On the behaviors of rough fractional type sublinear operators on

arXiv:1809.08851v1 [math.FA] 24 Sep 2018

ON THE BEHAVIORS OF ROUGH FRACTIONAL TYPE SUBLINEAR OPERATORS ON VANISHING GENERALIZED WEIGHTED MORREY SPACES ˙ GURB ¨ ¨ FERIT UZ Abstract. The aim of this paper is to get the boundedness of rough sublinear operators generated by fractional integral operators on vanishing generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. Also, rough fractional integral operator and a related rough fractional maximal operator which satisfy the conditions of our main result can be considered as some examples.

1. Introduction and useful informations 1.1. Background. The classical fractional integral (The classical fractional integral operator is also known as Riesz potential.) was introduced by Riesz in 1949 [6], defined by Iα f (x)

= =

−α

0 < α < n, (−∆) 2 f (x) Z 1 f (y) dy γ (α) |x − y|n−α Rn

with

n π 2 2α Γ α2 , γ (α) = Γ n2 − α2

where Γ (·) is the standard gamma function and Iα plays an important role in partial diferential equation as the inverse of power of Laplace operator. Especially, Its most significant feature is that Iα maps Lp (Rn ) continuously into Lq (Rn ), with 1 α n 1 q = p − n and 1 < p < α , through the well known Hardy-Littlewood-Sobolev imbedding theorem (see pp. 119-121,Theorem 1 and its proof in [7]) for Iα . Let Ω ∈ Ls (S n−1 ), 1 < s ≤ ∞, Ω(µx) = Ω(x) for any µ > 0, x ∈ Rn \ {0} and satisfy the cancellation condition Z Ω(x′ )dσ(x′ ) = 0, S n−1

where x′ =

x |x|

for any x 6= 0.

We first recall the definitions of rough fractional integral operator T Ω,α and a related rough fractional maximal operator MΩ,α . 2000 Mathematics Subject Classification. 42B20, 42B25, 42B35. Key words and phrases. Fractional type sublinear operator; rough kernel; vanishing generalized weighted Morrey space; commutator; A sp′ , sq′ weight. 1

˙ GURB ¨ ¨ FERIT UZ

2

Definition 1. Define IΩ,α f (x) =

Z

Ω(x − y) f (y)dy |x − y|n−α

0 < α < n,

Rn

MΩ,α f (x) = sup r>0

1 rn−α

Z

|Ω(x − y)| |f (y)| dy

0 < α < n.

|x−y|1 x∈Rn ϕ(x, t)

1 p

> 0.

From (1.6) and (1.7), we easily know that the bounded functions with compact support belong to V Mp,ϕ (w). On the other hand, the space V Mp,ϕ (w) is Banach space with respect to the following finite quasi-norm 1 kf kV Mp,ϕ (w) = sup kf kLp(B(x,r),w) , n ϕ(x, r) x∈R ,r>0 such that

1 kf kLp(B(x,r),w) = 0, r→0 x∈Rn ϕ(x, r) we omit the details. Moreover, we have the following embeddings: lim sup

V Mp,ϕ (w) ⊂ Mp,ϕ (w) ,

kf kMp,ϕ (w) ≤ kf kV Mp,ϕ (w) .

Henceforth, we denote by ϕ ∈ B (w) if ϕ(x, r) is a positive measurable function on Rn × (0, ∞) and positive for all (x, r) ∈ Rn × (0, ∞) and satisfies (1.6) and (1.7). The purpose of this paper is to consider the mapping properties for the rough fractional type sublinear operators TΩ,α satisfying the following condition Z |Ω(x − y)| (1.8) |TΩ,α f (x)| . |f (y)| dy, x∈ / supp f |x − y|n−α Rn

on vanishing generalized weighted Morrey spaces. Similar results still hold for the operators IΩ,α and MΩ,α , respectively. On the other hand, these operators have not also been studied so far on vanishing generalized weighted Morrey spaces and this paper seems to be the first in this direction. At last, here and henceforth, F ≈ G means F & G & F ; while F & G means F ≥ CG for a constant C > 0; and p′ and s′ always denote the conjugate index of any p > 1 and s > 1, that is, p1′ := 1 − 1p and s1′ := 1 − 1s and also C stands for a positive constant that can change its value in each statement without explicit


4

mention. Throughout the paper we assume that x ∈ Rn and r > 0 and also let B(x, r) denotes x-centred Euclidean ball with radius r, B C (x, r) denotes its complement. For any set E, χE denotes Z its characteristic function, if E is also measurable and w is a weight, w(E) :=

w(x)dx.

E

2. Main Results Our result can be stated as follows. n 1 , q = p1 − α Theorem 1. Suppose that 0 < α < n, 1 ≤ s′ < p < α n , 1 < q < ∞, n−1 Ω ∈ Ls (S ), 1 < s ≤ ∞, Ω(µx) = Ω(x) for any µ > 0, x ∈ Rn \ {0} such that TΩ,α is rough fractional type sublinear operator satisfying (1.8). For p > 1, s′ w (x) ∈ A sp′ , sq′ and s′ < p, the following pointwise estimate (2.1) Z∞ 1 − 1 dt q kTΩ,α f kLq (B(x0 ,r),wq ) . (w (B (x0 , r))) q kf kLp (B(x0 ,t),wp ) (wq (B (x0 , t))) q t 2r

n p q holds for any ball B (x0 , r) and for all f ∈ Lloc p,w (R ). If ϕ1 ∈ B (w ), ϕ2 ∈ B (w ) and the pair (ϕ1 , ϕ2 ) satisfies the following conditions

(2.2)

cδ :=

Z∞

sup x∈Rn

δ

ϕ1 (x, t)

1 dt < ∞ (wq (B (x, t))) t 1 q

for every δ > 0, and

(2.3)

Z∞ r

ϕ2 (x, r) 1 dt . 1 , t q (B (x, t))) (w (B (x, t))) q

ϕ1 (x, t) (wq

1 q

s′ then for p > 1, w (x) ∈ A sp′ , sq′ and s′ < p, the operator TΩ,α is bounded from V Mp,ϕ1 (wp ) to V Mq,ϕ2 (wq ). Moreover, (2.4)

kTΩ,α f kV Mq,ϕ

2 (w

q)

. kf kV Mp,ϕ

1 (w

p)

·

Proof. Since inequality (2.1) is the heart of the proof of (2.4), we first prove (2.1). For any x0 ∈ Rn , we write as f = f1 + f2 , where f1 (y) = f (y) χB(x0 ,2r) (y), f2 (y) = f (y) χ(B(x0 ,2r))C (y), r > 0 and χB(x0 ,2r) denotes the characteristic function of B (x0 , 2r). Then kTΩ,α f kLq (wq ,B(x0 ,r)) ≤ kTΩ,α f1 kLq (wq ,B(x0 ,r)) + kTΩ,α f2 kLq (wq ,B(x0 ,r)) . Let us estimate kTΩ,α f1 kLq (wq ,B(x0 ,r)) and kTΩ,α f2 kLq (wq ,B(x0 ,r)) , respectively.

VANISHING GENERALIZED WEIGHTED MORREY SPACES

5

Since f1 ∈ Lp (wp , Rn ), by the boundedness of TΩ,α from Lp (wp , Rn ) to Lq (wq , Rn ) (see Theorem 3.4.2 in [4]), (1.4) and since 1 ≤ s′ < p < q we get

kTΩ,α f1 kLq (wq ,B(x0 ,r))

≤ kTΩ,α f1 kLq (wq ,Rn ) . kf1 kLp (wp ,Rn ) = kf kLp (wp ,B(x0 ,2r)) . r

n−αs′

kf kLp (wp ,B(x0 ,2r))

Z∞

dt tn−αs′ +1

2r

s′

≈ kw kL q

s′

(B(x0 ,r)) kw

1

. (wq (B(x0 , r))) q

−s′

kL

( sp′ )′

(B(x0 ,r))

Z∞

kf kLp (wp ,B(x0 ,t))

2r

Z∞

kf kLp (wp ,B(x0 ,t)) kw−s kL

Z∞

′ kf kLp (wp ,B(x0 ,t)) kws kL

′

2r

1

. (wq (B(x0 , r))) q

2r

q

′ (B(x0 ,t))

( sp′ )

( sq′ )

(B(x0 ,t))

1 q

. (w (B(x0 , r))) Z∞ −1 1 × kf kLp (wp ,B(x0 ,t)) (wq (B(x0 , t))) q dt. t 2r

Now, let’s estimate the second part (= kTΩ,α f2 kLq (wq ,B(x0 ,r)) ). For the estimate used in kTΩ,α f2 kLq (wq ,B(x0 ,r)) , we first have to prove the below inequality:

(2.5)

|TΩ,α f2 (x)| .

Z∞

kf kLp (wp ,B(x0 ,t)) (wq (B(x0 , t)))

− q1

1 dt. t

2r

By [1] (see pp. 7 in the proof of Lemma2:), we get

(2.6)

|TΩ,α f2 (x)| .

Z∞

2r

dt tn−αs′ +1

kΩ (x − ·)kLs (B(x0 ,t)) kf kLs′ (B(x0 ,t))

dt tn+1−α

.

dt tn−αs′ +1

−1

1 dt t


6

On the other hand, by H¨ older’s inequality we have

kf kLs′ (B(x0 ,t))

=



≤



≤



 

B(x0 ,t)

 

Z

B(x0 ,t)

 

Z

B(x0 ,t)

=

(2.7)

 1′ s

Z

′

 s |f (y)| dy 

 p1 

  p p e |f (y)| |µ (y)| dy  

Z

|µ (y)|



−e p′

1 p e′ s′

 dy 

B(x0 ,t)

 p1

1 1 1  p p e −1 |f (y)| |µ (y)| dy  (wq (B(x0 , t))) q |B(x0 , t)| s′ + q − p

− q1


1

1

1

|B(x0 , t)| s′ + q − p ,

where in the second inequality we have used the following fact: By (1.4), we get the following:

  

Z B(x0 ,t)

|µ (y)|

−e p′



1 p e′ s′

 dy 

≈

kµkLq˜(B(x0 ,t))

=

1′ −1 s ′ 1 1 kws kLq˜(B(x0 ,t)) |B (x0 , t) |1+ q˜ − p˜

=

 Z   

B(x0 ,t)

(2.8)

i 1′ − 1′ h 1 1 s s |B (x0 , t) |1+ q˜ − p˜

=

− sq′

 q |w (y)| dy 

(wq (B(x0 , t)))

− q1

 s1′

s′ s′  |B (x0 , t) |1+ q − p   1

1

1

|B(x0 , t)| s′ + q − p .

At last, substituting (3.10) in [1] and (2.7) into (2.6), the proof of (2.5) is completed. Thus, by (2.5) we get 1

kTΩ,α f2 kLq (wq ,B(x0 ,r))

.

(wq (B(x0 , r))) q ×

Z∞

− q1


1 dt. t

2r

Combining all the estimates for kTΩ,α f1 kLq (wq ,B(x0 ,r)) and kTΩ,α f2 kLq (wq ,B(x0 ,r)) , we get (2.1).

VANISHING GENERALIZED WEIGHTED MORREY SPACES

7

Now, let’s estimate the second part (2.4) of Theorem 1. Indeed, by the definition of vanishing generalized weighted Morrey spaces, (2.1) and (2.3), we have kTΩ,α f kV Mq,ϕ2 (wq )

=

1 kTΩ,α f kLq (wq ,B(x0 ,r)) ϕ2 (x, r) 1 1 (wq (B (x0 , r))) q sup n ϕ (x, r) x∈R ,r>0 2 Z∞ 1 dt × kf kLp (B(x0 ,t),wp ) (wq (B (x0 , t)))− q t sup

x∈Rn ,r>0

.

r

sup

.

x∈Rn ,r>0

×

Z∞

1 1 (wq (B (x0 , r))) q ϕ2 (x, r)

− q1

(wq (B (x0 , t)))

r

.

kf kV Mp,ϕ

p 1 (w )

×

Z∞

h i dt −1 ϕ1 (x, t) ϕ1 (x, t) kf kLp (B(x0 ,t),wp ) t

1 1 (wq (B (x0 , r))) q x∈Rn ,r>0 ϕ2 (x, r)

sup

− q1

(wq (B (x0 , t)))

ϕ1 (x, t)

dt t

r

.

kf kV Mp,ϕ

1 (w

p)

.

At last, we need to prove that lim sup

r→0 x∈Rn

1 1 kTΩ,α f kLq (wq ,B(x0 ,r)) . lim sup kf kLp(wp ,B(x0 ,r)) = 0. r→0 x∈Rn ϕ1 (x, r) ϕ2 (x, r)

But, because the proof of above inequality is similar to Theorem 2 in [3], we omit the details, which completes the proof. Corollary 1. Under the conditions of Theorem 1, the operators MΩ,α and IΩ,α are bounded from V Mp,ϕ1 (wp ) to V Mq,ϕ2 (wq ). Corollary 2. For w ≡ 1, under the conditions of Theorem 1, we get the Theorem 2 in [3]. References [1] A. S. Balakishiyev, E. A. Gadjieva, F. G¨ urb¨ uz and A. Serbetci, Boundedness of some sublinear operators and their commutators on generalized local Morrey spaces, Complex Var. Elliptic Equ., 2017, 1-22. [2] F. G¨ urb¨ uz, On the behaviors of sublinear operators with rough kernel generated by Calder´ onZygmund operators both on weighted Morrey and generalized weighted Morrey spaces, Int. J. Appl. Math. & Stat., 57 (2) (2018): 33-42. [3] F. G¨ urb¨ uz, A class of sublinear operators and their commutators by with rough kernels on vanishing generalized Morrey spaces, arXiv:1804.01001v1 [math. FA]. [4] S. Z. Lu, Y. Ding and D. Yan, Singular integrals and related topics, World Scientific Publishing, Singapore, 2006. [5] B. Muckenhoupt, R.L. Wheeden, Weighted norm inequalities for singular and fractional integrals. Trans. Amer. Math. Soc., 161 (1971), 249-258. [6] M. Riesz, L´int´ egrale de Riemann-Liouville et le probl` eme de Cauchy, Acta Math. 81(1949), 1-222.

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[7] E.M. Stein, Singular integrals and Differentiability Properties of Functions, Princeton N J, Princeton Univ Press, 1970. HAKKARI UNIVERSITY, FACULTY OF EDUCATION, DEPARTMENT OF MATHEMATICS EDUCATION, HAKKARI, TURKEY E-mail address: [email protected]