Potential operators in modified Morrey spaces defined on Carleson curves

Potential operators in modified Morrey spaces defined on Carleson curves

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Original Article

Potential operators in modified Morrey spaces defined on Carleson curves I.B. Dadashova a , C. Aykol b , ∗, Z. Cakir b , A. Serbetci b a Baku State University, Baku, Azerbaijan b Ankara University, Department of Mathematics, Ankara, Turkey

Received 4 August 2017; received in revised form 26 September 2017; accepted 27 September 2017 Available online xxxx

Abstract In this paper we study the potential operator IΓα , 0 < 1 in the modified Morrey space ˜ L p,λ (Γ ) and the spaces B M O(Γ ) defined on Carleson curves Γ . We prove that for 1 < p < (1 − λ)/α the potential operator IΓα is bounded from the modified Morrey space ˜ L p,λ (Γ ) to ˜ L q,λ (Γ ) if and in the case of infinite curve only if α ≤ 1/ p − 1/q ≤ α/(1 − λ), and from the spaces ˜ L 1,λ (Γ ) to α . Furthermore, for the limiting case (1 − λ)/α ≤ p ≤ 1/α ˜ W L q,λ (Γ ) if and in the case of infinite curve only if α ≤ 1 − q1 ≤ 1−λ α is bounded from ˜ we show that if Γ is an infinite Carleson curve, then the modified potential operator ˜ IΓ L p,λ (Γ ) to B M O(Γ ), α ˜ and if Γ is a finite Carleson curve, then the operator IΓ is bounded from L p,λ (Γ ) to B M O(Γ ). c 2017 Published by Elsevier B.V. on behalf of Ivane Javakhishvili Tbilisi State University. This is an open access article under ⃝ the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Carleson curves; Modified Morrey spaces; B M O spaces; Potential operators; Sobolev–Morrey inequality

1. Introduction Morrey spaces were introduced by C.B. Morrey [1] in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variations (see [2,3]). Later, Morrey spaces found important applications to Navier–Stokes and Schr¨odinger equations, elliptic problems with discontinuous coefficients and potential theory (see [4–8]). Let Γ = {t ∈ C : t = t(s), 0 ≤ s ≤ l ≤ ∞} be a rectifiable Jordan curve in the complex plane with arc-length measure ν(t) = s, here l = νΓ = lengths of Γ . ∗ Corresponding author.

E-mail addresses: [email protected] (I.B. Dadashova), [email protected] (C. Aykol), [email protected] (Z. Cakir), [email protected] (A. Serbetci). Peer review under responsibility of Journal Transactions of A. Razmadze Mathematical Institute. https://doi.org/10.1016/j.trmi.2017.09.004 c 2017 Published by Elsevier B.V. on behalf of Ivane Javakhishvili Tbilisi State University. This is an open access article under the 2346-8092/⃝ CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: I.B. Dadashova, et al., Potential operators in modified Morrey spaces defined on Carleson curves, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.09.004.

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We denote Γ (t, r ) = Γ ∩ B(t, r ), t ∈ Γ , r > 0, where B(t, r ) = {z ∈ C : |z − t| < r }. A rectifiable Jordan curve Γ is called a Carleson curve (regular curve) if the condition νΓ (t, r ) ≤ c0r holds for all t ∈ Γ and r > 0, where the constant c0 > 0 does not depend on t and r . Definition 1. Let 1 ≤ p < ∞, 0 ≤ λ ≤ 1, [r ]1 = min{1, r }. We denote by L p,λ (Γ ) the Morrey space, and by ˜ L p,λ (Γ ) the modified Morrey space, the set of locally integrable functions f on Γ with the finite norms λ

∥ f ∥ L p,λ (Γ ) = sup sup r − p ∥ f ∥ L p (Γ (t,r )) , t∈Γ r >0

−λ

∥ f ∥˜L p,λ (Γ ) = sup sup[r ]1 p ∥ f ∥ L p (Γ (t,r )) t∈Γ r >0

respectively. Note that L p,0 (Γ ) = ˜ L p,0 (Γ ) = L p (Γ ), ˜ L p,λ (Γ )⊂≻ L p,λ (Γ ) ∩ L p (Γ )

max{∥ f ∥ L p,λ (Γ ) , ∥ f ∥ L p (Γ ) } ≤ ∥ f ∥˜L p,λ (Γ )

and

(1)

and if λ < 0 or λ > 1, then L p,λ (Γ ) = ˜ L p,λ (Γ ) = Θ, where Θ is the set of all functions equivalent to 0 on Γ . We denote by W L p,λ (Γ ) the weak Morrey space, and by W ˜ L p,λ (Γ ) the modified Morrey space, as the set of locally integrable functions f on Γ with finite norms ( )1/ p ∫ ∥ f ∥W L p,λ (Γ ) = sup β sup r −λ dν(τ ) , β>0

r >0, t∈Γ

∥ f ∥W ˜L p,λ (Γ ) = sup β sup β>0

{τ ∈Γ (t,r ): | f (τ )|>β}

(

r >0, t∈Γ

[r ]−λ 1

)1/ p dν(τ ) .

∫ {τ ∈Γ (t,r ): | f (τ )|>β}

Note that W L p (Γ ) = W L p,0 (Γ ), L p,λ (Γ ) ⊂ W L p,λ (Γ ) and ∥ f ∥W L p,λ (Γ ) ≤ ∥ f ∥ L p,λ (Γ ) . Definition 2. The space of functions with bounded mean oscillation B M O(Γ ) is defined as the set of locally integrable functions f with finite norm ∫ ∥ f ∥ B M O(Γ ) = sup (νΓ (t, r ))−1 | f (τ ) − f Γ (t,r ) |dν(τ ) < ∞, r >0, t∈Γ

Γ (t,r )

where −1



f Γ (t,r ) = (νΓ (t, r ))

f (τ )dν(τ ). Γ (t,r )

Maximal operators and potential operators in various spaces defined on Carleson curves have been widely studied by many authors (see, for example [9–16]). In Morrey spaces defined on quasimetric measure spaces, in particular Morrey spaces L p,λ (Γ ) defined on Carleson curves N. Samko [16] studied the boundedness of the maximal operator MΓ defined by ∫ −1 MΓ f (t) = sup(νΓ (t, r )) | f (τ )|dν(τ ) t>0

Γ (t,r )

and proved the following: Please cite this article in press as: I.B. Dadashova, et al., Potential operators in modified Morrey spaces defined on Carleson curves, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.09.004.

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Theorem A. Let Γ be a Carleson curve, 1 ≤ p < ∞, 0 < α < 1 and 0 ≤ λ < 1. (1) If 1 < p < ∞, then MΓ is bounded from L p,λ (Γ ) to L p,λ (Γ ). (2) If p = 1, then MΓ is bounded from L 1,λ (Γ ) to W L 1,λ (Γ ). In Morrey spaces defined on quasimetric measure spaces, in particular Morrey spaces L p,λ (Γ ) defined on Carleson curves V. Kokilashvili and A. Meskhi [14] studied the boundedness of the potential operator IΓα defined by ∫ f (τ )dν(τ ) IΓα f (t) = , 0<α<1 1−α Γ |t − τ | and proved the following Spanne type result: Theorem B. Let Γ be a Carleson curve, 1 < p < q < ∞, 0 < α < 1, 0 < λ1 < qp , the operator IΓα is bounded from the spaces L p,λ1 (Γ ) to L q,λ2 (Γ ).

λ1 p

=

λ2 q

and

1 p



1 q

= α. Then

In [17] Adams type Sobolev–Morrey inequalities for the potential operators in Morrey space defined on Carleson curves were proved. Theorem C. Let Γ be a Carleson curve, 0 < α < 1, 0 ≤ λ < 1 − α and 1 ≤ p < 1−λ . α 1 1 α (1) If 1 < p < 1−λ , then the condition − = is sufficient and in the case of infinite curve also necessary α p q 1−λ for the boundedness of IΓα from L p,λ (Γ ) to L q,λ (Γ ). α (2) If p = 1, then the condition 1 − q1 = 1−λ is sufficient and in the case of infinite curve also necessary for the α boundedness of IΓ from L 1,λ (Γ ) to W L q,λ (Γ ). In [18], in the case of infinite curve Γ for the modified potential operator defined by ∫ ( ) α ˜ I f (t) = |t − τ |α−1 − |t0 − τ |α−1 χΓ \Γ (t0 ,1) (τ ) f (τ )dν(τ ), t0 ∈ Γ Γ

the following theorem was proved. Theorem D. Let Γ be a Carleson curve, and let 0 < α < 1, 0 ≤ λ ≤ 1 − α. Then (1) If Γ is infinite, then the modified potential operator ˜ IΓα is bounded from L 1−λ ,λ (Γ ) to B M O(Γ ). α α (2) If Γ is finite, then the potential operator IΓ is bounded from L 1−λ ,λ (Γ ) to B M O(Γ ). α (3) MΓα is bounded from L 1−λ ,λ (Γ ) to L ∞ (Γ ). α

The main purpose of this paper is to establish the boundedness of the potential operator IΓα in the modified Morrey spaces ˜ L p,λ (Γ ) defined on Carleson curves Γ . We prove Sobolev–Morrey inequalities for the operator IΓα . In particular, we get the analog of the theorem given by Guliyev and others in [19] regarding the inequality for the potential operators in modified Morrey spaces defined on Carleson curves. We emphasize that in the infinite case of Γ the derived conditions are necessary and sufficient for appropriate inequalities. The paper is organized as follows. In Section 2, we give some lemmas that we will use in the sequel. In Section 3, we prove ˜ L p,λ (Γ )-boundedness of the maximal operator MΓ . In Section 4 we give main results. Our first main result is: “For 1 < p < (1 − λ)/α the operator IΓα is bounded from the modified Morrey space ˜ L p,λ (Γ ) to ˜ L q,λ (Γ ) if and ˜ in the case of infinite curve only if α ≤ 1/ p − 1/q ≤ α/(1 − λ), and from the spaces L 1,λ (Γ ) to W ˜ L q,λ (Γ ) if and in α the case of infinite curve only if α ≤ 1 − q1 ≤ 1−λ ”. Next, we give a corollary on the boundedness of the fractional α maximal operator MΓ in the modified Morrey spaces on Carleson curves. Our second main result is: “for the limiting case (1 − λ)/α ≤ p ≤ 1/α we show that if Γ is an infinite Carleson curve, then the modified potential operator ˜ IΓα is bounded from ˜ L p,λ (Γ ) to B M O(Γ ), and if Γ is a finite Carleson curve, then the operator IΓα is bounded from ˜ L p,λ (Γ ) to B M O(Γ )”. 2. Some auxiliary lemmas To prove our theorems we need the following lemmas. Please cite this article in press as: I.B. Dadashova, et al., Potential operators in modified Morrey spaces defined on Carleson curves, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.09.004.

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Lemma 1. Let Γ be an infinite Carleson curve, 1 ≤ p < ∞ and 0 ≤ λ ≤ 1. For r > 0 and t = t(s) ∈ Γ define r t = t(r s), fr (t) =: f (r t). Then ∥ fr ∥ L p,λ (Γ ) = r −

1−λ p

λ

1

∥ f ∥ L p,λ (Γ ) and ∥ fr ∥˜L p,λ (Γ ) = r − p [r ]1p ∥ f ∥˜L p,λ (Γ ) .

Proof. ∥ fr ∥ L p,λ (Γ ) = r

− 1p

( sup

t∈Γ , µ>0

µ

−λ



)1/ p

p

| f (τ )| dν(τ )

= r−

Γ (r t,r µ)

1−λ p

∥ f ∥ L p,λ (Γ )

and 1

∥ fr ∥˜L p,λ (Γ ) = r − p =r

− 1p

=r

− 1p

( sup

t∈Γ , µ>0

[µ]−λ 1



( [r µ] )λ/ p 1 sup µ>0 [µ]1 λ p

| f (τ )| p dν(τ )

)1/ p

Γ (r t,r µ)

( sup

t∈Γ , µ>0

[r ]1 ∥ f ∥˜L p,λ (Γ ) .

[r µ]−λ 1



| f (τ )| p dν(τ )

)1/ p

Γ (r t,r µ)



Lemma 2. Let 1 ≤ p < ∞, 0 ≤ λ ≤ 1. Then ˜ L p,λ (Γ ) = L p,λ (Γ ) ∩ L p (Γ ) and { } ∥ f ∥˜L p,λ (Γ ) = max ∥ f ∥ L p,λ (Γ ) , ∥ f ∥ L p (Γ ) . { } Proof. Let f ∈ ˜ L p,λ (Γ ). Then from (1) we have that f ∈ L p,λ (Γ ) ∩ L p (Γ ) and max ∥ f ∥ L p,λ (Γ ) , ∥ f ∥ L p (Γ ) ≤ ∥ f ∥˜L p,λ (Γ ) . Let now f ∈ L p,λ (Γ ) ∩ L p (Γ ). Then ( )1/ p ∫ p ∥ f ∥˜L p,λ (Γ ) = sup [r ]−λ | f (τ )| dν(τ ) 1 t∈Γ ,r >0 Γ (t,r ) { ( )1/ p (∫ )1/ p } ∫ −λ p p = max sup r | f (τ )| dν(τ ) , sup | f (τ )| dν(τ ) t∈Γ ,0
{

t∈Γ ,r >1

Γ (t,r )

Γ (t,r )

}

≤ max ∥ f ∥ L p,λ (Γ ) , ∥ f ∥ L p (Γ ) . ˜ Therefore, f ∈ ˜ L p,λ (Γ ) and the embedding L p,λ { (Γ ) ∩ L p (Γ )⊂≻ L p,λ (Γ } ) is valid. ˜ ∥ ∥ ∥ ∥ Thus L p,λ (Γ ) = L p,λ (Γ ) ∩ L p (Γ ) and max f L p,λ (Γ ) , f L p (Γ ) = ∥ f ∥˜L p,λ (Γ ) . □ The following statement can be proved analogously. Lemma 3. Let 1 ≤ p < ∞, 0 ≤ λ ≤ 1. Then W˜ L p,λ (Γ ) = W L p,λ (Γ ) ∩ W L p (Γ ) and } { ∥ f ∥W ˜L p,λ (Γ ) = max ∥ f ∥W L p,λ (Γ ) , ∥ f ∥W L p (Γ ) . Lemma 4 ([18]). Let Γ be a Carleson curve and 1 ≤ p < ∞. Then L p,1 (Γ ) = L ∞ (Γ ) and ∥ f ∥ L ∞ (Γ ) ≤ ∥ f ∥ L p,1 (Γ ) ≤ c01/ p ∥ f ∥ L ∞ (Γ ) . Please cite this article in press as: I.B. Dadashova, et al., Potential operators in modified Morrey spaces defined on Carleson curves, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.09.004.

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Lemma 5 ([18]). Let Γ be a Carleson curve, 1 ≤ p < ∞ and 0 ≤ λ < 1. If p =

1−λ , α

)



5

then



1/ p L p,λ (Γ ) ⊂ L 1,1−α (Γ ) and ∥ f ∥ L 1,1−α (Γ ) ≤ c0 ∥ f ∥ L p,λ (Γ ) ,

where 1/ p + 1/ p ′ = 1. Lemma 6. Let Γ be a Carleson curve, 0 < α < 1, 0 ≤ λ ≤ 1 − α. Then for

1−λ α

≤p≤

1 α



1/ p ˜ L p,λ (Γ )⊂≻ L 1,1−α (Γ ) and ∥ f ∥ L 1,1−α (Γ ) ≤ c0 ∥ f ∥˜L p,λ (Γ ) .

Proof. Let 0 < α < 1, 0 ≤ λ ≤ 1 − α, f ∈ ˜ L p,λ (Γ ) and ∫ ∥ f ∥ L 1,1−α (Γ ) = sup r α−1 | f (τ )|dν(τ ) t∈Γ , r >0

≤ c0 ≤ c0

1/ p ′

= c0 =

λ

sup r α− p [r ]1p

(

t∈Γ , r >0

1/ p ′

[r ]−λ 1

)1/ p | f (τ )| p dν(τ )

∫ Γ (t,r )

λ p

α− 1p

∥ f ∥˜L p,λ (Γ ) sup r [r ]1 r >0 { } 1−λ 1 ∥ f ∥˜L p,λ (Γ ) max sup r α− p , sup r α− p r >1

0
1/ p ′ c0

≤ p ≤ α1 . By the H¨older’s inequality we have

Γ (t,r ) 1

1/ p ′

1−λ α

∥ f ∥˜L p,λ (Γ ) .



For the 0 ≤ α < 1 we define the following fractional maximal functions on Γ )1/ p ( ∫ ( α ) −1+α α p 1/ p p MΓ , p f (t) ≡ MΓ | f | (t) = sup |Γ (t, r )| | f (τ )| dν(τ ) . r >0

In the case α = 0 we denote

MΓ0 , p

Γ (t,r )

f by MΓ , p f and in the case p = 1 we denote MΓα ,1 f by MΓα f .

Lemma 7. Let 1 ≤ p < ∞, 0 ≤ α < 1 and f ∈ L p,1−α (Γ ). Then MΓα , p f ∈ L ∞ (Γ ) and  α  M  Γ , p f L (Γ ) = ∥ f ∥ L p,1−α (Γ ) . ∞

Proof.  α M

Γ,p

 f

L ∞ (Γ )

( ∫ α−1 = sup r t∈Γ ,r >0

)1/ p | f (τ )| dν(τ ) = ∥ f ∥ L p,1−α (Γ ) . p

Γ (t,r )



Lemma 8. Let 1 ≤ p < ∞, 0 ≤ α < 1 and f ∈ ˜ L p,1−α (Γ ). Then MΓα , p f ∈ L ∞ (Γ ) and  α  M  L p,1−α (Γ ) . Γ , p f L (Γ ) ≤ ∥ f ∥˜ ∞

Proof.  α M

Γ,p

 f

L ∞ (Γ )

)1/ p | f (τ )| dν(τ )

( ∫ α−1 = sup r t∈Γ ,r >0

p

Γ (t,r )

( ) 1−α = sup r −1 [r ]1 p t∈Γ ,r >0

(

[r ]α−1 1



| f (τ )| p dν(τ )

)1/ p

Γ (t,r )

( ) 1−α ≤ ∥ f ∥˜L p,1−α sup r −1 [r ]1 p r >0

= ∥ f ∥˜L p,1−α (Γ ) .



In the case α = 0 from Lemma 7 for MΓ , p f the following property is valid. Please cite this article in press as: I.B. Dadashova, et al., Potential operators in modified Morrey spaces defined on Carleson curves, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.09.004.

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Corollary 1. Let 1 ≤ p < ∞ and f ∈ L ∞ (Γ ). Then MΓ , p f ∈ L ∞ (Γ ) and    MΓ , p f  = ∥ f ∥ L ∞ (Γ ) . L ∞ (Γ ) In the case p = 1 from Lemmas 5 and 7 for MΓα f the following property is valid. Corollary 2. Let 0 ≤ α < 1, 0 ≤ λ ≤ 1 − α and f ∈ L 1−λ ,λ (Γ ). Then MΓα f ∈ L ∞ (Γ ) and α

 α  1− α M f  = ∥ f ∥ L 1,1−α (Γ ) ≤ c0 1−λ ∥ f ∥ L 1−λ Γ L ∞ (Γ )

α ,λ

(Γ )

.

From Lemmas 6 and 8 for MΓα f the following property is valid. Corollary 3. Let 0 ≤ α < 1, 0 ≤ λ ≤ 1 − α and f ∈ ˜ L p,λ (Γ ). Then MΓα f ∈ L ∞ (Γ ) for  α  1/ p ′ M f  = ∥ f ∥ L 1,1−α (Γ ) ≤ c0 ∥ f ∥˜L p,λ (Γ ) . Γ L ∞ (Γ )

1−λ α

≤p≤

1 α

and

Lemma 9 ([15]). Let Γ be a Carleson curve and α > 0, then ∫ α |t − τ |α−1 dν(τ ) ≤ c2r α , c1 r ≤ Γ (t,r )

where positive constants c1 and c2 do not depend on t ∈ Γ and r > 0. Lemma 10 ([18]). Let t, τ, t0 ∈ C and 0 < α < 1. Then for 2|t − t0 | ≤ |t0 − τ | the following inequality is valid ⏐ ⏐ ⏐|t − τ |α−1 − |t0 − τ |α−1 ⏐ ≤ 21−α |t0 − τ |α−2 |t − t0 |. (2) ˜ p,λ (Γ )-boundedness of the maximal operator MΓ 3. L In this section we study the ˜ L p,λ (Γ )-boundedness of the maximal operator MΓ . Applying Theorem A, one obtains the following result. Theorem 1. Let Γ be a Carleson curve, 0 ≤ λ ≤ 1 and 1 ≤ p < ∞. (1) If 1 < p < ∞, then MΓ is bounded from ˜ L p,λ (Γ ) to ˜ L p,λ (Γ ). ˜ (2) If p = 1, then MΓ is bounded from L 1,λ (Γ ) to W ˜ L 1,λ (Γ ). Proof. It is obvious that (see Lemmas 2 and 3) { } ∥MΓ f ∥˜L p,λ (Γ ) = max ∥MΓ f ∥ L p,λ (Γ ) , ∥MΓ f ∥ L p (Γ ) for 1 < p < ∞ and { } ∥MΓ f ∥W ˜L 1,λ (Γ ) = max ∥MΓ f ∥W L 1,λ (Γ ) , ∥MΓ f ∥W L 1 (Γ ) for p = 1. Let 1 < p < ∞. By the boundedness of MΓ on L p (Γ ) (see, for example, [9,16]) and from Theorem A we get { } ∥MΓ f ∥˜L p,λ (Γ ) ≤ max C p , C p,λ ∥ f ∥˜L p,λ (Γ ) . Let p = 1. By the boundedness of MΓ from L 1 (Γ ) to W L 1 (Γ ) (see, for example, [9,16]) and from Theorem A we have { } ∥MΓ f ∥W ˜L 1,λ (Γ ) ≤ max C1 , C1,λ ∥ f ∥˜L 1,λ (Γ ) . □ 4. The Adams type Sobolev–Morrey inequalities for potential operator IΓα In this section we prove Adams type Sobolev–Morrey inequalities for the potential operators in the Modified Morrey spaces defined on Carleson curves. Please cite this article in press as: I.B. Dadashova, et al., Potential operators in modified Morrey spaces defined on Carleson curves, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.09.004.

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Theorem 2. Let Γ be a Carleson curve, 0 < α < 1, 0 ≤ λ < 1 − α and 1 ≤ p < 1−λ . α (1) If 1 < p < 1−λ , then the condition α ≤ 1/ p − 1/q ≤ α/(1 − λ) is sufficient and in the case of infinite curve α also necessary for the boundedness of IΓα from ˜ L p,λ (Γ ) to ˜ L q,λ (Γ ). (2) If p = 1, then the condition α ≤ 1 − 1/q ≤ α/(1 − λ) is sufficient and in the case of infinite curve also necessary for the boundedness of IΓα from ˜ L 1,λ (Γ ) to W ˜ L q,λ (Γ ). Proof. (1) Sufficiency. Let Γ be a Carleson curve, 0 < α < 1, 0 < λ < 1 − α, f ∈ L p,λ (Γ ) and 1 < p < we write (∫ ) ∫ α IΓ f (t) = + f (τ )|t − τ |α−1 dν(τ ) ≡ A1 (t, r ) + A2 (t, r ). Γ (t,r )

1−λ . α

Then

(3)

Γ \Γ (t,r )

For A1 (t, r ) we have ∫ |A1 (t, r )| ≤ | f (τ )||t − τ |α−1 dν(τ ) Γ (t,r )





∞ ∑ ( j=1 ∞ ∑

−j

2 r



)α−1

| f (τ )|dν(τ ) Γ (t,2− j+1 r )\Γ (t,2− j r )

(

2− j r

)α−1

νΓ (t, 2− j+1r ) MΓ f (t)

j=1

≤ 2c0r α MΓ f (t)

∞ ∑

2− jα .

j=1

Hence |A1 (t, r )| ≤ C1r α MΓ f (t) with C1 =

2c0 . 2α − 1

(4)

For A2 (t, r ) by the H¨older’s inequality we have (∫ )1/ p −β p |A2 (t, r )| ≤ |t − τ | | f (τ )| dν(τ ) Γ \Γ (t,r )

(∫

|t − τ |

×

(

β ′ p +α−1 p

)

)1/ p′ dν(τ ) = J1 · J2 .

Γ \Γ (t,r )

Let λ < β < 1 − αp. For J1 we get ∞ ∫ (∑ )1/ p J1 = | f (τ )| p |t − τ |−β dν(τ ) j=0

≤r

− βp

Γ (t,2 j+1 r )\Γ (t,2 j r )

∥ f ∥˜L p,λ (Γ )

∞ (∑

2−β j [2 j+1r ]λ1

)1/ p

j=0

β

= r − p ∥ f ∥˜L p,λ (Γ )

β

≈ r− p

⎧ 1 [log2 2r ⎪ ( ⎪ ∑] ⎪ ⎪ ⎪ 2λ r λ 2(λ−β) j + ⎪ ⎪ ⎨ j=0

∞ ∑

2−β j

)1/ p

, 0
1 ]+1 j=[log2 2r

⎪ ∞ ⎪ (∑ )1/ p ⎪ ⎪ ⎪ ⎪ 2−β j , ⎪ ⎩ j=0 ⎧( )1/ p 1 ⎪ ⎪ , 0
r≥

1 , 2

1 2

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)



⎧ λ−β ⎪ ⎪r p , 0 < r < 1 , ⎨ 2 ≈ ∥ f ∥˜L p,λ (Γ ) β ⎪ 1 − ⎪r p , ⎩ r≥ 2 λ

β

= [2r ]1p r − p ∥ f ∥˜L p,λ (Γ ) .

(5)

For J2 we obtain )1/ p′ (∫ ) ( β ′ p +α−1 p dν(τ ) J2 = |t − τ | Γ \Γ (t,r )

⎛ =⎝

∞ ∫ ∑ Γ (t,2 j+1 r )\Γ (t,2 j r )

j=1

⎛ ≤⎝

∞ ∑ (

(

(

2jr

)

β ′ p +α−1 p

|t − τ |

β ′ p +α−1 p

⎞1/ p′

)

dν(τ )⎠

⎞1/ p′

)

νΓ (t, 2 j+1r )⎠

j=1

⎛ ≲⎝

∞ ∑ (

(

2jr

)

β ′ p +α−1 p +1

)

⎞1/ p′ β

1

≲ r p +α− p .



(6)

j=1

Then from (5) and (6) we have λ

1

|A2 (t, r )| ≲ [r ]1p r α− p ∥ f ∥˜L p,λ (Γ ) .

(7)

Thus, from (4) and (7) we have λ ⏐ α ⏐ 1 ⏐ I f (t)⏐ ≲ r α MΓ f (t) + [r ] p r α− p ∥ f ∥˜L (Γ ) Γ 1 p,λ { } 1−λ α− 1p α ∥ f ∥˜L p,λ (Γ ) , r α MΓ f (t) + r α− p ∥ f ∥˜L p,λ (Γ ) . ≲ min r MΓ f (t) + r Minimizing with respect to r , at [ ] p/(1−λ) r = (MΓ f (t))−1 ∥ f ∥˜L p,λ (Γ ) and [ ]p r = (MΓ f (t))−1 ∥ f ∥˜L p,λ (Γ ) we have ⎧( ) pα ( )1− pα ⎫ ⎨ M f (t) 1− 1−λ ⎬ ⏐ α ⏐ MΓ f (t) Γ ⏐ I f (t)⏐ ≲ min ∥ f ∥˜L p,λ (Γ ) . , Γ ⎩ ∥ f ∥˜L p,λ (Γ ) ⎭ ∥ f ∥˜L p,λ (Γ ) Then ⏐ α ⏐ ⏐ I f (t)⏐ ≲ (MΓ f (t)) p/q ∥ f ∥1− p/q . Γ ˜ L (Γ ) p,λ

Hence, by Theorem C, we have ∫ ∫ ⏐ α ⏐ ⏐ I f (t)⏐q dν(τ ) ≲ ∥ f ∥q− p Γ ˜ L (Γ ) Γ (t,r ) ≲ [r ]λ1

(MΓ f (t)) Γ (t,r ) q [r ]λ1 ∥ f ∥˜L (Γ ) . p,λ

p,λ

p ∥ f ∥q− ∥ f ∥˜Lp ˜ L (Γ ) p,λ

p,λ (Γ )

=

p

dν(τ )

Therefore IΓα f ∈ ˜ L q,λ (Γ ) and  α   I f ˜ ≤ C6 ∥ f ∥˜L p,λ (Γ ) . Γ L (Γ ) q,λ

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9

Necessity. Let Γ be an infinite Carleson curve, 1 < p < 1−λ , and let IΓα be bounded from ˜ L p,λ (Γ ) to ˜ L q,λ (Γ ). α For r > 0 and t = t(s) ∈ Γ define r t = t(r s), fr (t) =: f (r t), [r ]1,+ = max{1, r }. Then ( )1/ p ∫ 1−λ − 1p −λ p ∥ fr ∥˜L p,λ (Γ ) = r sup [ν]1 | f (τ )| dν(τ ) = r − p ∥ f ∥˜L p,λ (Γ ) t∈Γ , ν>0

=r

− 1p

=r

− 1p

Γ (t,r ν)

[r ν]1 sup [ν]1 ν>0 (

)λ/ p sup

t∈Γ , ν>0

)1/ p | f (τ )| p dν(τ )

( ∫ [r ν]−λ 1

Γ (t,r ν)

λ p

[r ]1,+ ∥ f ∥˜L p,λ (Γ ) ,

and IΓα fr (t) = r −α IΓα f (r t). Thus we get  α   I f r ˜ Γ L

q,λ (Γ )

(

= r −α

sup

ν>0, t∈Γ

=r

−α− q1

=r

−α− q1

[ν]−λ 1



⏐ ⏐ α ⏐ I f (r t)⏐q dν(t) Γ

)1/q

Γ (t,ν)

) [r ν]1 λ/q sup [ν]1 ν>0 λ   q  I α f ˜ [r ]1,+ Γ L (

( sup

ν>0, t∈Γ

q,λ (Γ )

[r ν]−λ 1

∫ Γ (r t,r ν)

⏐ ⏐ α ⏐ I f (t)⏐q dν(t) Γ

)1/q

.

From the boundedness of IΓα 1

 α   I f ˜ Γ L

q,λ (Γ )

If

λ−λ

1

p q ∥ f ∥˜L p,λ (Γ ) . ≲ r α+ q − p [r ]1,+

< + α, then for all f ∈ L p,λ (Γ ), we obtain  α   I f ˜ = 0 as r → 0 Γ L

1 p

1 q

q,λ

which is impossible. Similarly, if 1p >  α   I f ˜ = 0 as r → ∞ Γ L (Γ )

1 q

+

α , 1−λ

then for all f ∈ ˜ L p,λ (Γ ), we obtain

q,λ

which is also impossible. Therefore α ≤ 1p − (2) Sufficiency. Let f ∈ ˜ L 1,λ (Γ ). We have ⏐ α ⏐ { } ν τ ∈ Γ (t, r ) : ⏐ IΓ f (τ )⏐ > 2β

1 q



α . 1−λ

≤ ν {τ ∈ Γ (t, r ) : |A1 (τ, r )| > β} + ν {τ ∈ Γ (t, r ) : |A2 (τ, r )| > β} . Taking into account inequality (4) and Theorem A we have ν {τ ∈ Γ (t, r ) : |A1 (τ, r )| > β} { ≤ ν τ ∈ Γ (t, r ) : MΓ f (τ ) > ≤

C7r α λ · r ∥ f ∥ L 1,λ (Γ ) , β

where C7 = C1 · C1,λ . For A2 (t, r ) we have ∫ |A2 (t, r )| ≤ ≤

β C1r α

}

| f (τ )||t − τ |α−1 dν(τ )

Γ \Γ (t,r ) ∞ ∑ ( j )α−1



2 r

j=0

| f (τ )|dν(τ ) Γ (t,2 j+1 r )\Γ (t,2 j r )

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∞ ∑ (

2jr

)



)α−1 [ j+1 ]λ 2 r 1 ∥ f ∥˜L 1,λ (Γ )

j=1

= r α−1 ∥ f ∥˜L 1,λ

⎧ 1 [log2 2r ⎪ ⎪ ∑] ⎪ ⎪ λ λ ⎪ 2(λ−1+α) j + 2 r ⎪ ⎪ ⎨ j=0

∞ ∑

2−(1−α) j , 0 < r <

1 ]+1 j=[log2 2r

⎪ ∞ ⎪ ∑ ⎪ ⎪ ⎪ ⎪ 2−(1−α) j , ⎪ ⎩ j=0 ⎧ 1 ⎪ ⎪ ⎨r λ + r 1−α , 0 < r < , 2 ≈ r α−1 ∥ f ∥˜L 1,λ ⎪ 1 ⎪ ⎩ 1, r≥ 2 ⎧ 1 ⎪ ⎪ ⎨r λ+α−1 , 0 < r < , 2 = [2r ]λ r α−1 ∥ f ∥ ≈ ∥ f ∥˜L 1,λ ˜ L 1,λ . 1 ⎪ 1 ⎪ ⎩ r α−1 , r≥ 2

r≥

1 , 2

1 2

Therefore if C [2r ]λ1 r α−1 ∥ f ∥˜L 1,λ (Γ ) = β, then |A2 (τ, r )| ≤ β and consequently, |{τ ∈ Γ (t, r ) : |A2 (τ, r )| > β}| = 0. Finally, { } 1 ν τ ∈ Γ (t, r ) : |IΓα f (τ )| > 2β ≲ [r ]λ1 r α ∥ f ∥˜L 1,λ (Γ ) β ( ) 1−λ ∥ f ∥˜L 1,λ (Γ ) 1−λ−α λ = [r ]1 , if 2r < 1 β and { } 1 ν τ ∈ Γ (t, r ) : |IΓα f (τ )| > 2β ≲ [r ]λ1 r α ∥ f ∥˜L 1,λ (Γ ) β ( ) 1 ∥ f ∥˜L 1,λ (Γ ) 1−α λ = [r ]1 , if 2r < 1. β Finally we have ⏐{ }⏐ ⏐ τ ∈ Γ (t, r ) : |I α f (τ )| > 2β ⏐ Γ ⎧ ⎫ 1−λ 1 ( ) 1−λ−α ( ) 1−α )q ⎬ ⎨( ∥ f ∥ ˜ ∥ ∥ f ˜ 1 L 1,λ (Γ ) L 1,λ (Γ ) ∥ f ∥˜L 1,λ (Γ ) . , ≲ [r ]λ1 ≲ [r ]λ1 min ⎭ ⎩ β β β Necessity. Let IΓα be bounded from ˜ L 1,λ (Γ ) to W ˜ L q,λ (Γ ). We have ( )1/q ∫  α  −λ  I fr  ˜ = sup β sup [ν] dν(τ ) Γ

W L q,λ (Γ )

β>0

τ ∈Γ , ν>0

( =r

−α

sup βr β>0

=r

−α− q1

1

= r −α− q

α

sup

τ ∈Γ , ν>0

[ν]−λ 1

1

α f (τ )|>β} {τ ∈Γ (t,ν) : |IΓ r

∫ α f (r τ )|>βr α } {τ ∈Γ (t,ν) : |IΓ

)1/q dν(τ )

( )1/q ) ∫ [r ν]1 λ/q −λ α sup sup βr sup [νr ]1 dν(τ ) α f (τ )|>βr α } [ν]1 ν>0 β>0 τ ∈Γ , ν>0 {τ ∈Γ (t,r ν) : |IΓ λ  α  q I f  ˜ [r ]1,+ . Γ W L (Γ ) (

q,λ

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11

From the boundedness of IΓα from ˜ L 1,λ (Γ ) to W ˜ L q,λ (Γ ) we get the following inequality 1

 α  I f  ˜ Γ WL

q,λ (Γ )

λ

q ≲ r −α− q [r ]1,+ ∥ f ∥˜L 1,λ (Γ ) .

L 1,λ (Γ ), we obtain If 1 < + α, then for all f ∈ ˜  α  I f  ˜ = 0 as r → 0 Γ W L (Γ ) 1 q

q,λ

which is impossible. Similarly, if 1 > q1 +  α  I f  ˜ = 0 as r → ∞ Γ W L (Γ )

α , 1−λ

then for all f ∈ ˜ L 1,λ (Γ ), we obtain

q,λ

which is also impossible. Therefore we obtain α ≤ 1 − Thus the proof of Theorem 2 is completed. □

1 q



α . 1−λ

α Let Γ be a Carleson curve and 0 < α < 1. Recall that, for all f ∈ L loc 1 (Γ ) the fractional maximal operator MΓ is defined by ∫ α α−1 MΓ f (t) = sup(νΓ (t, r )) | f (τ )|dν(τ ), t>0

Γ (t,r )

and for all t ∈ Γ the following inequality holds MΓα f (t) ≤ IΓα (| f |)(t).

(8)

As a consequence of Theorem 2 and (8) we have the following corollary. . Corollary 4. Let Γ be a Carleson curve, 0 < α < 1, 0 ≤ λ < 1 − α and 1 ≤ p < 1−λ α (1) If p = 1, then the condition α ≤ 1 − 1/q ≤ α/(1 − λ) is sufficient and in the case of infinite curve also necessary for the boundedness of MΓα from ˜ L 1,λ (Γ ) to W ˜ L q,λ (Γ ). 1−λ (2) If 1 < p < α , then the condition α ≤ 1/ p − 1/q ≤ α/(1 − λ) is sufficient and in the case of infinite curve also necessary for the boundedness of MΓα from ˜ L p,λ (Γ ) to ˜ L q,λ (Γ ). Proof. Sufficiency parts of the proofs of the statements (1) and (2) follow from Theorem 2 and inequality (8). Necessity. Let fr (τ ) =: f (r τ ) be defined as in Theorem 2. Then it is clear that λ    α  1  M fr  ˜ = r −α− q [r ] q  M α f  ˜ Γ

W L q,λ (Γ )

Γ

1,+

W L q,λ (Γ )

and  α   M f r ˜ Γ L

q,λ (Γ )

λ  α  1 q  M f ˜ = r −α− q [r ]1,+ Γ L

q,λ (Γ )

.

(1) Let MΓα be bounded from ˜ L 1,λ (Γ ) to W ˜ L q,λ (Γ ). Then we get  α  M f  ˜ Γ WL

q,λ (Γ )

1

−λ

 1 −λ  = r α+ q [r ]1,+q  MΓα fr W ˜L

q,λ (Γ )

1

λ− λ

≲ r α+ q [r ]1,+q ∥ fr ∥˜L 1,λ (Γ ) = r α−1+ q [r ]1,+q ∥ f ∥˜L 1,λ (Γ ) . If 1 < q1 + α, then for all f ∈ L 1,λ (Γ ) we obtain  α  M f  ˜ = 0 as r → 0 Γ W L (Γ ) q,λ

which is impossible. Similarly, if 1 > q1 +  α  M f  ˜ = 0 as r → ∞ Γ W L (Γ )

α , 1−λ

then for all f ∈ L 1,λ (Γ ) we obtain

q,λ

which is also impossible. Hence we get α ≤ 1 −

1 q



α . 1−λ

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(2) Let 1 < p <

1−λ , α



f ∈˜ L p,λ (Γ ), and let MΓα be bounded from ˜ L p,λ (Γ ) to ˜ L q,λ (Γ ). Then we get

 1 −λ  = r α+ q [r ]1,+q  MΓα fr ˜L

 α   M f ˜ Γ L

q,λ (Γ )

q,λ (Γ )

−λ

1

)

1

1

λ−λ

p q ≲ r α+ q [r ]1,+q ∥ fr ∥˜L p,λ (Γ ) = r α+ q − p [r ]1,+ ∥ f ∥˜L p,λ (Γ ) .

If

> q1 + α, then for all f ∈ ˜ L p,λ (Γ ) we obtain  α   M f ˜ = 0 as r → 0 Γ L (Γ )

1 p

q,λ

which is impossible. Similarly, if 1p <  α   M f ˜ = 0 as r → ∞ Γ L (Γ )

1 q

+

α , 1−λ

then for all f ∈ ˜ L p,λ (Γ ) we obtain

q,λ

which is also impossible, and therefore we get α ≤ Thus the proof of Corollary 4 is completed. □

1 p



1 q



α . 1−λ

Note that in the limiting case 1−λ ≤ p ≤ α1 statement (1) in Theorem 2 does not hold, when Γ is an infinite curve. α Moreover, there exists f ∈ L p,λ (Γ ) such that IΓα f (t) = ∞ for all t ∈ Γ . However, as will be proved, statement (1) in Theorem 2 holds for the modified potential operator ˜ IΓα if the space L ∞ (Γ ) is replaced by a wider space B M O(Γ ). In the following theorem we obtain conditions ensuring that the modified fractional integral operator ˜ IΓα is bounded from the space ˜ L p,λ (Γ ) to B M O(Γ ), when Γ is an infinite curve. ≤ p ≤ α1 . Then Theorem 3. Let Γ be a Carleson curve, and let 0 < α < 1, 0 ≤ λ ≤ 1 − α, 1−λ α α (1) If Γ is infinite, then the modified potential operator ˜ IΓ is bounded from ˜ L p,λ (Γ ) to B M O(Γ ). (2) If Γ is finite, then the potential operator IΓα is bounded from ˜ L p,λ (Γ ) to B M O(Γ ). (3) MΓα is bounded from ˜ L p,λ (Γ ) to L ∞ (Γ ). Proof. (1) Let α = 1 − λ, f ∈ L 1,1−α (Γ ) and let Γ be an infinite curve. For given r > 0 and for any t0 ∈ Γ we set f 1 (t) = f (t)χΓ (t0 ,2r ) (t),

f 2 (t) = f (t) − f 1 (t),

(9)

where χΓ (t0 ,2r ) is the characteristic function of the set Γ (t0 , 2r ). Then we get ˜ IΓα f (t) = ˜ IΓα f 1 (t) + ˜ IΓα f 2 (t) = F1 (t) + F2 (t), where ∫ F1 (t) =

(

Γ (t0 ,2r )

) |t − τ |α−1 − |t0 − τ |α−1 χΓ \Γ (t0 ,1) (τ ) f (τ )dν(τ )

and ∫ F2 (t) =

( Γ \Γ (t0 ,2r )

) |t − τ |α−1 − |t0 − τ |α−1 χΓ \Γ (t0 ,1) (τ ) f (τ )dν(τ ).

Note that the function f 1 has compact (bounded) support and thus ∫ a1 = − |t0 − τ |α−1 f (τ )dν(τ ) Γ (t0 ,2r )\Γ (t0 ,min{1,2r })

is finite. Note also that ∫ F1 (η) − a1 =

Γ (t0 ,2r )

|η − τ |α−1 f (τ )dν(τ )

∫ − Γ (t0 ,2r )\Γ (t0 ,min{1,2r })

|t0 − τ |α−1 f (τ )dν(τ )

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∫ + Γ (t0 ,2r )\Γ (t0 ,min{1,2r })



)



13

|t0 − τ |α−1 f (τ )dν(τ )

|η − τ |α−1 f 1 (τ )dν(τ ) = I α f 1 (η).

= Γ

Therefore we get ∫ |F1 (η) − a1 | ≤

Γ (t0 ,2r )

|η − τ |α−1 | f (τ )|dν(τ ).

From Lemmas 6 and 9 for all t ∈ Γ (t0 , r ) we have ∫ −1 |F1 (η) − a1 | dν(η) (νΓ (t, r )) Γ (t,r ) ) (∫ ∫ ≤ (νΓ (t, r ))−1 |η − τ |α−1 | f (τ )|dν(τ ) dν(η) Γ (t,r ) Γ (t0 ,2r ) ) (∫ ∫ −1 ≤r |η − τ |α−1 dν(η) | f (τ )|dν(τ ) Γ (t ,2r ) Γ (t,r ) ) ∫ 0 (∫ −1 ≤r |η − τ |α−1 dν(η) | f (τ )|dν(τ ) Γ (t0 ,3r ) Γ (τ,3r +|t−t0 |) ∫ −1 ≲ r (3r + |t − t0 |)α | f (τ )|dν(τ ) ≲ ∥ f ∥ L 1,1−α (Γ ) Γ (t0 ,3r )

≲ ∥ f ∥˜L p,λ (Γ ) .

(10)

Put ∫ a2 =

Γ (t0 ,max{1,2r })\Γ (t0 ,2r )

|t0 − τ |α−1 f (τ )dν(τ ).

Estimate |F2 (t) − a2 | for t ∈ Γ (t0 , r ): ∫ ⏐ ⏐ | f (τ )| ⏐|t − τ |α−1 − |t0 − τ |α−1 ⏐ dν(τ ). |F2 (t) − a2 | ≤ Γ \Γ (t0 ,2r )

Applying Lemma 10 we have 1−α

|F2 (t) − a2 | ≤ 2

∫ |t − t0 |

For I1 from Lemma 6 we get ∞ ∫ ∑ I1 = ≤

j=0 ∞ ∑

Γ (t,2 j+2 r )\Γ (t,2 j+1 r )

(

2 j+1r

j=0 1−α −1

≤2

r

)α−2

Γ \Γ (t0 ,2r )

| f (τ ) ∥ t0 − τ |α−2 dν(τ ) = 21−α |t − t0 |I1 .

| f (τ )||t0 − τ |α−2 dν(τ )

∫ | f (τ )|dν(τ ) Γ (t,2 j+2 r )

∥ f ∥ L 1,1−α (Γ ) ≲ r −1 ∥ f ∥˜L p,λ (Γ ) .

Then for all t ∈ Γ (t0 , r ) we have |F2 (t) − a2 | ≲ ∥ f ∥˜L p,λ (Γ ) .

(11)

Denote ∫ a f = a1 + a2 =

Γ (t0 ,max{1,2r })

|t0 − τ |α−1 f (τ )dν(τ ).

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Finally, from (10) and (11) we have ∫ ⏐ α ⏐ 1 ⏐˜ sup IΓ f (t) − a f ⏐ dν(t) ≲ ∥ f ∥˜L p,λ (Γ ) . r >0,t0 ∈Γ νΓ (t0 , r ) Γ (t0 ,r ) Thus  α  ˜ IΓ f  B M O ≤ 2 sup

r >0,t0 ∈Γ

1 νΓ (t0 , r )

∫ Γ (t0 ,r )

⏐ α ⏐ ⏐˜ IΓ f (t) − a f ⏐ dν(t)

≲ ∥ f ∥˜L p,λ (Γ ) . (2) Let f ∈ L p,λ (Γ ), 1 ≤ p = 1−λ and t0 ∈ Γ , and let Γ be a finite curve. Then there exists a natural number N α such that 2 N ≤ l = νΓ < 2 N +1 and IΓα f (t) = ˜ IΓα f (t) + C( f ), where ∫ C( f ) = Γ \Γ (t0 ,1)

|t0 − τ |α−1 f (τ )dν(τ ).

In this case of C( f ) is finite: ∫ |t0 − τ |α−1 | f (τ )|dν(τ ) |C( f )| ≤ Γ \Γ (t0 ,1)

=

N ∫ ∑ Γ (t,2 j+1 )\Γ (t,2 j )

j=0



N ∑ j=0

2

j(α−1)

|t0 − τ |α−1 | f (τ )|dν(τ )

∫ | f (τ )|dν(τ ) Γ (t,2 j+1 )\Γ (t,2 j ) 1/ p ′

≤ N 21−α ∥ f ∥ L 1,1−α (Γ ) ≤ N 21−α c0

∥ f ∥˜L p,λ (Γ ) .

MΓα

(3) The boundedness of the operator follows from the following inequalities ∫ ′ MΓα f (t) ≤ sup r α−1 | f (τ )|dν(τ ) = ∥ f ∥ L 1,1−α (Γ ) ≤ co1/ p ∥ f ∥ L 1−λ t∈Γ , r >0

Γ (t,r )

α ,λ

(Γ ) .

Therefore Theorem 3 is proved. □ References [1] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938) 126–166. [2] M.A. Ragusa, A. Tachikawa, On interior regularity of minimizers of p(x)-energy functionals, Nonlinear Anal. Theory Methods Appl. 93 (2013) 162–167. [3] M.A. Ragusa, A. Tachikawa, On continuity of minimizers for certain quadratic growth functionals, J. Math. Soc. Japan 57 (3) (2005) 691–700. [4] V.S. Guliyev, M.N. Omarova, M.A. Ragusa, A. Scapellato, Commutators and generalized local morrey spaces, J. Math. Anal. Appl. 457 (2) (2018) 1388–1402. http://dx.doi.org/10.1016/j.jmaa.2016.09.070. [5] S. Leonardi, Gradient estimates below duality exponent for a class of linear elliptic systems, Nonlinear Differential Equations Appl. 18 (3) (2011) 237–254. [6] S. Leonardi, J. Stara, Regularity up to the boundary for the gradient of solutions of linear elliptic systems with VMO coefficients and L 1,λ data, Complex Var. Elliptic Equ. 56 (12) (2011) 1085–1098. [7] A. Scapellato, On some qualitative results for the solution to a Dirichlet problem in Local Generalized Morrey Spaces, in: S. Sivasundaram (Ed.), AIP Conference Proceedings, vol. 1798, Article Number: UNSP 020138, 2017. http://dx.doi.org/10.1063/1.4972730. [8] A. Scapellato, Some properties of integral operators on generalized Morrey spaces, in: T.E. Simos, Ch. Tsitouras (Eds.), AIP Conference Proceedings, vol. 1863, 21 Article number 510004, 2017. http://dx.doi.org/10.1063/1.4992662. [9] A. Böttcher, Yu.I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, Birkhäuser Verlag, Basel, Boston, Berlin, 1997, p. 397. [10] A. Böttcher, Yu.I. Karlovich, Toeplitz operators with PC symbols on general Carleson Jordan curves with arbitrary Muckenhoupt weights, Trans. Amer. Math. Soc. 351 (1999) 3143–3196.

Please cite this article in press as: I.B. Dadashova, et al., Potential operators in modified Morrey spaces defined on Carleson curves, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.09.004.

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[11] A. Yu. Karlovich, Maximal operators on variable Lebesgue spaces with weights related to oscillations of Carleson curves, Math. Nachr. 283 (1) (2010) 85–93. [12] V. Kokilashvili, Fractional integrals on curves, Trudy Tbliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 95 (1990) 56–70 (in Russian). [13] V. Kokilashvili, A. Meskhi, Fractional integrals on measure Spaces, Fract. Calc. Appl. Anal. 4 (4) (2001) 1–24. [14] V. Kokilashvili, A. Meskhi, Boundedness of maximal and singular operators in morrey spaces with variable exponent, Arm. J. Math. (Electron.) 1 (1) (2008) 18–28. [15] V. Kokilashvili, S. Samko, Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent, Acta Math. Sin. (Engl. Ser.) 24 (11) (2008) 1775–1800. [16] N. Samko, Weighted hardy and singular operators in Morrey spaces, J. Math. Anal. Appl. 350 (1) (2009) 56–72. [17] A. Eroglu, I.B. Dadashova, Potential operators on carleson curves in morrey spaces, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 67 (2) (2018) 1–8. [18] J.I. Mamedkhanov, I.B. Dadashova, Some properties of the potential operators in Morrey spaces defined on Carleson curves, Complex Var. Elliptic Equ. 55 (8–10) (2010) 937–945. [19] V.S. Guliyev, J.J. Hasanov, Y. Zeren, Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces, J. Math. Inequal. 5 (4) (2011) 491–506.

Further reading [1] V. Kokilashvili, S. Samko, Sobolev theorem for potentials on Carleson curves in variable Lebesgue spaces, Mem. Differential Equations Math. Phys. 33 (2004) 157–158.

Please cite this article in press as: I.B. Dadashova, et al., Potential operators in modified Morrey spaces defined on Carleson curves, Transactions of A. Razmadze Mathematical Institute (2017), https://doi.org/10.1016/j.trmi.2017.09.004.