Anal.Math.Phys. https://doi.org/10.1007/s13324-018-0208-x

On the boundedness of a certain class of maximal functions on product spaces and extrapolation Mohammed Al-Dolat1 · Mohammed Ali1 · Imad Jaradat1 · Khaldoun Al-Zoubi1

Received: 29 November 2016 / Revised: 28 April 2017 / Accepted: 3 January 2018 © Springer International Publishing AG, part of Springer Nature 2018

Abstract In this paper, we establish L p estimates for certain class of maximal functions on product domains with rough kernels in L q (Sn−1 × Sm−1 ) (n, m ≥ 2). These estimates are used in an extrapolation argument to obtain some new improved and extended results on maximal functions. Keywords L p boundedness · Maximal functions · Rough kernels Mathematics Subject Classification 47A05 · 47A10 · 47A12

1 Introduction Let n, m ≥ 2, and let S N −1 (N = n or m) be the unit sphere in R N equipped with the normalized Lebesgue surface measure dσ = dσ (·). Also, let x  = x/|x| for x ∈ Rn \{0}, y  = y/|y| for y ∈ Rm \{0}. Let p  denote to the exponent conjugate to p.

B

Mohammed Al-Dolat [email protected] Mohammed Ali [email protected] Imad Jaradat [email protected] Khaldoun Al-Zoubi [email protected]

1

Department of Mathematics and Statistics, Jordan University of Science and Technology, P.O.Box 3030, Irbid 22110, Jordan

M. Al-Dolat et al.

Let K ,h (x, y) = (x  , y  )|x|−n |y|−m h(|x|, |y|), where h is a measurable function on R+ × R+ and  is a homogeneous function of degree zero on Rn × Rm with  ∈ L 1 (Sn−1 × Sm−1 ) satisfying the cancellation conditions:  Sn−1

(x  , .)dσ (x  ) =

 Sm−1

(., y  )dσ (y  ) = 0.

(1.1)

Let P : Rn → R, Q : Rm → R be real valued polynomials, for f ∈ S(Rn × Rm ). Define the maximal function M,P,Q by        i P(u) i Q(v)  M,P,Q ( f )(x, y) = sup  e e f (x − u, y − v)K ,h (u, v)dudv  , h∈U   n m R ×R

(1.2) where U is the class of all h ∈ L 2 (R+ × R+ , drr ds ) with h dr ds ≤ 1. + + 2 L (R ×R , r s ) s When P(u) = 0 = Q(v), then the operator M,P,Q is the classical maximal integral operator on product domains, denoted by M,c , which was studied by many mathematicians. For example, Ding in [9] gave the L 2 boundedness of M,c provided that  ∈ L(log L)2 (Sn−1 × Sm−1 ). This result was improved in [6] in which the author showed that M,c is bounded on L p for all 2 ≤ p < ∞ when the function  ∈ L(log L)(Sn−1 × Sm−1 ). He also proved the optimality of the condition in the sense that the space L(log L)(Sn−1 × Sm−1 ) cannot be replaced by L(log L)1−ε (Sn−1 × Sm−1 ) for any 0 < ε < 1. For more information about the importance and the recent advances on the study of such operators, the readers are refereed to [5], and for the corresponding results in the one parameter cases, see for instance [3,7,8,10] and references therein. The primary focus of this paper is to establish the L p estimates of M,P,Q for p ≥ 2 with rough kernels in L q (Sn−1 × Sm−1 ) for some q > 1; then apply the extrapolation argument used in [2] to obtain new improved results. Our main result is formulated as follows: Theorem 1.1 Let  ∈ L q (Sn−1 ×Sm−1 ) for some q > 1 be a homogeneous function of degree zero on Rn ×Rm satisfies the condition (1.1) with 1 ≤ 1. Let P : Rn → R, Q : Rm → R be two real valued polynomials of degree d1 , d2 , respectively. Then     M,P,Q ( f ) ≤ C p Aq 1 + wq ()  f  p p for all p ≥ 2, where Aq = (

1  2 /q )2 1/q  2 −1

(1.3)

and ωq () = log(e + q ).

The power of our theorem lies in using its conclusion and an extrapolation argument (see [2]) to obtain improved results. In particular, Theorem 1.1 and extrapolation lead to the following theorem. Theorem 1.2 Suppose that  is a homogeneous function of degree zero on Rn × Rm satisfies the condition (1.1). Let f ∈ L p (Rn × Rm ) with p ≥ 2.

On the boundedness of a certain class of maximal... (0,0)

(i) If  belong to the block space Bq

(Sn−1 × Sm−1 ) for some q > 1, then

    M,P,Q f  ≤ C p Aq  f  p 1 +  (0,0) n−1 m−1 ; p B (S ×S ) q

(ii) If  ∈ L(log L)(Sn−1 × Sm−1 ), then     M,P,Q f  ≤ C p Aq  f  p 1 +  L(logL)(Sn−1 ×Sm−1 ) . p Throughout this paper, we let the letters C and C p denote bounded positive constants that may vary at each occurrence but independent of the essential variables. Also we let dσn,m (u  , v  ; z  , w  ) = dσn (u  )dσn (z  )dσm (v  )dσm (w  ), dσn,m (u  , v  ) = dσn (u  )dσm (v  ).

2 Preliminary results In this section, we give some auxiliary lemmas used in the sequel. In order to prove Theorem 1.1, we need to the following lemmas. Lemma 2.1 Suppose that  ∈ L 1 (Sn−1 × Sm−1 ) satisfying the condition (1.1). Let M : Rn × Rm → R be given by  M f (x, y) = sup

i, j∈Z

  | f (x − u, y − v)| |u|−n |v|−m (u  , v  ) dudv,

i, j

(2.1)

where i, j = {(u, v) ∈ Rn × Rm : 2i ≤ |u| < 2i+1 , 2 j ≤ |v| < 2 j+1 }. Then for p ∈ (1, ∞), there exists a constant C p > 0 such that M f  p ≤ C p 1  f  p

(2.2)

for every f ∈ L p (Rn × Rm ). Proof It is trivial to obtain  M f (x, y) ≤ C

Sn−1 ×Sm−1

|(u, v)| sup

t,s∈R+

1 ts



t



s

| f (x − ru, y − lv)| dr dldudv.

t/2 s/2

Hence, Minkowski’s inequality for integrals yields that M f  p ≤ C p 1 Mv Mu (| f |) p , t where Mu f (x) = supt∈R 1t 0 f (x − tu)dt is the Hardy–Littlewood maximal function of f in the direction u, Mu f (x, y) = Mu f (·, y)(x), Mv f (x, y) =

M. Al-Dolat et al.

Mv f (x, ·)(y), and denotes the composition of operators. Therefore, since the Hardy-Littlewood maximal function is bounded on L p (Rn ) for p > 1, then by the latter inequality we immediately get (2.2). 

(Sn−1 × Sm−1 ) for some q > 1 with 1 ≤ 1. Let P(x) = Lemma 2.2 Let  ∈ L q

α β |α|≤d1 aα x , Q(y) = |β|≤d2 bβ y be two real-valued polynomials of degree d1 > d 1 1, d2 > 1, respectively such that |x| and |y|d2 are not one of their terms. For k, s ∈ Z, let Ak, : [1, ωq ()] × P(Sn−1 ) × R → C, Bs, : [1, ωq ()] × Q(Sm−1 ) × R → C and Jk,s, : Rn × Rm → R be given by 



  −i P(2−(k+1)ωq () r1 u  )+2−(k+1)ωq () tr1 Ak, r1 , P(u  ), t = e ,  −(s+1)ω () −(s+1)ω ()  q q   −i Q(2 r2 v )+2 zr2 Bs, r2 , Q(v  ), z = e , and Jk,s, (ξ, η)  2                  dr1 dr2 = (u , v )Ak, (r1 , P(u ), ξ.u )Bs, (r2 , Q(v ), η.v )dσn,m (u , v ) ,    r1 r2  ,q Sn−1 ×Sm−1

where  

 ,q = (r1 , r2 ) ∈ R2 : (r1 , r2 ) ∈ 1, 22ωq () × 1, 22ωq () . Then ⎛

sup (ξ,η)∈Rn ×Rm

Jk,s, (ξ, η) ≤

⎞−ε2/q     bβ ⎠ ⎝

⎞−ε1/q  ⎛

 (k+s+2)/4q  ⎝ |aα |⎠ C p wq2 ()2 |α|=d1

|β|=d2

for some positive constants ε1 , ε2 < 1, where C p is a constant that depend on the degree of the polynomials P and Q but not on the function , q, the diminsion and the coefficients of the polynomials P and Q. Proof Since  is integrable on Sn−1 × Sm−1 , then it is easy to get Jk,s, (ξ, η) ≤ Cwq2 ().

(2.3)

On the other hand, 

Jk,s, (ξ, η)

q 

≤

2q  q



      R(u  , z  )q T (v  , w  )q dσn,m (u  , v  ; z  , w  ),

(Sn−1 ×Sm−1 )2

(2.4)

On the boundedness of a certain class of maximal...

where 





22ωq ()

R(u , z ) =

Ak, (r1 , P(u  ), ξ.u  )Ak, (r1 , P(z  ), ξ.z  )

1

dr1 , r1

and 

T (v  , w  ) =

22ωq ()

Bk, (r2 , Q(v  ), η.v  )Bk, (r2 , Q(w  ), η.w  )

1

dr2 . r2

It is clear that P(γ r1 u  ) + γ (ξ.u  )r1 − P(γ r1 z  ) + γ (ξ.z  )r1 ⎧ ⎫ ⎨  ⎬  α α = γ d1 r1d1 aα u  − z + γ ξ.(u  − z  )r1 + Hk (r1 , u  , z  , ξ ), ⎩ ⎭ |α|=d1

with

d d1 Hk dr d1

|α|=d1

= 0, γ = 2−(k+1)wq () . Thus, by Van der-Corput Lemma, we get −1/d1     R(u  , z  ) ≤ γ d1 (P(u  ) − P(z  )) ,

(2.5)

which when combined with the trivial estimates |R(u  , z  )| ≤ Cwq (), gives  −1/4q  d1     R(u  , z  ) ≤ C γ d1 (P(u  ) − P(z  )) (wq ())1−(1/4q ) .

(2.6)

Similarly, we derive  −1/4q  d2    T (v  , w  ) ≤ C 2−(s+1)d2 wq () (Q(v  ) − Q(w  )) (wq ())1−(1/4q ) . (2.7) Hence, by (2.4), (2.6), (2.7) and [11, (3.11)], we reach ⎛ Jk,s, (ξ, η) ≤ C2

(k+s+2)wq ()/4q 

q2 ⎝



⎞−1/4d2 q     bβ ⎠ ⎝ .

⎞−1/4d1 q  ⎛ |aα |⎠

|α|=d1

|β|=d2

(2.8) Hence, by (2.3) and (2.8), we get ⎛

Jk,s, (ξ, η) ≤

for some positive constants ε1 , ε2 < 1.

⎞−ε2/q     bβ ⎠ ⎝

⎞−ε1/q  ⎛

 (k+s+2)/4q  2 |aα |⎠ C2 wq ()⎝ |α|=d1

|β|=d2



M. Al-Dolat et al.

The following lemma can be obtained by applying similar arguments used in [4]. Lemma 2.3 Let q > 1, and let  ∈ L q (Sn−1 × Sm−1 ) be a homogenous function of degree zero on Rn × Rm that satisfies the condition (1.1) with 1 ≤ 1. Then there exists a constant C p (independent of the function  and the index q) such that   M,c ( f ) ≤ C p wq ()Aq  f  p p

(2.9)

holds for all p ≥ 2. Proof Let {wk,ωq () }k∈Z be a smooth partition of unity on (0, ∞) adapted to the intervals Ik,ωq () = [2−(k+1)ωq () , 2−(k−1)ωq () ]. More precisely, we require the following: wk,ωq () ∈ C ∞ , 0 ≤ wk,ωq () ≤ 1,



wk,ωq () (t) = 1,

(*)

k∈Z

supp wk,ωq ()

   C  di w k,ωq () (t)  i  ⊆ Ik,ωq () , and  ≤ i ,   dt i t

where Ci is independent of ωq (). Define the multiplier operators G k,s on Rn ×Rm by n m ˆ  (G k,s f )(ξ, η) = wk,ωq () (|ξ |)ws,ω

q () (|η|) f (ξ, η). Then for any f ∈ S(R × R ) G k+i,s+ j ( f )(x, y). Therefore, and k, s ∈ Z, we have f (x, y) = i, j∈Z

M,c ( f )(x, y) ≤



E i, j ( f )(x, y),

(2.10)

i, j∈Z

where E i, j ( f )(x, y) ⎛  ⎜ =⎝ s,k∈Z ([1,ωq

 2 ⎞ 21      dr dr 1 2⎟   (u  , v  )G k+i,s+ j ( f )(ax,k,r1 ,u  , b y,s,r2 ,v  )dσn,m (u  , v  )  ⎠ ,   r1 r2   ()])2 Sn−1 ×Sm−1

ax,k,r1 ,u  = x − 2kωq () r1 u  , b y,s,r2 ,v  = y − 2sωq () r2 v  .

Let us first compute the L 2 -norm of E i, j . The same arguments used in [1] and [8] lead to    E i, j ( f ) ≤ C2−δ(|i|+| j|)/q  wq () f 2 , (2.11) 2 and

   E i, j ( f ) ≤ Cwq () f  p , p

(2.12)

for some 0 < δ < 1 for all 2 < p < ∞. Thus by interpolation between (2.11) and (2.12), we have    E i, j ( f ) ≤ C2−ε(|i|+| j|)/q  wq () f  p (2.13) p

On the boundedness of a certain class of maximal...

for some 0 < ε < 1 and for all 2 ≤ p < ∞. Consequently, the inequality (2.9) follows immediately by (2.10) and (2.13). This completes the proof of Lemma 2.3.



3 Proof of Theorem 1.1 We prove Theorem 1.1 by using the mathematical induction on the degrees of the polynomials P and Q. If d1 = d2 = 0, then it is easy to see M,P,Q ( f )(x, y) ≤ CM,c ( f )(x, y).

(3.1)

Thus, by Lemma 2.3 we get that   M,P,Q ( f ) ≤ C p Aq wq () f  p p   ≤ C p Aq 1 + wq ()  f  p

(3.2)

for any p ≥ 2. The inequality (1.5) is directly obtained, for the cases d1 = 0 or d2 = 0, by following the same approach used in [4]. Let us prove (1.5) for the cases d1 = 1 or d2 = 1, so we can write P(x) = a.x for some a ∈ Rn and Q(y) = b.y for some b ∈ Rm . Hence by the latter inequality, we have   M,P,Q ( f ) ≤ C p Aq wq ()W  p p   ≤ C p Aq 1 + wq ()  f  p ,

(3.3)

where W (x, y) = ei P(x) ei Q(y) f (x, y). Next, assume that (1.5) holds for all polynomials G and

Q such that degr ee(G) < d1 +1 and degr ee(Q) = d2 where d1 , d2 > 1. Let P(x) = |α|≤d1 +1 aα x α be polynomials of degree d1 + 1 and let ⎛∞∞ ⎞1 2   2 dr1 dr2    ⎝ ⎠ E P,Q,, f (r1 , r2 , x, y) , M,P,Q ( f )(x, y) = r1 r2 0

0

where E P,Q,, f (r1 , r2 , x, y)    = ei P(r1 u ) ei Q(r2 v ) (u  , v  ) f (ax,0,r1 ,u  , b y,0,r2 ,v  )dσn,m (u  , v  ). Sn−1 ×Sm−1

We may assume that P does not contain |x|d+1 as one of its terms, and Q is not contains |y|d+1 as one of its terms. We may also assume by dilation invariance that  |α|=d1 +1

|aα | =

 |β|=d2 +1

  bβ  = 1.

(3.4)

M. Al-Dolat et al.

Let {ψk }k∈Z and {ws }s∈Z be collection of C ∞ functions defined on (0, ∞) that satisfy the conditions in (*). Set 0 

η∞ (u) =

ψk (u), η0 (u) =

k=−∞

∞ 

ψk (u),

k=1

and 0 

λ∞ (v) =

ws (v), λ0 (v) =

s=−∞

∞ 

ws (v).

s=1

Then η∞ (u) + η0 (u) = λ∞ (v) + λ0 (v) = 1;

supp (η∞ (u)), supp (λ∞ (v)) ⊆ [2− log(e+q ) , ∞);

and supp (η0 (u)), supp (λ0 (v)) ⊆ (0, 1]. N,P,Q,∞ ( f ),

Define the operators N,P,Q,∞,0 ( f ) by ⎛

∞

⎜ N,P,Q,∞ ( f )(x, y) = ⎝

∞

2−ωq () 2−ωq ()

N,P,Q,0 ( f ),

N,P,Q,0,∞ ( f )

and

⎞1 2 2 dr1 dr2 ⎟  λ∞ (r2 )η∞ (r1 )E P,Q,, f (r1 , r2 , x, y) ⎠ ; r1 r2

⎛ 1 1 ⎞ 21   2 dr1 dr2  λ0 (r2 )η0 (r1 )E P,Q,, f (r1 , r2 , x, y) ⎠ ; N,P,Q,0 ( f )(x, y) = ⎝ r1 r2 0

⎛ 1 ⎜ N,P,Q,0,∞ ( f )(x, y) = ⎝

0

∞

0 2−wq ()

⎞1 2 2 dr2 dr1 ⎟  λ∞ (r2 )η0 (r1 )E P,Q,, f (r1 , r2 , x, y) ⎠ ; r2 r1

and ⎛ ⎜ N,P,Q,∞,0 ( f )(x, y) = ⎝

∞ 1

2−wq () 0

⎞1 2

  λ0 (r2 )η∞ (r1 )E P,Q,, f (r1 , r2 , x, y)2 dr1 dr2 ⎟ ⎠ . r1 r2

Thus, it is clear to see M,P,Q ( f )(x, y) ≤ N,P,Q,0 ( f )(x, y) + N,P,Q,∞ ( f )(x, y) +N,P,Q,0,∞ ( f )(x, y) + N,P,Q,∞,0 ( f )(x, y).

On the boundedness of a certain class of maximal...

Let us first estimate the L p -norm of N,P,Q,0 ; let g(x) =



(1)

|α|≤d1

aα x α , with

deg(g) = l1 and 0 ≤ l1 ≤ d1 . Define the operators N,P,Q,0 ( f )(x, y) and (2) N,g,Q,0 ( f )(x, y) by

⎞ 21 ⎛ 1 1      (1)  E P,Q,, f − E g,Q,, f (x, y, r1 , r2 )2 dr1 dr2 ⎠ N,P,Q,0 ( f )(x, y) = ⎝ r1 r2 0

0

and ⎛ (2) N,g,Q,0 ( f )(x, y) = ⎝

1 1 0

0

⎞ 21

   E g,Q,, f (x, y, r1 , r2 )2 dr1 dr2 ⎠ . r1 r2

By Minkowski’s inequality, it is easy to get (1)

(2)

N,P,Q,0 ( f )(x, y) ≤ N,P,Q,0 ( f )(x, y) + N,g,Q,0 ( f )(x, y).

(3.5)

Since the deg(g) ≤ d1 , then by the induction assumption, for all p ≥ 2 we have      (2)  N,g,Q,0 ( f )(x, y) ≤ C p Aq 1 + wq ()  f  p . p





(3.6)



Let I (r1 , r2 ) = ei Q(r2 v ) (ei P(r1 u ) − eig(r1 u ) ). Then it is clear that        |I (r1 , r2 )| = ei ( Q(r2 v )+P(r1 u )−g(r1 u )) − 1 ≤ r2d2 + r1d1 +1 . Hence, by Cauchy–Schwarz inequality we obtain that (1) N,P,Q,0 ( f )(x, y) ⎞1 ⎛ 2 1 1    2   dr1 dr2 ⎟ ⎜    2     ≤⎝ (u , v ) |I (r1 , r2 )| f (ax,0,r1 ,u  , b y,0,r2 ,v  ) dσn,m (u , v ) ⎠ r1 r2

⎛ ≤⎝

0 Sn−1 ×Sm−1

0

−1  

2d2 j + 2(d1 +1)i

2

i, j=−∞ 2i+1 2j+1   

× 2i

2 j Sn−1 ×Sm−1 1

⎞1 2     dr dr (u  , v  )  f (ax ,0,r ,u  , b y,0,r ,v  )2 dσn,m (u  , v  ) 1 2 ⎟ ⎠ 2 1 r1 r2

≤ C(M (| f |2 )) 2 (x, y).

M. Al-Dolat et al.

Therefore, by Lemma 2.1 and the fact that 1 ≤ 1, we reach    (1)  N,P,Q,0  ≤ C p  f  p .

(3.7)

p

Consequently, by (3.5)–(3.7) we deduce   N,P,Q,0 ( f ) ≤ C p Aq (1 + wq ()) f  p p

(3.8)

for all p ≥ 2. Next, we estimate the L p norm of N,P,Q,∞ ; by generalized Minkowski’s inequality we have that N,P,Q,∞ ( f )(x, y) ⎞1 ⎛ 2 ∞ ∞ 2 dr1 dr2 ⎟  ⎜ λ∞ (r2 )η∞ (r1 )E ,P,Q, f (x, y, r1 , r2 ) =⎝ ⎠ r1 r2 2−ωq () 2−ωq ()

⎞1 ⎛ 2  2  ∞ ∞   0  dr1 dr2 ⎟  ⎜  =⎝ ψk (r2 )ws (r1 )E ,P,Q, f (x, y, r1 , r2 ) ⎠   r1 r2 s,k=−∞ 0

0

0 

≤

N,P,Q,∞,s,k ( f )(x, y),

(3.9)

s,k=−∞

where N,P,Q,∞,s,k ( f )(x, y)  −ωq ()(s−1)  −ωq ()(k−1)  2 2 2 dr1 dr2   E ,P,Q, f (x, y, r1 , r2 ) = . r1 r2 2−ωq ()(s+1) 2−ωq ()(k+1) On one hand, for the case p = 2, the L 2 norm of N,P,Q,∞,s,k can be estimated as follows: using Plancherel’s theorem, Fubini’s theorem and Lemma 2.2, we deduce   N,P,Q,∞,s,k ( f )2 2    2  ˆ =  f (ξ, η) Jk,s, (ξ, η)dξ dη Rm

Rn

≤ C p2

(k+s+2)/4q  2 wq () f 22 .

On the other hand, for the case p > 2, we choose h ∈ L ( h( p/2) = 1 such that

p/2) (Rn

(3.10)

× Rm ) with

On the boundedness of a certain class of maximal...

  N,P,Q,∞,s,k ( f )2 p ⎛   ⎜ = ⎝

     Ak, (r1 , P(u  ), 0)Bs, (r2 , Q(v  ), 0)   Rn ×Rm ([1,22ωq () ])2 Sn−1 ×Sm−1 2 ⎞   dr dr 1 2⎟  × f (ax,1,r1 ,u  , b y,1,r2 ,v  )dσn,m (u  , v  ) ⎠  r1 r2 

× |h(x, y)| dσn,m (x, y)   | f (t, z)|2 ≤ Rn ×Rm



   (u  , v  ) h(t + 2−ωq () r1 u  , z + 2−ωq () r2 v  )

([1,22ωq () ])2 Sn−1 ×Sm−1

dr1 dr2 dσn,m (t; z) r r 1 2   ˜ z) ≤ Cwq2 ()  f 2p M,P,Q h(t,  p , ( /2) × dσn,m (u  , v  )

˜ z) = h(−t, −z). Hence, where h(t,   N,P,Q,∞,s,k ( f ) ≤ Cwq () f  p , p

(3.11)

which when combined with (3.10) gives   N,P,Q,∞,s,k ( f ) ≤ Cwq ()2(k+s+2)κ/2q   f  p , p

(3.12)

for some 0 < κ < 1 and for all p ≥ 2. Therefore, by Minkowski’s inequality, (3.9) and (3.12), we get that for all p ≥ 2,   N,P,Q,∞ ( f ) ≤ C p wq ()Aq  f  p p   ≤ C p Aq 1 + wq ()  f  p ,

(3.13)

By following the same above approaches, we get that

and

    N,P,Q,0,∞  ≤ C p Aq 1 + wq ()  f  p p

(3.14)

    N,P,Q,∞,0  ≤ C p Aq 1 + wq ()  f  p p

(3.15)

for all p ≥ 2. Consequently, by Minkowski’s inequality and (3.8) together with (3.13)(3.15); we prove Theorem 1.1 for all polynomials P and G such that deg(P) = d1 + 1 and deg(Q) = d2 . In the same manner, we show that (1.3) is true for all polynomials P and G, with deg(P) = d1 and deg(G) = d2 + 1. This completes the proof of Theorem 1.1.

M. Al-Dolat et al. Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest.

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