 + |x|)−2n−1  −2n−1

(1 + j)

Z

∞ X j=1



λj |aj (x)| dx

∞ X

λj |aj (x)| dx

|x|≤j j=1

j=1

X ∞

−2n/p−1 2n/p+1 λj χQj φ(j)(1 + j) . sup (1 + |x|) |κ(x)|

x∈Rn

j=1 j=1 M1,φ

X

∞ 2n/p+1

. sup (1 + |x|) |κ(x)| λj χ Q j .

x∈Rn

j=1 ∞ X

M1,φ

Thus, f is represented by an L1loc (Rn )-functions and satisfies f (x) =

∞ X j=1

almost all x ∈ Rn . 33

λj aj (x) for

3.3

Applications to the boundedness of the singular integral operators

Going through the same argument as [22, Theorem 5.5] and [23, Theorem 5.5], we can prove the following theorem; Theorem 3.7. Let φ satisfy (49). Let k ∈ S(Rn ). Write Am ≡ sup |x|n+m |∇m k(x)| x∈Rn

(m ∈ N ∪ {0}).

Define a convolution operator T by T f (x) ≡ k ∗ f (x)

(f ∈ S ′ (Rn )).

Then, T , restricted to HMp,φ (Rn ), is an HMp,φ (Rn )-bounded operator and the norm depends only on kFkkL∞ and a finite number of collections A1 , A2 , . . . , AN with N depending only on φ. Once Theorem 3.7 is proved, we can obtain the Littlewood-Paley decomposition in the same way as [22, Theorem 5.7] and [23, Theorem 5.10]. Theorem 3.8. Let 0 0

j=−∞

for ξ ∈

Rn

\ {0}. Then the following norm is an equivalent norm of HMp,φ (Rn ):

 1/2

∞

X



2 kf kE˙ 0 ≡ |ϕj (D)f | , f ∈ S ′ (Rn ). (75)

p,φ,2

j=−∞

Mp,φ

Once we obtain Theorem 3.8, we can prove the wavelet decomposition and the atomic decomposition as in [26, 30]. Finally, we point out a mistake in our earlier paper [17]. Remark 3.9. The function Aj,k in [17, p. 162] is not in L∞ (Rn ) unless f ∈ L1loc (Rn ). Thus, the proof of [17, Theorem 1.3] is valid only of f ∈ HMpq (Rn ) ∩ L1 (Rn ), where HMpq (Rn ) = HMq,φ (Rn ) with φ(t) = t−n/p . The gap will be closed by the technique described above.

4

Applications to the Olsen inequality

This is a bilinear estimate of Iα , which is nowadays called the Olsen inequality [24]. Recall that we define the fractional integral operator Iα with 0 < α < n by; Z f (y) Iα f (x) = dy n−α Rn |x − y| 34

for all suitable functions f on Rn . Olsen’s inequality is the inequality of the form kg · Iα f kZ . kf kX kgkY , where X, Y, Z are suitable quasi-Banash spaces. There is a vast amount of literatures on Olsen inequalities; see [5, 34, 32, 33, 36, 37, 38, 41, 44] for theoretical aspects and [9, 10, 11] for applications to PDEs. Here we will prove the following theorem. Theorem 4.1. Let 0 1, then kIα f kMq,λ . kf kHMp,λ for all f ∈ HMp,λ . Proof. Argue as we did in [31] by using Theorem 3.8. Theorem 4.2. Let 0 0 such that kg · Iα f kHMp,λ . kgkM1,n−α · kf kHMp,λ for all f ∈ HMp,λ . Proof. We argue as we did in [17, Theorem 1.7].

5

Acknowledgement

The research of V. Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2013-9(15)46/10/1 and by the grant of Presidium Azerbaijan National Academy of Science 2015. A. Akbulut was partially supported by the grant of Ahi Evran University Scientific Research Projects (PYO.FEN.4003.13.004) and (PYO.FEN.4003/2.13.006). This paper is written during the stay of Y. Sawano in Ahi Evran University and Beijing Normal University. Y. Sawano is thankful to Ahi Evran University and Beijing Normal University for this support of the stay there. The authors are indepted to Mr. S. Nakamura for his hint to improve (48). 35

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