Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces

Research Article

Vagif Guliyev, Hatice Armutcu* and Tahir Azeroglu

Characterizations for the potential operators

on Carleson curves in local generalized

Morrey spaces

https://doi.org/10.1515/math-2020-0102

received June 21, 2020; accepted September 25, 2020

Abstract: In this paper, we give a boundedness criterion for the potential operatorα_{in the local general} -ized Morrey spaceLM{ }p φt,0( )Γ and the generalized Morrey space Mp φ, ( )Γ defined on Carleson curves Γ, re-spectively. For the operatorα_{, we establish necessary and sufficient conditions for the strong and weak} Spanne-type boundedness onLM{ }p φt,0( )Γ and the strong and weak Adams-type boundedness onMp φ, ( )Γ. Keywords: Carleson curve, local generalized Morrey space, potential operator, Adams-type inequalities MSC 2020: 26A33, 42B25, 42B35, 47B38

1 Introduction

LetΓ= { ∈t : t= ( )   ≤t s, 0 s≤ ≤ ∞}l be a rectifiable Jordan curve in the complex plane  with arc-length measure ( ) =ν t s, where =l νΓ=lengths of Γ. We denote

(t r) ≔ ∩ (B t r) t∈ r>

Γ , Γ , , Γ, 0,

whereB t r(, ) = { ∈z  :| − | < }z t r. We also denote for brevityν t rΓ ,( ) = | (Γ ,t r)|. A rectifiable Jordan curve Γ is called a Carleson curve if the condition

( ) ≤

ν t rΓ , c r0

holds for all ∈t Γ andr>0, where the constantc0>0does not depend on t and r.

Let f∈L₁loc( )Γ_{. The maximal operator and the potential operator }α_{on Γ are defined by}

 ( ) = ( ( ))

∫

| ( )| ( ) > − ( ) f t sup Γ ,ν t r f τ dν τ t t r 0 1 Γ , and  ( ) =

∫

( ) ( ) | − |− < < f t f τ ν τ t τ α d , 0 1, α α Γ 1 respectively.

Vagif Guliyev: Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan; Department of Mathematics, Dumlupinar University, Kutahya, Turkey; Institute of Mathematics and Mechanics, Baku, Azerbaijan, e-mail: vagif@guliyev.com 

* Corresponding author: Hatice Armutcu, Gebze Technical University, Kocaeli, Turkey, e-mail: haticexarmutcu@gmail.com

Tahir Azeroglu: Gebze Technical University, Kocaeli, Turkey, e_{-mail: aliyev@gtu.edu.tr}

Open Access. © 2020 Vagif Guliyev et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License.

Maximal operator and potential operator in various spaces, in particular, defined on Carleson curves have been widely studied by many authors(see, for example, [1–14]).

The main purpose of this paper is to establish the boundedness of potential operatorα_,0<α<1 in local generalized Morrey spacesLMp φ{ }x,0( )Γ defined on Carleson curves Γ. We shall give characterizations

for the strong and weak Spanne-type boundedness of the operator α _{from LM}{ }_{( )Γ}

p φx,0₁ to LMq φ{ }x,0₂( )Γ, <p<q< ∞

1 , / − / =1 p 1 q α and from the space _LM{ }( )_Γ

φ x

1,0₁ to the weak space WLMq φ{ },x0₂( )Γ, 1<q< ∞,

− / =q α

1 1 . Also, we study Adams-type boundedness of the operator α _{from generalized Morrey spaces}

( )

M_{p φ, p}1 Γ toM ( )Γ

q φ, q1 ,1<p<q< ∞, and from the spaceM1,φ( )Γ to the weak spaceWMq φ, q1( )Γ,1<q < ∞. We

shall give characterizations for the Adams-type boundedness of the operator α _{in generalized Morrey}

spaces, including weak versions.

ByA≲B we mean thatA≤CB with some positive constant C independent of appropriate quantities. If

≲

A B andB≲A, we writeA≈B and say that A and B are equivalent.

2 Preliminaries

Morrey spaces were introduced by C. B. Morrey[15] in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variations. Later, Morrey spaces found important applications to Navier-Stokes and Schrödinger equations, elliptic problems with discontinuous coefficients, and poten-tial theory.

LetL Γp( ),1≤p< ∞be the space of measurable functions on Γ withfinite norm        

∫

∥ ∥ ( ) = | ( )| ( ) / f L f t pdν t . p Γ Γ 1 p

Definition 2.1. Let ≤ < ∞1 p ,0≤λ≤1, [ ] =r1 min 1,{ r}. We denote byLp λ,( )Γ the Morrey space, and by ( )

∼

Lp λ, Γ the modified Morrey space, the set of locally integrable functions f on Γ with the finite norms

∥ ∥ ( ) = ∥ ∥ ∥ ∥ = [ ] ∥ ∥ ∈ > − ( ( )) ( ) ∈ > − ( ( )) ∼ f L sup r f , f sup r f , t r L t r L t r L t r Γ Γ, 0 Γ , Γ Γ, 0 1 Γ , p λ λ p p p λ λ p p , , respectively.

Note that(see [16,17])Lp,0( ) =Γ L˜p,0( ) =Γ Lp( )Γ,

( ) = ( ) ∩ ( ) ∥ ∥∼ ( )= {∥ ∥ ( ) ∥ ∥ ( )}

L˜p λ, Γ Lp λ, Γ Lp Γ and f Lp λ, Γ max f Lp λ, Γ, f LpΓ

and ifλ<0 or λ>1, thenLp λ, ( ) =Γ L˜p λ, ( ) =Γ Θ,whereΘ is the set of all functions equivalent to 0 on Γ. We denote byWLp λ, ( )Γ the weak Morrey space, and byWL˜p λ, ( )Γ the modified Morrey space, as the set of locally integrable functions f on Γ withfinite norms

               

∫

∫

∥ ∥ = ( ) ∥ ∥ = [ ] ( ) ( ) > ∈ > − { ∈ ( ) | ( ) |> } / ( ) > ∈ > − { ∈ ( ) | ( ) |> } / f β r ν τ f β r ν τ sup sup d , sup sup d . WL β t r λ τ t r f τ β p WL β t r λ τ t r f τ β p Γ 0 Γ, 0 Γ , : 1 ˜ Γ 0 Γ, 0 1 Γ , : 1 p λ p λ , ,

Samko [14] studied the boundedness of the maximal operator  defined on quasimetric measure spaces, in particular on Carleson curves in Morrey spacesLp λ, ( )Γ:

Theorem A. Let Γ be a Carleson curve, < < ∞1 p ,0<α<1 and0≤λ<1. Then is bounded fromLp λ, ( )Γ toLp λ, ( )Γ.

Kokilashvili and Meskhi [18] studied the boundedness of the operator α _{defined on quasimetric}

measure spaces, in particular on Carleson curves in Morrey spaces and proved the following: Theorem B. Let Γ be a Carleson curve, < < < ∞1 p q ,0<α<1,0<λ < p

q 1 , = λ p λ q 1 2 _and − = α p q 1 1 . Then the operatorα_{is bounded from the spacesL ( )Γ}

p λ,1 toLq λ, 2( )Γ .

The following Adams boundedness(see [19]) of the operator α_{in Morrey space defined on Carleson}

curves was proved in[20].

Theorem C. Let Γ be a Carleson curve, < <0 α 1,0≤λ<1−αand1≤p< 1−_αλ. (1) If < <_{1 p} −λ

α

1

, then the condition − =

−

p q

α λ

1 1

1 is sufficient and in the case of infinite curve also necessary for the boundedness of the operatorα_{fromL ( )Γ}

p λ, toLq λ, ( )Γ.

(2) If =p 1, then the condition − =

− 1 q α λ 1

1 is sufficient and in the case of infinite curve also necessary for the boundedness of the operatorα _{fromL ( )Γ}

λ

1, toWLq λ,( )Γ.

The following Adams boundedness of the operator α _{in modified Morrey space L˜ ( )Γ}

p λ, defined on

Carleson curves was proved in[16], see also [17].

Theorem D. Let Γ be a Carleson curve, < <0 α 1,0≤λ<1−αand1≤p< −λ

α 1 . (1) If < <_{1 p} −λ α 1

, then the condition α≤ − ≤ ₋

p q

α λ

1 1

1 is sufficient and in the case of infinite curve also

necessary for the boundedness of the operatorα_{fromL˜ ( )Γ}

p λ, toL˜q λ, ( )Γ . (2) If =p 1, then the conditionα≤1− ≤ ₋

q α

λ 1

1 is sufficient and in the case of infinite curve also necessary for the boundedness ofα_{fromL˜ ( )Γ}

λ

1, toWL˜q λ, ( )Γ.

We use the following statement on the boundedness of the weighted Hardy operator:

∫

( ) ≔ ( ) ( ) < < ∞ ∞ H g tw g s w s d ,s 0 t , t where w is a weight.

The following theorem was proved in[21].

Theorem 2.1. Let v1, v2and w be weights on ( ∞)0, and ( )v t1 be bounded outside a neighborhood of the origin. The inequality

( ) ( ) ≤ ( ) ( )

> >

v t H g t C v t g t

ess sup ess sup

t w t 0 2 0 1 (2.1)

holds for someC>0 for all non-negative and non-decreasing g on ( ∞)0, if and only if

∫

≔ ( ) ( ) ( ) < ∞ > ∞ < <∞ B v t w s s v τ sup d ess sup . t t s τ 0 2 1

3 Local generalized Morrey spaces

Wefind it convenient to define the local generalized Morrey spaces in the form as follows, see [21,22]. Definition 3.2. Let ≤ < ∞1 p and (φ t r, )be a positive measurable function onΓ× (0,∞). Fixedt0∈Γ, we denote by LM{ }p φt,0( )Γ (WLMp φ{ }t,0( ))Γ the local generalized Morrey space(the weak local generalized Morrey space), the space of all functions ∈f Lploc( )Γ withfinite quasinorm

      ∥ ∥ = ( ) ( ( )) ∥ ∥ ∥ ∥ = ( ) ( ( )) ∥ ∥ ( ) > ( ( )) ( ) > ( ( )) { } { } f φ t r ν t r f f φ t r ν t r f sup 1 , 1 Γ , sup 1 , 1 Γ , . LM r L t r WLM r WL t r Γ 0 0 0 Γ , Γ 0 0 0 Γ , p φt _p p p φt p p ,0 1 0 ,0 1 0

Definition 3.3. Let ≤ < ∞1 p and (φ t r, )be a positive measurable function onΓ× (0,∞). The generalized Morrey spaceMp φ, ( )Γ is defined as the set of all functions ∈f Lploc( )Γ by thefinite norm

∥ ∥ = ( ) ( ( )) ∥ ∥ ∈ > ( ( )) f φ t r ν t r f sup 1 , 1 Γ , . M t r L t r Γ, 0 Γ , p φ p p , 1

Also, the weak generalized Morrey spaceWMp φ, ( )Γ is defined as the set of all functions ∈f Lploc( )Γ by the finite norm ∥ ∥ = ( ) ( ( )) ∥ ∥ ∈ > ( ( )) f φ t r ν t r f sup 1 , 1 Γ , . WM t r WL t r Γ, 0 Γ , p φ p p , 1

It is natural,first the set of all, to find conditions ensuring that the spacesLM{ }_{p φ}t ( )Γ

,0 andMp φ, ( )Γ are non -trivial, that is, consist not only of functions equivalent to 0 on Γ.

Lemma 3.1. [23] Let ∈t0 Γ andφ t r(, )be a positive measurable function onΓ× (0,∞). If

( ) ( ( )) = ∞ > < <∞φ t r ν t r for some r sup 1 , 1 Γ , 0, r τ 0 0 p1 (3.2) thenLM_{p φ}{ }t ( ) =Γ Θ ,0 .

Remark 3.1. We denote by Ωp,locthe set of all positive measurable functionsφ on × (Γ 0,∞)such that for all > r 0, ( ) ( ( )) _{∞( ∞)} < ∞ φ t τ ν t τ 1 , 1 Γ , _{L r} . 0 0 p , 1

In what follows, keeping in mind Lemma 1, for the non-triviality of the spaceLMp φ{ }t,0( )Γ we always assume thatφ∈Ωp,loc.

Lemma 3.2. [23] Let (φ t r, )be a positive measurable function onΓ× (0,∞). (i) If

( ) ( ( )) = ∞ > ∈

< <∞φ t τ ν t τ

for some r and for all t

sup 1 , 1 Γ , 0 Γ, r τ p1 (3.3) thenMp φ, ( ) =Γ Θ. (ii) If ( ) = ∞ > ∈ < < −

φ t τ for some r and for all t

sup , 0 Γ, τ r 0 1 (3.4) thenMp φ, ( ) =Γ Θ.

Remark 3.2. We denote by Ωpthe sets of all positive measurable functionsφ onΓ× (0,∞)such that for all > r 0, ( ) ( ( )) < ∞ ∥ ( ) ∥ < ∞ ∈ _{( ∞)} ∈ − ( ) ∞ ∞ φ t τ ν t τ φ t τ sup 1 , 1 Γ , and sup , , t _{L r} t L r Γ _, Γ 1 0, p 1

respectively. In what follows, keeping in mind Lemma 2, we always assume thatφ∈Ωp.

A functionφ : 0,( ∞) → (0,∞)is said to be almost increasing(resp. almost decreasing) if there exists a constantC>0 such that

( ) ≤ ( ) ( ( ) ≥ ( )) ≤

φ r Cφ s resp. φ r Cφ s for r s.

Let1≤p< ∞. Denote by p the set of all almost decreasing functions φ : 0,( ∞) → (0,∞)such that

∈ ( ∞) ↦ ( ) ∈ ( ∞)

t 0, t φ tp1 0, is almost increasing.

Seemingly, the requirementφ∈pis superfluous but it turns out that this condition is natural. Indeed, Nakai established that there exists a functionρ such that ρ itself is decreasing, that ( )_{ρ t t}n p/ ≤ ( )_{ρ T T}n p/ _for all0<t≤T< ∞and thatLM_{p φ}{ }t ( ) =Γ LM{ }( )Γ

p ρt

,0 ,0 ,Mp φ, ( ) =Γ Mp ρ,( )Γ.

By elementary calculations we have the following, which shows particularly that the spaces_LM{ } p φt,0, { }

WLM_{p φ}t

,0,Mp φ, ( )Γ andWMp φ, ( )Γ are not trivial, see, for example,[23–25].

Lemma 3.3. [23] Letφ∈p,1≤p< ∞,Γ0= (Γt0,r0)and χΓ0be the characteristic function of the ball Γ0, then

∈ { }( ) ∩ ( )

χΓ₀ LMp φt,0 Γ Mp φ, Γ. Moreover, there existsC>0 such that

( ) ≤ ∥ ∥ { } ≤ ∥ ∥ { }≤ ( ) φ r χ χ C φ r 1 WLM LM 0 Γ p φ Γ 0 t p φt 0 ,0 0 ,0 and ( ) ≤ ∥ ∥ ≤ ∥ ∥ ≤ ( ) φ r χ χ C φ r 1 . WM M 0 Γ Γ 0 p φ p φ 0 , 0 ,

4 Maximal operator in the spaces

LM

_{p φ}{ }t_,0

_{( )}

Γ

_and

_M

_Γ

p φ,

( )

We denote byL∞,v( ∞)0, the set of all functions ( )g t , >t 0 withfinite norm

∥ ∥ ( ∞) = ( ) ( ) > ∞ g L ess supv t g t t 0, 0 v ,

andL∞(0,∞) ≡L∞,1(0, ∞). LetM( ∞)0, be the set of all Lebesgue-measurable functions on ( ∞)0, and M (+_0,∞)_{its subset consisting of all non-negative functions on (0, ∞). We denote byM (}+_{0, ∞ ↑); the} cone of all functions inM (+_0, ∞)_{which are non-decreasing on (0, ∞)and}

M

 = { ∈ +( ∞ ↑) ( ) = }

→ +

φ 0, ; : lim φ t 0 .

t 0

Let u be a continuous and non-negative function on (0, ∞). We define the supremal operator S¯u on

M

∈ ( ∞)

g 0, by

(S g t¯u )( ) ≔ ∥ug∥L∞( ∞)t, , t∈ (0,∞). The following theorem was proved in[26].

Theorem 4.2. Let v1, v2be non-negative measurable functions satisfying < ∥ ∥0 v1 L∞( ∞)t, < ∞for any >t 0 and let u be a non-negative continuous function on ( ∞)0, .

Then the operator S¯uis bounded fromL∞,v1(0,∞)toL∞,v2(0,∞)on the cone if and only if

∥v S2¯u(∥ ∥v1 −L1∞(⋅ ∞), )∥L∞( ∞)0, < ∞. (4.5)

The following Guliyev-type local estimate for the maximal operator  is true, see for example, [27,28]. Lemma 4.4. Let Γ be a Carleson curve, ≤ < ∞1 p andt0∈Γ. Then forp>1 and anyr>0 the inequality

 ∥ ∥ ( ( ))≲ ∥ ∥ ( ( ))+ ∥ ∥ > − ( ( )) f L t r f L t r r supτ f τ r L t τ Γ , Γ ,2 2 1 Γ , p 0 p 0 p 1 1 0 (4.6)

holds for all f ∈L_ploc( )Γ_.

Moreover, forp=1 the inequality  ∥ ∥ ( ( ))≲ ∥ ∥ ( ( ))+ ∥ ∥ > − ( ( )) f WL t r f L t r rsupτ f τ r L t τ Γ , Γ ,2 2 1 Γ , 1 0 1 0 1 0 (4.7)

holds for all f ∈L1loc( )Γ .

Proof. Let < < ∞1 p . For arbitrary ball (Γ t0,r)letf=f1+f2, where f1 =fχΓ₍t0,2r₎andf2 =fχ∁( (Γ 0,2t r)).

  

∥ f∥Lp( (Γt r0,)) ≤ ∥ f1∥Lp( (Γt r0,)) + ∥ f2∥Lp( (Γt r0,)). By the continuity of the operator :Lp( ) →Γ Lp( )Γ from Theorem A we have



∥ f1∥Lp( (Γt r0,))≲ ∥ ∥f Lp( (Γt0,2r)).

Let y be an arbitrary point from (Γ t0,τ). IfΓ ,(y τ) ∩ ( (∁Γt0, 2r)) ≠ ∅, thenτ>r. Indeed, if z∈ (Γ ,y τ) ∩

( ( ))

∁_{Γt , 2r ,}

0 thenτ> | − | ≥ | − | − | − | >y z t z t y 2r−r= r.

On the other hand,Γ ,(y τ) ∩ ( (∁Γt0, 2r)) ⊂ (Γ t0, 2τ). Indeed,z∈ (Γ ,y τ) ∩ ( (∁Γ t0, 2r)), then we get| − | ≤t z | − | + | − | <y z t y τ+r<2 .τ Hence,  ( ) ≤

∫

∫

∫

( ) | ( )| ( ) = ( ) | ( )| ( ) ≤ | ( )| ( ) > ( ) > ( ) > − ( ) f y ν t τ f z ν z ν t τ f z ν z τ f z ν z 2 sup 1 Γ , 2 d 2 sup 1 Γ , d 2 sup d . τ r t τ τ r t τ τ r t τ 2 0 Γ ,2 2 0 Γ , 2 1 Γ , 0 0 0

Therefore, for ally∈ (Γt0,τ)we have

 ( ) ≤

∫

| ( )| ( ) > − ( ) f y 2 supτ f z dν z. τ r Γ t τ 2 2 1 , 0 (4.8) Thus,         

_∫

∥ ∥ ( ( ))≲ ∥ ∥ ( ( )) + | ( )| ( ) > − ( ) f L t r f L t r r supτ f z dν z . τ r t τ Γ , Γ ,2 2 1 Γ , p 0 p 0 p 1 0 Letp=1. It is obvious that for any ball (Γt0,r)

  

∥ f∥WL1( (Γt r0,))≤ ∥ f1∥WL1( (Γt r0,))+ ∥ f2∥WL1( (Γt r0,)). By the continuity of the operator :L1( ) →Γ WL1( )Γ from Theorem A we have



∥ f1∥WL1( )Γ ≲ ∥ ∥f L1( (Γt0,2r)).

Lemma 4.5. Let Γ be a Carleson curve, ≤ < ∞1 p andt0∈Γ. Then forp>1 and any >r 0 in Γ, the inequality  ∥ ∥ ( ( ))≲ ∥ ∥ > − ( ( )) f L t r r supτ f τ r L t τ Γ , 2 Γ , p 0 p p p 1 1 0 (4.9)

holds for all f∈Lploc( )Γ .

Moreover, forp=1 the inequality  ∥ ∥ ( ( )) ≲ ∥ ∥ > − ( ( )) f WL t r rsupτ f τ r L t τ Γ , 2 1 Γ , 1 0 1 0 (4.10)

holds for all f∈L₁loc( )Γ_. Proof. Let < < ∞1 p . Denote

 ≔

∫

| ( )| ( )  ≔ ∥ ∥ > − ( ) ( ( )) r supτ f z dν z , f . τ r t r L t r 1 2 1 Γ , 2 Γ ,2 p p 1 0 0

Applying Hölder’s inequality, we get

         ≲  

_∫

| ( )| ( ) > ( ) r supτ f z dν z . τ r t τ p 1 2 Γ , p p p 1 1 0 1

On the other hand,

            

∫

| ( )| ( ) ≳ ∥ ∥ ≈ > ( ) > ( ( )) r supτ f z dν z r supτ f . τ r t τ p τ r L t r 2 Γ , 2 Γ ,2 2 p p p p p p 1 1 0 1 1 1 0 Since by Lemma 4.4    ∥ f∥Lp( (Γt r0,)) ≤ 1+ 2, we arrive at(4.9).

Let p=1. The inequality(4.10) directly follows from (4.7). □

The following theorem is valid.

Theorem 4.3. Let Γ be a Carleson curve, ≤ < ∞1 p ,t0∈Γand (φ φ1, 2)satisfies the condition

( ) ≤ ( )

< <∞ −

< <∞

τ φ t s s Cφ t r

sup ess inf , , ,

r τ τ s 1

0 2 0

p p

1 1

(4.11) where C does not depend on r. Then forp>1 the operator is bounded fromLM{ }p φt,0₁( )Γ toLM{ }p φt,0₂( )Γ and for

=

p 1 the operator is bounded from_LM{ }( )_Γ φ t

1,0₁ toWLM{ }φ( )Γ t 1,0₂ .

Proof. By Theorem 4.2 and Lemma 4.5, we get  ∥ ∥ _{( )} ≲ ( ) ∥ ∥ ≲ ( ) ∥ ∥ = ∥ ∥ > − > − ( ( )) > − − ( ( )) ( ) { } { }

f _LM supφ t,r supτ f supφ t r, r f f

r τ r L t τ r L t r LM Γ 0 2 0 1 Γ , 0 1 1 Γ , Γ p φt p p p p p φt , 20 1 0 1 0 _{, 1}0 ifp∈ ( ∞)1, and  ∥ ∥ _{( )} ≲ ( ) ∥ ∥ ≲ ( ) ∥ ∥ = ∥ ∥ > − > − ( ( )) > − − ( ( )) ( ) { } { }

f _WLM supφ t ,r supτ f supφ t r, r f f

r τ r L t r r L t r LM Γ 0 2 0 1 1 Γ , 0 1 1 1 Γ , Γ p φt, 20 1 0 1 0 _{1, 1}tφ0 ifp=1. □

Corollary 4.1. Let Γ be a Carleson curve, ≤ < ∞1 p andφ φ1, 2 ∈Ωp satisfies the condition ( ) ≤ ( ) < <∞ − < <∞ τ φ t s s Cφ t r

sup ess inf , , ,

r τ τ s 1 2

p p

1 1

(4.12) where C does not depend on t and r. Then forp>1 the operator is bounded fromMp φ, ₁( )Γ toMp φ, ₂( )Γ and for

=

p 1 the operator is bounded fromM1,φ₁( )Γ toWM1,φ₂( )Γ .

Corollary 4.2. Let Γ be a Carleson curve, ≤ < ∞1 p andφ∈p. Then forp>1 the operator is bounded on ( )

Mp φ, Γ and for p=1 the operator is bounded fromM1,φ( )Γ toWM1,φ( )Γ.

5 Fractional integral operator in the spaces

LM

_{p φ}{ }t_,0

_{( )}

Γ

_and

_M

_Γ

p φ,

( )

5.1 Spanne

-type results

The following local estimate is true, see for example,[28].

Theorem 5.4. Let Γ be a Carleson curve, ≤ < ∞1 p ,t0∈Γ,0<α< _p 1

,_q1 = _p1 − αand f∈L_ploc( )Γ_{. Then for} > p 1 

∫

∥ ∥ ( ( )) ≤ ∥ ∥ ∞ − − ( ( )) f Cr τ f dτ α L t r r L t τ Γ , 2 1 Γ , q 0 q q p 1 1 0 (5.13) and for p=1 

∫

∥ ∥ ( ( ))≤ ∥ ∥ ∞ − − ( ( )) f Cr τ f d ,τ α WL t r r L t τ Γ , 2 1 Γ , q 0 q q 1 1 1 0 (5.14)

where C does not depend on f,t0∈Γandr>0.

Proof. For a given ball (Γt0,r), we split the function f as f=f1+ f2, where f1 =fχΓ₍t0,2r₎, f2 =fχ∁( (Γ t0,2r)), and then αf t( ) =αf t( ) +αf t .( ) 1 2 Let1<p< ∞,0<α< p 1

,_q1 = _p1 − α. Sincef1∈L Γp( ), by the boundedness of the operatorαfromL Γp( ) toL Γq( )(see Theorem B) it follows that



∫

∥ ∥ ( ) ≤ ∥ ∥ ( ) = ∥ ∥ ( ( ))≤ ∥ ∥ ∞ − − ( ( )) f C f C f Cr τ f d ,τ α L L L t r r L t τ 1 Γ 1 Γ Γ ,2 2 1 Γ , q p p 0 q q p 1 1 0 (5.15)

where the constant C is independent of f.

Observe that the conditionsz∈ (Γ t0,r),y∈ ( (∁ Γt0, 2r))imply | − | ≤ | − | ≤z y t y | − |t z 1

2

3

2 .

Then for allz∈ (Γt0,r)we get

      

∫

| ( )| ≤ | − | | ( )| ( ) − ∁ − ( ( )) f z 3 t y f y ν y 2 d . α α α 2 1 1 t r Γ 0,2

By Fubini’s theorem, we have

∫

∫

∫

∫ ∫

∫ ∫

| − | | ( )| ( ) ≈ | ( )| ( )∞ ≈ | ( )| ( ) ≲ | ( )| ( ) ∁ − ∁ | − | − ∞ ≤| − |< − ∞ ( ) − ( ( )) ( ( )) t y f y ν y f y ν y τ τ f y ν y τ τ f y ν y τ τ d d d d d d d . α t y α r r t y τ α r t τ α 1 2 2 2 2 2 Γ , 2 t r t r Γ 0,2 Γ 0,2 0 Applying Hölder’s inequality, we get

∫

| − | | ( )| ( ) ≲

∫

∥ ∥ ∁ − ∞ ( ( )) − − ( ( )) t yα f y dν y f τ dτ r L t τ 1 2 Γ , 1 t r p q Γ 0,2 0 1

and for allz∈ (Γt0,r)



_∫

| ( )| ≲ ∥ ∥ ∞ ( ( )) − − f z f τ d .τ α r L t τ 2 2 Γ , 1 p 0 q 1 (5.16) Moreover, for allp∈ [ ∞)1, the inequality



∫

∥ ∥ ( ( ))≲ ∥ ∥ ∞ − − ( ( )) f r τ f dτ α L t r r L t τ 2 Γ , 2 1 Γ , q 0 q q p 1 1 0 (5.17)

is valid. Thus, from(5.15) and (5.17) we get inequality (5.13).

Finally, in the casep=1 by the weak (1,q)-boundedness of the operator α_{(see Theorem B) it follows} that 

∫

∥ ∥ ( ( ))≤ ∥ ∥ ( ) ≤ ∥ ∥ ∞ − − ( ( )) f C f Cr τ f d ,τ α WL t r L r L t τ 1 Γ , 1 Γ 2 1 Γ , q 0 1 q q 1 1 1 0 (5.18)

where C does not depend ont0and r. Then from(5.17) and (5.18) we get inequality (5.14). □

Theorem 5.5. Let Γ be a Carleson curve, ≤ < ∞1 p ,t0∈Γ,0<α< _p 1

,_q1 = _p1 − α,φ₁∈Ωp,loc,φ₂ ∈Ωq,locand the pair (φ φ₁, ₂)satisfy the condition

∫

( ) ≤ ( ) ∞ < <∞ φ t s s τ τ τ Cφ t r ess inf , d , , r τ s 1 0 2 0 p q 1 1 (5.19)

where C does not depend ont0and r. Then forp>1 the operatorαis bounded fromLMp φ{ }t,0₁( )Γ toLMq φ{ }t,0₂( )Γ and forp=1 the operatorα_{is bounded fromLM}{ }_{( )Γ}

φ t

1,0₁ toWLMq φ{ }t,0₂( )Γ. Proof. By Theorems 2.1 and 5.4 with ( ) =_{v r φ t}( _,r)−

2 2 0 1, ( ) =v r1 φ t1(0,r)−1r−p 1

and_{w r}( ) =_r−q1_{we have for}p_>1



_∫

∥ ∥ ( ) ≲ ( ) ∥ ∥ ≲ ( ) ∥ ∥ = ∥ ∥ > − ∞ − − ( ( )) > − − ( ( )) ( ) { } { } f supφ t ,r s f ds supφ t ,r r f f α LM r r L t s r L t r LM Γ 0 2 0 1 1 Γ , 0 1 0 1 Γ , Γ q φt q p p p p φt , 20 1 0 1 0 _{, 1}0 and forp=1 

∫

∥ ∥ _{( )} ≲ ( ) ∥ ∥ ≲ ( ) ∥ ∥ = ∥ ∥ > − ∞ − − ( ( )) > − − ( ( )) ( ) { } { } f supφ t,r s f ds supφ t ,r r f f . □ α WLM r r L t s r Q L t r LM Γ 0 2 0 1 1 Γ , 0 1 0 1 Γ , Γ q φt q φ t , 20 1 1 0 1 0 _{1, 1}0

Corollary 5.3. Let Γ be a Carleson curve, ≤ < ∞1 p ,0<α< p 1

, _q1 = _p1 − α,φ₁∈Ωp,φ2∈Ωq and the pair

(φ φ₁, ₂)satisfy the condition

∫

( ) ≤ ( ) ∞ < <∞ φ t s s τ τ τ Cφ t r ess inf , d , , r τ s 1 2 p q 1 1 (5.20)

where C does not depend on t and r. Then forp>1 the operatorα_{is bounded fromM ( )Γ}

p φ, ₁ toMq φ, ₂( )Γ and for =

p 1 the operatorα_{is bounded fromM ( )Γ}

φ

1, ₁ toWMq φ, ₂( )Γ. For proving our main results, we need the following estimate.

Lemma 5.6. Let Γ be a Carleson curve andΓ0≔ (Γt0,r0), thenr0α≲αχΓ0( )t for every ∈t Γ0.

Proof. If t y, ∈Γ0, then | − | ≤ | −t y t t0| + |t0− | <y 2r0. Since 0< α<1, we get r0α−1≤ 21−α| − |t yα Q− . Therefore, α_χ ( ) =_t

∫

_χ ( )| − |_{y t y}α− _{dν y}( ) =

∫

| − |_{t y}α−_{dν y}( ) ≥_{c 2}−α α_{r . □} Γ Γ Γ 1 Γ 1 0 1 0 0 0 0

The following theorem is one of our main results.

Theorem 5.6. Let Γ be a Carleson curve, < <0 α 1,t0∈Γand p q, ∈ [ ∞)1, . 1. If1≤p<

α

1

and_q1 = _p1 − α, then condition(5.20) is sufficient for the boundedness of the operator α_from ( )

{ }

LMp φt,0₁Γ toWLMq φ{ }t,0₂( )Γ . Moreover, if1<p< _α

1

, condition(5.20) is sufficient for the boundedness of the operatorα _fromLM{ }_{( )Γ}

p φt,0₁ toLMq φ{ }t,0₂( )Γ . 2. If the functionφ1∈p, then the condition

( ) ≤ ( )

r φ rα Cφ r ,

1 2 (5.21)

for allr>0, whereC>0 does not depend on r, is necessary for the boundedness of the operatorα_from ( )

{ } LM_{p φ}t Γ

,0₁ toWLMq φ{ }t,0₂( )Γ andLMp φ{ }t,0₁( )Γ toLMq φ{ }t,0₂( )Γ.

3. Let1≤p< _α1 and_q1 = 1_p − α. Ifφ1∈psatisfies the regularity condition

∫

( ) ≤ ( ) ∞ − s φ s ds Cr φ r, r α 1 α 1 1 (5.22)

for allr>0, whereC>0 does not depend on r, then condition(5.21) is necessary and sufficient for the boundedness of the operatorα_fromLM{ }_{( )Γ}

p φt,0₁ toWLMq φ{ }t,0₂( )Γ. Moreover, if1<p< Q

α, then condition(5.21) is necessary and sufficient for the boundedness of the operator α_fromLM{ }_{( )Γ}

p φt,0₁ toLMq φ{ }t,0₂( )Γ. Proof. The first part of the theorem is proved in Theorem 5.3.

We shall now prove the second part. LetΓ0= (Γt0,r0)and ∈t Γ0. By Lemma 5.6, we haver0α≤CαχΓ0( )r. Therefore, by Lemmas 3.3 and 5.6

  ≲ ( ( )) ∥ ∥ ≲ ( )∥ ∥ ≲ ( )∥ ∥ ≲ ( ) ( ) − ( ) r ν χ φ r χ φ r χ φ r φ r Γ α α L α M M 0 0 Γ Γ 2 0 Γ 2 0 Γ 2 0 1 0 p q q φ p φ 1 0 0 0 , 2 0 , 1 or ≲ ( ) ( ) > ⇔ ( ) ≲ ( ) > r φ r

φ r for all r 0 r φ r φ r for all r 0.

α α

0 2 0 1 0

0 0 1 0 2 0 0

Since this is true for everyr0>0, we are done.

Remark 5.3. If we take ( ) = −

φ r₁ rλp1 and ( ) =

−

φ r₂ rμq

1

at Theorem 5.6, then conditions(5.22) and (5.21) are equivalent to0<λ<1−αpand λ_p = _qμ, respectively. Therefore, we get Theorem C from Theorem 5.6.

5.2 Adams

-type results

The following pointwise estimate plays a key role where we prove our main results. Theorem 5.7. Let Γ be a Carleson curve, ≤ < ∞1 p ,0<α<1 andf∈L_ploc( )Γ_{. Then}

 

_∫

| ( )| ≤ ( ) + ∥ ∥ ∞ − − ( ( )) f t Cr f t C s f d ,s α α r α L t s 1 Γ , p p 1 (5.23) where C does not depend on f, ∈t Γ andr>0.

Proof. Write = +f f1 f2, where f1=fχΓ ,2₍t r₎and f2 =fχ∁( (Γ ,2t r)). Then αf t( ) =αf t( ) +αf t .( )

1 2

For ( )αf t

1 , following Hedberg’s trick (see for instance [29, p. 354]), for all ∈z Γ we obtain∣αf z1( )∣ ≤  ( )

C rα f z .

1 For ( )αf z2 withz∈D from(5.16) we have



_∫

_∫

∣ ( )∣ ≤ | − | | ( )| ≤ ∥ ∥ ∁ − ∞ − − ( ( )) ( ( )) f z t y f y dy C s f d ,s α α r α L t s 2 1 2 1 Γ , t r p p Γ ,2 1 (5.24) which proves(5.23). □

The following is a result of Adams type for the fractional integral on Carleson curves(see [28]). Theorem 5.8. (Adams-type result) Let Γ be a Carleson curve, ≤ < < ∞1 p q ,0<α<

p 1

and letφ∈Ωpsatisfy condition ( ) ≤ ( ) < <∞ − < <∞ τ φ t s s Cφ t r

sup ess inf , , ,

r τ τ s 1 (5.25) and

∫

( ) ≤ ∞ − − − τ φ t τ, dτ Cr , r α 1 _p1 _{q p}αp (5.26) where C does not depend ont∈Γ andr>0. Then for p> 1 the operatorα _{is bounded fromM ( )Γ}

p φ, p1 to

( )

M Γ

q φ, q1 and for p=1 the operator

α_{is bounded from} ( )

M1,φΓ toWM_{q φ,} q1( )Γ.

Proof. Let ≤ < ∞1 p andf∈Mp φ, ( )Γ. By Theorem 5.7, inequality(5.23) is valid. Then from condition (5.26) and inequality(5.23) we get

   

∫

∫

| ( )| ≲ ( ) + ∥ ∥ ≤ ( ) + ∥ ∥ ( ) ≤ ( ) + ∥ ∥ ∞ − − ( ( )) ( ) ∞ − − ( ) − f t r f t s f s r f t f s φ t s s r f t r f d , d . α α r α L t s α M r α α M 1 Γ , Γ 1 Γ p p p φ p p αp q p p φ p 1 , 1 1 , 1 (5.27)

Hence, choosing        = ( ) ( ) r M f t Γ p φ p, 1

for every ∈t Γ, we have

  | αf t( )| ≲ ( f t( )) ∥ ∥f − _{( )}. M Γ 1 p q p φ p p q , 1

Hence, the statement of the theorem follows in view of the boundedness of the maximal operator in

( )

Mp φ, Γ provided by Theorem 3, by virtue of condition(5.25).

   ∥ ∥ ( ) ≲ ∥ ∥ ( ) ( ) ∥ ∥ ≲ ∥ ∥ ∥ ∥ ≲ ∥ ∥ − ∈ > − − ( ( )) ( ) − ( ) ( ) f f sup φ t r, r f f f f α M M t r L t r M M M Γ Γ 1 Γ, 0 Γ , Γ 1 Γ Γ q φq p φ p p q _qp _q p p q p φ p p q p φ p p q p φ p , 1 , 1 1 , 1 , 1 , 1 if1<p<q< ∞and    ∥ ∥ ( ) ≲ ∥ ∥ ( ) ( ) ∥ ∥ ≲ ∥ ∥ ∥ ∥ ≲ ∥ ∥ − ∈ > − − ( ( )) ( ) − ( ) ( ) f f sup φ t r, r f f f f α WM M t r WL t r M M M Γ Γ 1 Γ, 0 Γ , Γ 1 Γ Γ q φq φ q _{q q} q φ q φ q φ , 1 _1, 1 _{1 1} 1 1 1, 1 1, 1 1, ifp=1<q < ∞. □

The following theorem is another of our main results.

Theorem 5.9. Let Γ be a Carleson curve, < <0 α 1,1≤p<q< ∞andφ∈Ωp.

1. If (φ t r, )satisfies condition (5.25), then condition (5.26) is sufficient for the boundedness of the operator α

fromM ( )Γ

p φ, p1 toWMq φ, q1( )Γ. Moreover, if 1<p<q < ∞, then condition(5.26) is sufficient for the

bound-edness of the operatorα_{fromM ( )Γ}

p φ, p1 toMq φ, q1( )Γ.

2. Ifφ∈p, then the condition

( ) ≤ − − r φ rα p Cr , αp q p 1 (5.28) for allr>0, whereC>0 does not depend on r, is necessary for the boundedness of the operatorα_from

( )

M_{p φ, p}1 Γ toWM ( )Γ

q φ, q1 and fromMp φ, 1p( )Γ toMq φ, q1( )Γ.

3. Ifφ∈psatisfies the regularity condition

∫

( ) ≤ ( ) ∞ − s φ s ds Cr φ r , r α 1 _p1 α _p1 (5.29) for allr>0, whereC>0 does not depend on r, then condition (5.28) is necessary and sufficient for the boundedness of the operatorα_{fromM ( )Γ}

p φ, p1 toWMq φ, q1( )Γ. Moreover, if 1<p<q< ∞, then condition

(5.28) is necessary and sufficient for the boundedness of the operator α _{fromM ( )Γ}

p φ, p1 toMq φ, q1( )Γ.

Proof. The first part of the theorem is a corollary of Theorem 5.8.

We shall now prove the second part. LetΓ0= (Γt0,r0)and ∈t Γ0. By Lemma 5.6, we haver0α≲αχΓ0( )t. Therefore, by Lemmas 3.3 and 5.6 we have

  ≲ ( ( )) ∥− ∥ ≲ ( ) ∥ ∥ ≲ ( ) ∥ ∥ ≲ ( ) ( ) ( ) ( ) − rα νΓ αχ φ r χ φ r χ φ r L α M M Γ 0 0 q Γ qΓ 0q Γ Γ 0 Γ 0 q φq q p φ p q p 1 0 0 1 0 , 1 1 0 , 1 1 1 or ( ) − ≲ > ⇔ ( ) ≲ − − r φ rα 1 for all r 0 r φ rα r . 0 0 p q 0 0 0p 0 αp q p 1 1 1

Since this is true for every ∈t Γ andr0>0, we are done.

The third statement of the theorem follows fromfirst and second parts of the theorem. □

Theorem 10. (Adams-type result). Let Γ be a Carleson curve, < <0 α 1,1≤p<q< ∞andφ∈Ωpsatisfy condition(5.25) and

∫

( ) + ( ) ≤ ( ) ∞ − r φ t rα , s φ t s, ds Cφ t r, , r α 1 qp (5.30)

where C does not depend ont∈Γ andr>0. Then for p> 1 the operatorα _{is bounded fromM ( )Γ} p φ, p1 to

( )

M_{q φ, q}1 Γ and for p=1 the operatorαis bounded fromM_1,φ( )Γ toWM ( )Γ q φ, q1 .

Proof. Let ≤ < ∞1 p andf∈Mp φ, ( )Γ. By Theorem 5.7, inequality(5.23) is valid. Then from condition (5.26) and inequality(5.23), we get

 

∫



∫

| ( )| ≲ ( ) + ∥ ∥ ≤ ( ) + ∥ ∥ ( ) ∞ − − ( ( )) ( ) ∞ − f t r f t s f ds r f t f s φ t s, d .s α α r α L t s α M r α 1 Γ , Γ 1 p p p φ 1 , (5.31)

Thus, by(5.30) and (5.31) we obtain

   

{

}

{

}

| ( )| ≲ ( ) ( ) ( ) ∥ ∥ ≲ ( ) ∥ ∥ = ( ( )) ∥ ∥ − ( ) > − ( ) ( ) − f t φ t r f t φ t r f r f t r f f t f

min , , , sup min ,

, α β M r M M 1 Γ 0 1 Γ Γ 1 p q p φ p q p q p φ p q p φ p q , , , (5.32) where we have used that the supremum is achieved when the minimum parts are balanced. From Theorem 4.3 and(5.32), we get   ∥ ∥ ( ) ≲ ∥ ∥ ( )∥ ∥ ≲ ∥ ∥ − ( ) ( ) f f f f , α M Γ M Γ M M 1 Γ Γ q φq p φ p p q p φ p p q p φ p , 1 , 1 , 1 , 1 if1<p<q< ∞and   ∥ ∥ ( ) ≲ ∥ ∥ ( )∥ ∥ ≲ ∥ ∥ − ( ) ( ) f f f f , α WM Γ M Γ M Γ M 1 Γ q φq φ q φ q φ , 1 _1, 1 1, 1 1, ifp=1<q< ∞. □

The following theorem is another of our main results.

Theorem 5.11. Let Γ be a Carleson curve, < <0 α 1,1≤p<q< ∞andφ∈Ωp.

1. If (φ t r, )satisfies condition (5.25), then condition (5.30) is sufficient for the boundedness of the operator α

fromM ( )Γ

p φ, p1 toMq φ, q1( )Γ. Moreover, if 1<p< q< ∞, then condition(5.30) is sufficient for the

bounded-ness of the operatorα_{fromM ( )Γ}

p φ, p1 toMq φ, q1( )Γ.

2. Ifφ∈p, then the condition

( ) ≤ ( )

r φ rα _p1 Cφ r ,_q1

(5.33) for allr>0, whereC>0 does not depend on r, is necessary for the boundedness of the operatorα_from

( )

M_{p φ, p}1 Γ toWM ( )Γ

q φ, q1 and fromMp φ, 1p( )Γ toMq φ, q1( )Γ.

3. Ifφ∈p satisfies the regularity condition (5.29), then condition (5.33) is necessary and sufficient for the

boundedness of the operatorα _{from M ( )Γ}

p φ, p1 toWMq φ, 1q( )Γ. Moreover, if1<p<q< ∞, then condition

(5.33) is necessary and sufficient for the boundedness of the operator α_{fromM ( )Γ}

p φ, p1 toMq φ, q1( )Γ .

Proof. The first part of the theorem is a corollary of Theorem 5.10.

We shall now prove the second part. LetΓ0= (Γt0,r0)and ∈t Γ0. By Lemma 5.6 we haver0α≤ CαχΓ0( )t. Therefore, by Lemmas 3.3 and 5.6 we have

  ≲ ( ( )) ∥− ∥ ≲ ( ) ∥ ∥ ≲ ( ) ∥ ∥ ≲ ( ) ( ) ( ) ( ) − rα νΓ αχ φ r χ φ r χ φ r L α M M 0 0 q Γ qΓ 0q Γ Γ 0 Γ Γ 0 q φq q p φ p q p 1 0 0 1 0 , 1 1 0 , 1 1 1

or

( ) − ≲ > ⇔ ( ) ≲ ( )

r φ r0α 0 1p 1q 1 for all r0 0 r φ r0α 0p1 φ r0q1. Since this is true for every ∈t Γ andr0>0, we are done.

The third statement of the theorem follows fromfirst and second parts of the theorem. □

Remark 5.4. If we take ( ) =_{φ r r}λ 1− _{in Theorem 5.9, then condition(5.29) is equivalent to < < −0 λ 1 αpand}

condition(5.28) is equivalent to − = − p q α λ 1 1

1 . Therefore, from Theorem 5.9 we get Theorem C.

Remark 5.5. If we take ( ) = [ ]_{φ r r}λ−

1 1in Theorem 5.9, then condition(5.29) is equivalent to < < −0 λ 1 αand

condition(5.28) is equivalent to ≤ − ≤ − α p q α λ 1 1

1 . Therefore, from Theorem 5.9 we get Theorem D.

Acknowledgments: The authors thank the referee(s) for carefully reading the paper and useful comments.

The research of V. Guliyev was supported by cooperation Program 2532 TUBITAK RFBR(RUSSIAN

FOUN-DATION FOR BASIC RESEARCH) under grant no. 119N455 and by the Grant of 1st Azerbaijan-Russia

Joint Grant Competition (Agreement Number no. EIF-BGM-4-RFTF-1/2017-21/01/1-M-08). The research of

H. Armutcu was supported by program TUBİTAK-BİDEB 2211-A (general domestic doctoral scholarship)

under the application no. 1649B031800400.

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