Commutator of fractional integral with Lipschitz functions associated

Guliyev and Akbulut Boundary Value Problems (2018) 2018:80 https://doi.org/10.1186/s13661-018-1002-2

RESEARCH

Open Access

Commutator of fractional integral with Lipschitz functions associated with Schrödinger operator on local generalized Morrey spaces Vagif S. Guliyev1,2,3 and Ali Akbulut1* *

Correspondence: [email protected] 1 Department of Mathematics, Ahi Evran University, Kirsehir, Turkey Full list of author information is available at the end of the article

Abstract Let L = – + V be a Schrödinger operator on Rn , where n ≥ 3 and the nonnegative potential V belongs to the reverse Hölder class RHq1 for some q1 > n/2. Let b belong to a new Campanato space θν (ρ ) and IβL be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators α ,V,{x } [b, IβL ] with b ∈ θν (ρ ) on local generalized Morrey spaces LMp,ϕ 0 , generalized Morrey spaces Mp,α ,Vϕ and vanishing generalized Morrey spaces VMp,α ,Vϕ associated with Schrödinger operator, respectively. When b belongs to θν (ρ ) with θ > 0, 0 < ν < 1 and (ϕ1 , ϕ2 ) satisfies some conditions, we show that the commutator operator [b, IβL ] α ,V,{x }

α ,V,{x0 }

are bounded from LMp,ϕ1 0 to LMq,ϕ2 VMq,α ,Vϕ2 , 1/p – 1/q = (β + ν )/n.

, from Mp,α ,Vϕ1 to Mq,α ,Vϕ2 and from VMp,α ,Vϕ1 to

MSC: 42B35; 35J10; 47H50 Keywords: Schrödinger operator; Fractional integral; Commutator; Lipschitz function; Local generalized Morrey space

1 Introduction and main results Let us consider the Schrödinger operator on Rn , n ≥ 3,

L = – + V

where V is a nonnegative, V = 0, and belongs to the reverse Hölder class RHq for some q ≥ n/2, i.e., there exists a constant C > 0 such that the reverse Hölder inequality

1 |B(x, r)|

B(x,r)

1/q V q (y) dy ≤

C |B(x, r)|

V (y) dy

(1.1)

B(x,r)

holds for every x ∈ Rn and 0 < r < ∞, where B(x, r) denotes the ball centered at x with radius r. In particular, if V is a nonnegative polynomial, then V ∈ RH∞ . © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Guliyev and Akbulut Boundary Value Problems (2018) 2018:80

Page 2 of 14

As in [29], for a given potential V ∈ RHq with q ≥ n/2, we define the auxiliary function ρ(x) :=

1 1 = sup r : n–2 V (y) dy ≤ 1 . mV (x) r>0 r B(x,r)

It is well known that 0 < ρ(x) < ∞ for any x ∈ Rn . Let θ > 0 and 0 < ν < 1, in view of [22], the Campanato class, associated with the Schrödinger operator θν (ρ) consists of the locally integrable functions b such that 1 |B(x, r)|1+ν/n

θ b(y) – bB dy ≤ C 1 + r ρ(x) B(x,r)

(1.2)

for all x ∈ Rn and r > 0. A seminorm of b ∈ θν (ρ), denoted by [b]θβ , is given by the infimum of the constants in the inequality above. Note that if θ = 0, θν (ρ) is the classical Campanato space; if ν = 0, θν (ρ) is exactly the space BMOθ (ρ) introduced in [5]. α,V (Rn ) (including the We now present the definition of generalized Morrey spaces Mp,ϕ weak version) associated with a Schrödinger operator, which was introduced by the first author in [18]. The classical Morrey spaces Lp,λ (Rn ) was introduced by Morrey in [24] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the reader to [9–12, 24, 35]. The generalized Morrey spaces are defined with rλ replaced by a general nonnegative function ϕ(x, r) satisfying some assumptions (see, for example, [15, 23, 25, 30]). For brevity, in the sequel we use the notations r α –n/p α,V Ap,ϕ (f ; x, r) := 1 + r ϕ(x, r)–1 f Lp (B(x,r)) ρ(x) and W ,α,V (f ; x, r) := A,ϕ

r α –n/p 1+ r ϕ(x, r)–1 f WLp (B(x,r)) . ρ(x)

Definition 1.1 Let ϕ(x, r) be a positive measurable function on Rn × (0, ∞), 1 ≤ p < ∞, α,V ,{x } α,V ,{x } α ≥ 0, and V ∈ RHq , q ≥ 1. For any fixed x0 ∈ Rn we denote by LMp,ϕ 0 = LMp,ϕ 0 (Rn ) the local generalized Morrey space associated with Schrödinger operator, the space of all n functions f ∈ Lloc p (R ) with finite norm α,V (f ; x0 , r). f LMα,V ,{x0 } = sup Ap,ϕ p,ϕ

r>0

α,V ,{x }

α,V ,{x }

Also WLMp,ϕ 0 = WLMp,ϕ 0 (Rn ) we denote the weak local generalized Morrey space n associated with Schrödinger operator, the space of all functions f ∈ WLloc p (R ) with W ,α,V (f ; x0 , r) < ∞. f WLMα,V ,{x0 } = sup Ap,ϕ p,ϕ

r>0


α,V ,{x0 }

The local spaces LMp,ϕ the norm

α,V ,{x0 }

(Rn ) and WLMp,ϕ

α,V (f ; x0 , r), f LMα,V ,{x0 } = sup Ap,ϕ p,ϕ

Page 3 of 14

r>0

(Rn ) are Banach spaces with respect to

W ,α,V f WLMα,V ,{x0 } = sup Ap,ϕ (f ; x0 , r), p,ϕ

r>0

respectively. Remark 1.1 α,V ,{x } (i) When α = 0, and ϕ(x, r) = r(λ–n)/p , LMp,ϕ 0 (Rn ) is the local (central) Morrey space {0} LMp,λ (Rn ) studied in [4]. α,V ,{x } {x } (ii) When α = 0, LMp,ϕ 0 (Rn ) is the local generalized Morrey space VMp,ϕ0 (Rn ) were introduced by the first author in [13]; see also [14, 16, 21] etc. Definition 1.2 The vanishing generalized Morrey space associated with the Schrödinger α,V α,V (Rn ) is defined as the spaces of functions f ∈ Mp,ϕ (Rn ) such that operator VMp,ϕ α,V lim sup Ap,ϕ (f ; x, r) = 0.

(1.3)

r→0 x∈Rn

The vanishing weak generalized Morrey space associated with the Schrödinger operator α,V α,V (Rn ) is defined as the spaces of functions f ∈ WMp,ϕ (Rn ) such that VWMp,ϕ W ,α,V lim sup Ap,ϕ (f ; x, r) = 0.

r→0 x∈Rn

α,V α,V The vanishing spaces VMp,ϕ (Rn ) and VWMp,ϕ (Rn ) are Banach spaces with respect to the norm

α,V f VMα,V ≡ f Mα,V = sup Ap,ϕ (f ; x, r), p,ϕ

p,ϕ

x∈Rn ,r>0

α,V f VWMα,V ≡ f WMα,V = sup AW ,p,ϕ (f ; x, r), p,ϕ

p,ϕ

x∈Rn ,r>0

respectively. α,V (Rn ) is the vanishing Morrey space VMp,λ In the case α = 0, and ϕ(x, r) = r(λ–n)/p VMp,ϕ introduced in [33], where applications to PDE were considered. We refer to [3, 20, 27, 28] for some properties of vanishing generalized Morrey spaces. Definition 1.3 Let L = – + V with V ∈ RHq1 , q1 > n/2. The fractional integral associated with L is defined by

IβL f (x) = L–β/2 f (x) =

∞

e–tL (f )(x)t β/2–1 dt

0

for 0 < β < n. The commutator of IβL is defined by

b, IβL f (x) = b(x)IβL f (x) – IβL (bf )(x).


Page 4 of 14

Note that, if L = – is the Laplacian on Rn , then IβL and [b, IβL ] are the Riesz potential Iβ and the commutator of the Riesz potential [b, Iβ ], respectively, that is, Iβ f (x) =

Rn

f (y) dy, |x – y|n

[b, Iβ ]f (x) =

Rn

b(x) – b(y) f (y) dy. |x – y|n

When b ∈ BMO, Chanillo proved in [8] that [b, Iβ ] is bounded from Lp (Rn ) to Lq (Rn ) with 1/q = 1/p – β/n, 1 < p < n/β. When b belongs to the Campanato space ν , 0 < ν < 1, Paluszynski in [26] showed that [b, Iβ ] is bounded from Lp (Rn ) to Lq (Rn ) with 1/q = 1/p – (β + ν)/n, 1 < p < n/(β + ν). When b ∈ BMOθ (ρ), Bui in [6] obtained the boundedness of [b, IβL ] from Lp (Rn ) to Lq (Rn ) with 1/q = 1/p – β/n, 1 < p < n/β. Inspired by the above results, we are interested in the boundedness of [b, IβL ] on generα,V α,V (Rn ) and the vanishing generalized Morrey spaces VMp,ϕ (Rn ), alized Morrey spaces Mp,ϕ θ when b belongs to the new Campanato class ν (ρ). In this paper, we consider the boundedness of the commutator of IβL on the local genα,V ,{x } α,V eralized Morrey spaces LMp,ϕ 0 , the generalized Morrey spaces Mp,ϕ (Rn ) and the vanα,V ishing generalized Morrey spaces VMp,ϕ (Rn ). When b belongs to the new Campanato α,V ,{x } α,V ,{x } space θν (ρ), 0 < ν < 1, we show that [b, IβL ] are bounded from LMp,ϕ1 0 to LMq,ϕ2 0 , α,V α,V α,V α,V from Mp,ϕ (Rn ) to Mq,ϕ (Rn ) and from VMp,ϕ (Rn ) to VMq,ϕ (Rn ) with 1/q = 1/p – (β + ν)/n, 1 < p < n/(β + ν). Our main results are as follows. Theorem 1.1 Let x0 ∈ Rn , b ∈ θν (ρ), V ∈ RHq1 , q1 > n/2, 0 < ν < 1, α ≥ 0, 1 ≤ p < n/(β + α,V ν), 1/q = 1/p – (β + ν)/n and let ϕ1 , ϕ2 ∈ p,loc satisfy the condition

∞

r

n

ess inft 1 n–β–ν

b, I L f

β

α,V ,{x } LMq,ϕ2 0

to

2

≤ C[b]θν f LMα,V ,{x0 } , p,ϕ1

and for p = 1

b, I L f

β

WLM

α,V ,{x0 } n n–β–ν ,ϕ2

≤ C[b]θν f LMα,V ,{x0 } , 1,ϕ1

where C does not depend on f . Corollary 1.1 Let b ∈ θν (ρ), V ∈ RHq1 , q1 > n/2, 0 < ν < 1, α ≥ 0, 1 ≤ p < n/(β + ν), 1/q = 1/p – (β + ν)/n and let ϕ1 ∈ pα,V , ϕ2 ∈ qα,V satisfy the condition r

∞

n

ess inft 1 n–β–ν

b, I L f

α,V Mq,ϕ 2

β

2

≤ C[b]θ f Mα,V , p,ϕ1

and for p = 1

b, I L f

WMα,Vn

β

n–β–ν ,ϕ2

≤ Cf Mα,V , 1,ϕ1

where C does not depend on f . Theorem 1.2 Let b ∈ θν (ρ), V ∈ RHq1 , q1 > n/2, 0 < ν < 1, α ≥ 0, b ∈ θν (ρ), 1 < p < n/(β + α,V α,V ν), 1/q = 1/p – (β + ν)/n, and let ϕ1 ∈ p,1 , ϕ2 ∈ q,1 satisfy the conditions cδ :=

∞

sup ϕ1 (x, t)

δ

x∈Rn

dt 0, and

∞

ϕ1 (x, t) r

dt ≤ C0 ϕ2 (x, r), t 1–β–ν

(1.6)

where C0 does not depend on x ∈ Rn and r > 0. Then the operator [b, IβL ] is bounded from α,V α,V α,V to VMq,ϕ for p > 1 and from VM1,ϕ to VWMα,Vn ,ϕ . VMp,ϕ 1 1 2 n–β–ν

2

Remark 1.2 Note that, in the case of V ≡ 0, ν = 0 Corollary 1.1 and Theorem 1.2 were proved in [19, Corollary 5.5 and 7.5] and in the case of ϕ(x, r) = r(λ–n)/p , ν = 0 in [32, Theorems 1.3 and 1.4]. In this paper, we shall use the symbol A B to indicate that there exists a universal positive constant C, independent of all important parameters, such that A ≤ CB. A ≈ B means that A B and B A.

2 Some technical lemmas and propositions We would like to recall the important properties concerning the critical function. Lemma 2.1 ([29]) Let V ∈ RHq1 with q1 > n/2. For the associated function ρ there exist C and k0 ≥ 1 such that k0 |x – y| –k0 |x – y| 1+k0 ≤ ρ(y) ≤ Cρ(x) 1 + C ρ(x) 1 + ρ(x) ρ(x) –1

for all x, y ∈ Rn . Lemma 2.2 ([2]) Suppose x ∈ B(x0 , r). Then for k ∈ N we have 1 (1 +

2k r N ) ρ(x)

1 (1 +

2k r N/(k0 +1) ) ρ(x0 )

.

(2.1)


Page 6 of 14

According to [5], the new BMO space BMOθ (ρ) with θ ≥ 0 is defined as a set of all locally integrable functions b such that 1 |B(x, r)|

θ b(y) – bB dy ≤ C 1 + r ρ(x) B(x,r)

1 for all x ∈ Rn and r > 0, where bB = |B| B b(y) dy. A norm for b ∈ BMOθ (ρ), denoted by [b]θ , is given by the infimum of the constants in the inequalities above. Clearly, BMO ⊂ BMOθ (ρ). Let θ > 0 and 0 < ν < 1, a seminorm on the Campanato class θν (ρ) is denoted by [b]θν ,

[b]θν := sup

1 |B(x,r)|1+ν/n

x∈Rn ,r>0

B(x,r) |b(y) – bB | dy r θ (1 + ρ(x) )

< ∞.

The Lipschitz space, associated with the Schrödinger operator (see [22]), consists of the functions f satisfying f Lipθν (ρ) := sup

x∈Rn ,r>0

|f (x) – f (y)| |x – y|ν (1 + |x–y| ρ(x)

+

|x–y| θ ) ρ(y)

< ∞.

It is easy to see that this space is exactly the Lipschitz space when θ = 0. Note that if θ = 0 in (1.2), θν (ρ) is exactly the classical Campanato space; if ν = 0, θν (ρ) is exactly the space BMOθ (ρ); if θ = 0 and ν = 0, it is exactly the John–Nirenberg space BMO. The following relations between Lipθν (ρ) and θν (ρ) were proved in [22, Theorem 5]. Lemma 2.3 ([22]) Let θ > 0 and 0 < ν < 1. Then following embedding is valid: θν (ρ) ⊆ Lipθν (ρ) ⊆ ν(k0 +1)θ (ρ), where k0 is the constant appearing in Lemma 2.1. We give some inequalities about the Campanato space, associated with the Schrödinger operator θν (ρ). Lemma 2.4 ([22]) Let θ > 0 and 1 ≤ s < ∞. If b ∈ θν (ρ), then there exists a positive constant C such that

1 |B|

1/s r θ θ ν b(y) – bB s dy ≤ C[b]ν r 1 + ρ(x) B

for all B = B(x, r), with x ∈ Rn and r > 0, where θ = (k0 +1)θ and k0 is the constant appearing in (2.1). Let Kβ be the kernel of IβL . The following result gives the estimate on the kernel Kβ (x, y).


Page 7 of 14

Lemma 2.5 ([6]) If V ∈ RHq1 with q1 > n/2, then, for every N , there exists a constant C such that Kβ (x, y) ≤

C (1 +

|x–y| N ) ρ(x)

1 . |x – y|n–β

(2.2)

Finally, we recall a relationship between an essential supremum and an essential infimum. Lemma 2.6 ([34]) Let f be a real-valued nonnegative function and measurable on E. Then

–1

ess inf f (x)

= ess sup

x∈E

x∈E

1 . f (x) α,V ,{x0 }

It is natural, first of all, to find conditions ensuring that the spaces LMp,ϕ are nontrivial, that is, consist not only of functions equivalent to 0 on Rn .

α,V and Mp,ϕ

Lemma 2.7 Let x0 ∈ Rn , ϕ(x, r) be a positive measurable function on Rn × (0, ∞), 1 ≤ p < ∞, α ≥ 0, and V ∈ RHq , q ≥ 1. If sup 1 + t 0, hence f LMα,V ,{x0 } ≥ sup 1 + p,ϕ

t n/2 and b ∈ θν (ρ) with 0 < ν < 1. Let 0 < β +ν < n and let 1 ≤ p < q < ∞ satisfy 1/q = 1/p – (β + ν)/n. Then for all f in Lp (Rn ) we have

b, I L f

β

Lq (Rn )

f Lp (Rn )

when p > 1, and also

b, I L f

β

WLq (Rn )

f L1 (Rn )

when p = 1. In order to prove Theorem 1.1, we need the following. Theorem 3.1 Suppose V ∈ RHq1 with q1 > n/2, b ∈ θν (ρ), θ > 0, 0 < ν < 1. Let 0 < β + ν < n and let 1 ≤ p < q < ∞ satisfy 1/q = 1/p – (β + ν)/n then the inequality

b, I L f

β

Lq (B(x0 ,r))

Iβ+ν |f | Lq (B(x ,r)) 0 ∞ f Lp (B(x0 ,t)) dt n rq n t 2r tq

p

holds for any f ∈ Lloc (Rn ). Moreover, for p = 1 the inequality

b, I L f

β

WL

n (B(x0 ,r)) n–β–ν

Iβ+ν |f | WL rn–β

∞ 2r

n (B(x0 ,r)) n–β–ν

f L1 (B(x0 ,t)) dt t n–β–ν t

holds for any f ∈ L1loc (Rn ). Proof For arbitrary x0 ∈ Rn , set B = B(x0 , r) and λB = B(x0 , λr) for any λ > 0. We write f as f = f1 + f2 , where f1 (y) = f (y)χB(x0 ,2r) (y), and χB(x0 ,2r) denotes the characteristic function of B(x0 , 2r). Then

b, I L f

β

Lq (B(x0 ,r))

Iβ+ν |f | Lq (B(x

0 ,r))

≤ Iβ+ν f1 Lq (B(x0 ,r)) + Iβ+ν f2 Lq (B(x0 ,r)) .


Page 10 of 14

Since f1 ∈ Lp (Rn ) and from the boundedness of Iβ+ν from Lp (Rn ) to Lq (Rn ) (see [31]) it follows that Iβ+ν f1 Lq (B(x0 ,r)) f Lp (B(x0 ,2r))

n

r q f Lp (B(x0 ,2r))

∞

2r

dt t

n +1 q

n

∞

rq

2r

f Lp (B(x0 ,t)) dt . n t tq

(3.1)

To estimate Iβ+ν f2 Lp (B(x0 ,r)) , the obverse of x ∈ B, y ∈ (2B)c implies |x – y| ≈ |x0 – y|. Then by (2.2) we have supIβ+ν f2 (x)

∞

–n+β |f (y)| 2k+1 r dy n–β–ν |x0 – y|

(2B)c

x∈B

k=1

f (y) dy. 2k+1 B

By Hölder’s inequality we get ∞ –1– np +β f Lp (2k+1 B) 2k+1 r supIβ+ν f2 (x) x∈B

dt 2k r

k=1

∞

2k+1 r

2k r

k=1

f Lp (B(x0 ,t)) dt n t tq

2k+1 r

∞

2r

f Lp (B(x0 ,t)) dt . n t tq

(3.2)

Then Iβ+ν f2 Lq (B(x0 ,r)) r

n q

∞


2r

(3.3)

holds for 1 ≤ p < n/β. Therefore, by (3.1) and (3.3) we get

Iβ+ν |f |

Lq (B(x

n

0

rq ,r))

∞

2r


(3.4)

for 1 < p < n/β. n (Rn ), we get When p = 1, by the boundedness of Iβ+ν from L1 (Rn ) to WL n–β–ν Iβ+ν f1 WL

n n–β–ν

n–β–ν (B(x0 ,r)) f L1 (B(x0 ,2r)) r

∞

2r

f L1 (B(x0 ,t)) dt . t n–β–ν t

By (3.3) we have Iβ+ν f2 WL

n (B(x0 ,r)) n–β–ν

≤ Iβ+ν f2 L rn–β–ν

n (B(x0 ,2r)) n–β–ν

∞


2r

Then

Iβ+ν |f |

WL

n (B(x0 n–β–ν

r ,r))

n–β–ν

∞

2r



Page 11 of 14

Proof of Theorem 1.1 From Lemma 2.6, we have 1 ess inft