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Eroglu Boundary Value Problems 2013, 2013:70 http://www.boundaryvalueproblems.com/content/2013/1/70

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Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces Ahmet Eroglu* *

Correspondence: [email protected] Department of Mathematics, Nigde University, Nigde, Turkey

Abstract In this paper, it is proved that both oscillatory integral operators and fractional oscillatory integral operators are bounded on generalized Morrey spaces Mp,ϕ . The corresponding commutators generated by BMO functions are also considered. MSC: Primary 42B20; 42B25; 42B35 Keywords: generalized Morrey space; oscillatory integral; commutator; BMO spaces

1 Introduction and main results The classical Morrey spaces, were introduced by Morrey [] in , have been studied intensively by various authors and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces. Morrey spaces Mp,λ (Rn ) are defined as the set of all functions f ∈ Lp (Rn ) such that λ

f Mp,λ ≡ f Mp,λ (Rn ) = sup r– p f Lp (B(x,r)) < ∞. x,r>

Under this definition, Mp,λ (Rn ) becomes a Banach space; for λ = , it coincides with Lp (Rn ) and for λ =  with L∞ (Rn ). n We also denote by W Mp,λ the weak Morrey space of all functions f ∈ WLloc p (R ) for which λ

f W Mp,λ ≡ f W Mp,λ (Rn ) = sup r– p f WLp (B(x,r)) < ∞, x∈Rn ,r>

where WLp denotes the weak Lp -space. Definition  Let ϕ(x, r) be a positive measurable function on Rn × (, ∞) and  ≤ p < ∞. We denote by Mp,ϕ ≡ Mp,ϕ (Rn ) the generalized Morrey space, the space of all functions n f ∈ Lloc p (R ) with finite quasinorm  –  f Mp,ϕ ≡ f Mp,ϕ (Rn ) = sup ϕ(x, r)– B(x, r) p f Lp (B(x,r)) . x∈Rn ,r>

© 2013 Eroglu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Also, by WMp,ϕ ≡ WMp,ϕ (Rn ), we denote the weak generalized Morrey space of all funcn tions f ∈ WLloc p (R ) for which  –  f WMp,ϕ ≡ f WMp,ϕ (Rn ) = sup ϕ(x, r)– B(x, r) p f WLp (B(x,r)) < ∞. x∈Rn ,r>

According to this definition, we recover the spaces Mp,λ and WMp,λ under the choice ϕ(x, r) = r

λ–n p

 Mp,ϕ 

: λ–n

ϕ(x,r)=r p

 WMp,ϕ 

= Mp,λ ,

λ–n

ϕ(x,r)=r p

= WMp,λ .

The theory of boundedness of classical operators of the real analysis, such as the maximal operator, fractional maximal operator, Riesz potential and the singular integral operan tors etc., from one weighted Lebesgue space to another one is well studied. Let f ∈ Lloc  (R ). The fractional maximal operator Mα and the Riesz potential Iα are defined by  –+ α Mα f (x) = supB(x, t) n  Iα f (x) =

t>

Rn



  f (y) dy,

 ≤ α < n,

B(x,t)

f (y) dy , |x – y|n–α

 < α < n.

If α = , then M ≡ M is the Hardy-Littlewood maximal operator. In [], Chiarenza and Frasca obtained the boundedness of M on Mp,λ (Rn ). In [], Adams established the boundedness of Iα on Mp,λ (Rn ). Here and subsequently, C will denote a positive constant which may vary from line to line but will remain independent of the relevant quantities. The Calderón-Zygmund singular integral operator is defined by  Tf (x) = p.v.

 Rn

K(x – y)f (y) dy,

(.)

where K is a Calderón-Zygmund kernel (CZK). We say a kernel K ∈ C  (Rn \ {}) is a CZK if it satisfies   K(x) ≤ C , |x|n   ∇K(x) ≤ C |x|n+

(.) (.)

and  K(x) dx = ,

(.)

a



t 

dr < ∞, ess inf

r

f 

dt

p

Lp (B(x,t – n ))



 p –  sup ϕ x, r– n rf 

p

Lp (B(x,r– n ))

x∈Rn ,r>

dt

= f Mp,ϕ

if p ∈ (, ∞), and  Tf WM,ϕ  sup ϕ (x, r)– x∈Rn ,r>

∞



r–n

≈ sup ϕ (x, r)– x∈Rn ,r>

=



  – sup ϕ x, r– n

x∈Rn ,r>

f L (B(x,t)) t ––n dt

r

f 



r

f 

if p = .



dt

L (B(x,t – n ))



  –  sup ϕ x, r– n rf  x∈Rn ,r>



L (B(x,t – n ))



L (B(x,r– n ))

dt

= f M,ϕ 

Proof of Theorem . The proof of Theorem . follows from Theorem F and the following observation:     Sα f (x) ≤ Iα |f | (x).



4 Commutators of fractional oscillatory integral operators in the spaces Mp,ϕ (Rn ) Let T be a Calderón-Zygmund singular integral operator and b ∈ BMO(Rn ). A well known result of Coifman, Rochberg and Weiss [] states that the commutator operator [b, T]f = T(bf ) – bTf is bounded on Lp (Rn ) for  < p < ∞. The commutator of Calderón-Zygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order (see, for example, [, , ]). First, we recall the definition of the space BMO(Rn ).

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n Definition  Suppose that f ∈ Lloc  (R ), let

 f ∗ = sup n |B(x, r)| x∈R ,r>



  f (y) – fB(x,r)  dy < ∞,

B(x,r)

where fB(x,r) =

 |B(x, r)|

 f (y) dy. B(x,r)

Define    n

BMO Rn = f ∈ Lloc  R : f ∗ < ∞ . If one regards two functions whose difference is a constant as one, then space BMO(Rn ) is a Banach space with respect to norm  · ∗ . Remark  () The John-Nirenberg inequality: there are constants C , C > , such that for all f ∈ BMO(Rn ) and β >    

  x ∈ B : f (x) – fB  > β  ≤ C |B|e–C β/f ∗ ,

∀B ⊂ Rn .

() The John-Nirenberg inequality implies that  f ∗ ≈ sup

x∈Rn ,r>

 |B(x, r)|

 p   f (y) – fB(x,r) p dy



(.)

B(x,r)

for  < p < ∞. () Let f ∈ BMO(Rn ). Then there is a constant C >  such that |fB(x,r) – fB(x,t) | ≤ Cf ∗ ln

t r

for  < r < t,

(.)

where C is independent of f , x, r and t. Lemma . Let  ≤ p < ∞, b ∈ BMO(Rn ), K is a SCZK and the Calderón-Zygmund singular integral operator  S is of type (L (Rn ), L (Rn )). Then for  < p < ∞ and any polynomial P(x, y) the inequality n

Sb f Lp (B(x ,r))  b∗ r p



∞

r

n

f Lp (B(x ,t)) t –– p dt

n holds for any ball B(x , r) and for all f ∈ Lloc p (R ).

Proof Let p ∈ (, ∞). For arbitrary x ∈ Rn , set B = B(x , r) for the ball centered at x and radius r, B = B(x , r). We represent f as f = f + f ,

f (y) = f (y)χB (y), f (y) = f (y)χ(B) (y)

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and have Sb f Lp (B) ≤ Sb f Lp (B) + Sb f Lp (B) . It is known that (see [], see also [, , ]), if K is a SCZK and the operator  S is of type (L (Rn ), L (Rn )), then for  < p < ∞ and any polynomial P(x, y) the commutator operator Sb is bounded on Lp (Rn ). Since f ∈ Lp (Rn ), Sf ∈ Lp (Rn ) and boundedness of Sb in Lp (Rn ) (see []) it follows that Sb f Lp (B) ≤ Sb f Lp (Rn ) ≤ Cb∗ f Lp (Rn ) = Cb∗ f Lp (B) , where constant C >  is independent of f . For x ∈ B, we have   Sb f (x) 

 Rn

 ≈

 |b(y) – b(x)|   dy f (y) |x – y|n

 (B)

 |b(y) – b(x)|  f (y) dy. n |x – y|

Then p  p  |b(y) – b(x)|   dy dx f (y)  (B) |x – y|n B   p  p  |b(y) – bB |    f (y) dy dx  (B) |x – y|n B   p  p  |b(x) – bB |   + f (y) dy dx  (B) |x – y|n B  

Sb f Lp (B) 

= I + I . Let us estimate I . 

 |b(y) – bB |  f (y) dy n  (B) |x – y|      ∞ n b(y) – bB f (y) ≈ rp n

I ≈ r p

 (B)

n



r

|x –y|

r

   b(y) – bB f (y) dy dt t n+ r≤|x –y|≤t

r

   b(y) – bB f (y) dy dt . t n+ B(x ,t)

≈ rp

n p

∞

dt dy t n+



∞

Applying Hölder’s inequality and by (.), (.), we get 

∞

   b(y) – bB(x ,t) f (y) dy dt  t n+ r B(x ,t)   ∞     n bB(x ,r) – bB(x ,t)  f (y) dy dt +rp   t n+ B(x ,t) r n

I  r p

Eroglu Boundary Value Problems 2013, 2013:70 http://www.boundaryvalueproblems.com/content/2013/1/70

r

n p



∞ 

    p dt b(y) – bB(x ,t) p dy f Lp (B(x ,t)) n+  t B(x ,t)

r n

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+rp

∞

 n bB(x ,r) – bB(x ,t) f L (B(x ,t)) t –– p dt p   

r

 b∗ r

n p



∞

 n t  + ln f Lp (B(x ,t)) t –– p dt. r

r

In order to estimate I note that 

  b(x) – bB p dx

I =

 p   (B)

B

|f (y)| dy. |x – y|n

By (.), we get I  b∗ r

n p

  (B)

|f (y)| dy. |x – y|n

Thus, by (.) I  b∗ r

n p



∞

r

n

f Lp (B(x ,t)) t –– p dt.

Summing up I and I , for all p ∈ (, ∞) we get n

∞



Sb f Lp (B)  b∗ r p

 + ln r

 n t f Lp (B(x ,t)) t –– p dt. r

(.)

Finally, n

Sb f Lp (B)  b∗ f Lp (B) + b∗ r p



∞

 + ln r

 n t f Lp (B(x ,t)) t –– p dt, r

and statement of Lemma . follows by (.).



Proof of Theorem . The statement of Theorem . follows by Lemma . and Theorem G in the same manner as in the proof of Theorem G.  Proof of Theorem . The proof of Theorem . follows from the Theorem . in [] and the following observation:     Sα,b f (x) ≤ Iα,b |f | (x).

Competing interests The author declares that they have no competing interests. Received: 25 July 2012 Accepted: 17 March 2013 Published: 2 April 2013



Eroglu Boundary Value Problems 2013, 2013:70 http://www.boundaryvalueproblems.com/content/2013/1/70

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doi:10.1186/1687-2770-2013-70 Cite this article as: Eroglu: Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces. Boundary Value Problems 2013 2013:70.

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