Scholarly article on topic 'Multilinear Commutators of Calderón-Zygmund Operator on Generalized Weighted Morrey Spaces'

Multilinear Commutators of Calderón-Zygmund Operator on Generalized Weighted Morrey Spaces Academic research paper on "Mathematics"

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Academic research paper on topic "Multilinear Commutators of Calderón-Zygmund Operator on Generalized Weighted Morrey Spaces"

Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 710542, 9 pages http://dx.doi.org/10.1155/2014/710542

Research Article

Multilinear Commutators of Calderon-Zygmund Operator on Generalized Weighted Morrey Spaces

Vagif S. Guliyev1,2 and Farida Ch. Alizadeh2

1 Department of Mathematics, Ahi Evran University, 40200 Kirsehir, Turkey

2 Institute of Mathematics and Mechanics, ANAS, AZ1141 Baku, Azerbaijan

Correspondence should be addressed to Vagif S. Guliyev; vagif@guliyev.com Received 3 December 2013; Accepted 15 December 2013; Published 21 January 2014 Academic Editor: Yoshihiro Sawano

Copyright © 2014 V. S. Guliyev and F. Ch. Alizadeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The boundedness of multilinear commutators of Calderon-Zygmund operator Tg on generalized weighted Morrey spaces Mp v(w) with the weight function w belonging to Muckenhoupt's class A p is studied. When 1 < p < m and b = (b1>..., bm), bt e BMO, i = 1,... ,m, the sufficient conditions on the pair ((pl, (p2) which ensure the boundedness of the operator T^ from Mp (w) to Mf (f2 (w) are found. In all cases the conditions for the boundedness of Tg are given in terms of Zygmund-type integral inequalities on ((pl ,<p2), which do not assume any assumption on monotonicity of (pl (x, r), <p2(x, r) in r.

1. Introduction

Let T be a Calderón-Zygmund singular integral operator and b e BMO(r"). A well known result of Coifmanetal. [1] states that if be BMO(r") and T is a Calderón-Zygmund operator, then the commutator operator [b,T]f = T(bf) - bTf is bounded on L^(r") for 1 < p < >x>. The commutators of Calderoón-Zygmund operator play an important role in studying the regularity of solutions of elliptic, parabolic and ultraparabolic partial differential equations of second order (see, [2-7]).

The classical Morrey spaces Mp X were originally intro-ducedbyMorreyin[8] 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 readers to [2-4, 8, 9].

Let b = (b1,..., bm), fcj,and 1 < j <m be locally integrable functions when we consider multilinear commutators as defined by

Rn n (h (x) - h (y))K (x> y) f (y) dy> (i)

where K(x, y) is Calderon-Zygmund kernel. That is, for all distinct x,y e r",andallz with 2\x-z\ < \x-y\, there exist positive constant C and y such that

(i) \K(x,y)\<C\x-y\-n

(ii) \K(x, y) - K(z, y)\ < C(\x - z\yl\x - y\n+y); and

(iii) \K(y, x) - K(y, z)\<C(\x- z\vl\x - y\n+y)

when m = 1, it is the classical commutator which was introduced by Coifman et al. in [1]. It is well known that Calderon-Zygmund operators play an important role in harmonic analysis (see [10-12]).

We define the generalized weighed Morrey spaces as follows.

Definition 1. Let 1 < p < m, f be a positive measurable function on r" x (0, m) and w be non-negative measurable function on r". We denote by Mp f(w) the generalized weighted Morrey space, the space of all functions f e l^(r") with finite norm

WfK,(„) = sup rM-'MBMr^WfW ,

M xeR" ,r> 0 p'

where Lpw(B(x,r)) denotes the weighted Lp-space of measurable functions f for which

,,(B(x,r)) - ll/X£(x,r)ll

f \f(y)\Pw(y)dy]

JB(x,r) !

JB(x,r)

Furthermore, by WMpf(w) we denote the weak general-

nf fnnrtinnc f \Ai] , p,wv

ized weighted Morrey space of all functions f e WLlppCw(Rn)

for which

= sup <p{x,r) 1w(B(x,r)) l/ph

xeR" ,r>0

nWL „ w(B(x,r))

where WLpw(B(x,r)) denotes the weak L^ ^-space of measurable functions f for which

llWL „ w(B(x,r))

X,B(x,r)llwL „

= sup t

t>0 \ J {yeB(x,r)-.\f(y)\>t}

w(y)dy

Up (5)

Remark 2. (1) If w = 1, then Mpf(1) = Mpf is the generalized Morrey space.

(2) If <p(x,r) = w(B(x,r)fK-1)/p, then Mpf(w) = Lp^(w) is the weighted Morrey space.

(3) If <p(x,r) = v(B(x,r)fpw(B(x,r))-1/p, then Mpf(w) = LPyK(v, w) is the two weighted Morrey space.

(4) If w = 1 and <p(x, r) = r(X-n)/p with 0 < X < n, then Mp f(w) = Lis the classical Morrey space and WMp tp(w) = WLpÀ(Rn) is the weak Morrey space.

(5) If f(x,r) = w(B(x,r))-1/p, then Mp,f(w) = L^(R") is the weighted Lebesgue space.

The commutators are useful in many nondivergence elliptic equations with discontinuous coefficients, [2-5]. In the recent development of commutators, Perez and Trujillo-Gonzalez [13] generalized these multilinear commutators and proved the weighted Lebesgue estimates. The weighted Morrey spaces Lp K(w) was introduced by Komori and Shirai [14]. Moreover, they showed that some classical integral operators and corresponding commutators are bounded in weighted Morrey spaces. Feng in [15] obtained the boundedness of the multilinear commutators in weighted Morrey spaces Lp K(w)

for 1 < p < œ> and 0 < k < 1, where the symbol b belongs to bounded mean oscillation (BMO)". Furthermore, was given the weighted weak type estimate of these operators in weighted Morrey spaces of Lp K(w) for p = 1 and 0 < k < 1.

Recently, the generalized weighted Morrey spaces Mpf(w) introduced by Guliyev [16,17]. Moreover, in [16,17] he studied the boundedness of the sublinear operators and their higher order commutators generated by Calderon-Zygmund operators and Riesz potentials in these spaces (see, also [18-20]).

The following statement was proved in [18].

Theorem A. Let 1 < p < <x>, we A p and (f1, <p2) satisfies the condition

[œ essinft<s<mfi (x,s)w(B(x,s))1/p dt

w(B(x,t))up < C<p2 (x, r),

where C does not depend on x and r. Then the operator T is bounded from Mpf (w) to Mpf (w) for p > 1 and from MUfi (w) to WMhf2(w).

Remark 3. Note that, Theorem A was proved in the case w = 1 in [21] and in the case w e Ap and f1(x,r) = (p2(x>r) =

w(B(x, r))(K-1)/p in [14].

Definition 4. BMO(r") is the Banach space modulo constants with the norm || • ||„ defined by

WH, = sup / ,, [ \b{y)-bB( r)\dy<™, (7) *6R» ,r>0 \B(x,r)\ JB(x,r)

where b e Ll°c(Rn) and

B(x,r)

f b (y) dy-

X,r)\ JB(x,r)

In this paper, we prove the boundedness of the multilinear commutators of Calderon-Zygmund operator T-^ from one generalized weighted Morrey space Mp fi (w) to another

Mpv2(w) for 1 < p < rn and b = (b1,...,bm), bt e BMO, i = 1,.. ,,m.

By A < B we mean that A < CB with some positive constant C independent of appropriate quantities. If A < B and B < A, we write A ~ B and say that A and B are equivalent.

2. Main Results

In the following, main results are given. First, we present some estimates which are the main tools to prove our theorems, for the boundedness of the multilinear commutator operators T-^ on the generalized weighted Morrey spaces.

Theorem 5. Let 1 < p < >x>, we A p, b = (bx,... ,bm), bt e BMO, i = 1,...,m, and T^ be a multilinear commutators defined as (6). Then

llTlfl

¡(x0,r))

<C l\bllw(B(x0,r))1/p

/ f i m I

X I ln"

e + -r

w(B(xo,t))-1/p ^

¡(x0,t))

holds for any ball B = B(x0,r) and for all f e L ), where C does not depend on f, x0 e r" and r > 0.

Theorem 6. Let w e A1, b = (b1,...,bm), b¡ e BMO, i = 1,.. ,,m, and T^ be a multilinear commutators defined as (6). Then

WL lJB(x0,r))

< c||S||4W(£ (x0,r)) t

X \ ln"

jB(Xo,t)) <B(x0,t)y

_d£ t

holds for any ball B = B(x0, r) and for all f e L11oc(r"), where C does not depend on f, x0 e r" and r > 0, where O(i) = tlnm(e + t) and \\f\\L^ = II^G/DII^•

Now we give a theorem about the boundedness of the multilinear commutator operator T-^ on the generalized weighted Morrey spaces.

Theorem 7. Let 1 < p < >x>, we A p, and (f1,f2) satisfies the condition

t\ essinft<s<mf1 (x,s)w(B(x,s))1/p dt

w(B(x,t))1/p

< Cf2 (x, r),

where C does not depend on x and r. Let b = (b1,..bm), bt e BMO, i = 1,...,m. Then the operator T^ is bounded from Mnr„ (w) to Mnr„ (w). Moreover,

P'Tl P>Y2 v '

MP,V2 {W)

< \\b\\

Theorem 8. Let w e A1, and (<p1, <p2) satisfies the condition j™ lnm • t) essinf t<s<m(Pi (x,s)w(B(x,s)) dt

t\ ess inf f e + -r /

w(B(x,t))

< C(?2 (x, r) ,

where C does not depend on x and r. Let b = {h1,..., bm), bt e BMO, i = 1,...,m. Then the operator T^ is bounded from M® (w) to WM1f2(w). Moreover,

WM 1>„2 (w)

< \\b\\

"Ms, (w)>

where ®(t) = t\nm(e + t) and \\f\\M^(w) = \\0(|/|)\\^^

When ^1(x,r) = f2(x,r) = w(B(x,r))(K-1)/p, from Theorem 7 we also get the following new result.

Corollary 9. Let 1 < p < rn, 0 < k < 1, we A p,

b = (b1,..., bm), bt e BMO, i = 1,...,m. Then the operator T^ is bounded on Lp K(w) for p > 1 and from L$ K(w) to WM1k(w) forp= 1,where<$(t) = t lnm(e+t) and \\f\\L (w) =

U^lfDKw.

Proof. Let 1 < p < <m, we A p, 0 < k < 1 and b = (b1,...,bm), bt e BMO, i = l,...,m. Then the pair (w(B(x,r))(K-1)/p,w(B(x,r))(K-1)/p) satisfies the condition (11) for p > 1 and the condition (13) for p = 1. Indeed, by Lemma 10 there exists C > 0 and S > 0 such that for all x e r" and t> r:

i t\ nS

w(B(x,t))>C(-) w(B(x,r)). (15)

lnm (e+í) ess inf t<s<^w(B (x> S))K'P dt

r) w(B(x,t))1/p t

'lnm (e+t-\w(B(x,t))(K-1)/p-

j00, m ( t\((t\M „\(Kl)/pdt

< \ lnm(e + -\((-) w(B(x,r))

w(B(x,r))(K-1)/p j™ lnm (e+t-)^)

t\ ft\ "s((k-1)/p) dt

lnm (e + r)T

rJ \r/ t

nS((K-1)/p)<-_

w(B(x,r))(K-1)/p.

(16) □

Note that from Corollary 9 was proved in [15].

3. Some Lemmas

Let R" be the n-dimensional Euclidean space of points x = (x1t..., xn) with norm \x\ = (Yü¡=1 x2)1/2. For x e R" and r > 0, denote B(x, r) the open ball centered at x of radius r. Let ^B(x, r) be the complement of the ball B(x, r), and \B(x, r)\ be the Lebesgue measure of B(x, r).

A weight function is a locally integrable function on R" which takes values in (0, rn) almost everywhere. For a weight w and a measurable set E, we define w(E) = jE w(x)dx, the Lebesgue measure of E by |£|, and the characteristic function of E by xE. Given a weight w, we say that w satisfies the doubling condition if there is a constant D > 0 such that w(2B) < Dw(B) for any ball B. When w satisfies the doubling condition, we denote w e A 2, for short.

If w is a weight function, then we denote the weighted Lebesgue space by Lp(w) = L^(R", w) with the norm

Hi =(j \KX)\P w(x)dx)UP

p,™ VJr» )

when 1 < p < rn = esssupxgR„\f(x)\w(x) when p = >x>.

We recall that a weight function w is in the Mucken-houpt's class Ap, 1 < p < œ>, if

lw]Ap := supiw]a„(b)

= sUp (]fiB w(x)dx) (18)

x ( -— f w(x)1-p dx) < <x>,

V|£| Jb J

where the sup is taken with respect to all the balls B and l/p + 1/p' = 1. Note that, for all balls B we have

[<%) = W-1\\<U\w1%(B) *1 (19)

by Holder's inequality. For p = 1, the class A1 is defined by the condition Mw(x) < Cw(x) with [w]A = supxeR„(Mw(x)/w(x)), and for p = m we define Aœ =

Ul<p<œ A p.

Lemma 10 (see [22]). We have the following:

(1) If w e Ap for some 1 < p < œ>, then w e A 2.

Moreover, for all X > 1 we have

w(XB) < Xnp[w]A w(B).

(2) If w e Athen weh 2. Moreover, for all X > 1 we have

w(XB) < l [w]XA w(B).

(3) If w e Ap for some 1 < p < >x>, then there exist C > 0 and S > 0 such that for any ball B and a measurable set S c B,

w(B)~ V I В|

The following results are proved by Perez and Trujillo-Gonzalez [13].

Lemma 11. Let 1 < p < >x> and w e Ap and suppose that

b = (b1,..bm), bt e BMO, i = 1,...,m, then there exists a constant C > 0 such that

\Ttf(x)\pw(x)dx<c[ \f(x)\pw(x)dx. (23)

Jr" JR"

Although the commutators with BMO function are not of weak type (1,1), they have the following inequality.

Lemma 12. Let w e A TO and suppose that b = (b1,...,bm), bt e BMO, i = 1,...,m, then there exists a constant C > 0 such that

sup^^w(xe r" :\Tif(x)\>t)

Ф(1Ц)

< С sup ,

t>0F Ф(1/^

w(xe rn : |МФ (||S||/)(x)| > t),

where Ф(t) = t lnm(e + t).

Lemma 13. Let w e A1 and suppose that b = (b1,...,bm), bt e BMO, i = 1,...,m, then there exists a constant C > 0 such that

w(xe r" : \Jif(x)\ > X) <C I ®(\f(x)\)w(x)dx,

where Ф(t) = t \nm(e + t).

In this paper, we need the following statement on the boundedness of the Hardy type operator

(H1g)(t) := 1 J lnm (e+ g (r) d^(r), 0 < t < ж,

where ^ be a non-negative Borel measure on (0, ж). Theorem 14. The inequality

ess sup w(t)H1g(t)<c ess sup v (t)g(t) (27)

t>0 t>0 ^ '

holds for all non-negative and non-increasing д on (0, ж) if and only if

w(t) Ç m( t\ dp(r) A, := sup- I ln (e + - )

1 > t Jo V r)

r) esssup 0<s<r V (s)

(21) and с ~ A p

Note that, Theorem 14 is proved analogously to Theorem 4.3 in [21].

Lemma 15 (see [23, Theorem 5, page 236]). Let w e ATO. Then the norm of BMO(R") is equivalent to the norm of BMO(w), where

BMO (w) = \b :\\b\\tw = sup

,r>ow(B(x, r))

\ lb(y)-bB(x,r),wlw(y)dy <ж\,

JB(x,r) J

'B(x,r),w

w (В (x, r

[ b{y)w{y)dy. (x,r)) JB(x,r)

Remark 16. (1) The John-Nirenberg inequality: there are constants C1, C2 > 0, such that for all b e BMO(r") and ¡3>0

| [xeB: | b(x) - bB| > p}| < C1 |B| e-C2l}m', VB c r".

(2) For 1 < p < rn the John-Nirenberg inequality implies that

sup^ \b(y)-bBlPdy) (31)

and for 1 < p < >x> and w e A TO

\\b\l - sup(-l) f \b(y)-bB\pw(y)dy) . (32)

Indeed, it follows from the John-Nirenberg inequality and using Lemma 10 (3), we get

w({xeB:\b (x) -bB\> ¡3})< Cw (B) e for some S > 0. Hence, this inequality implies that

\ \b(y)-bB\Pw(y)dy

= p\ ¡3p-1w({xeB:\b(x)-bB\> I3})dp (34) Jo

< Cw (B) \ pP-1e-C2l3sm>dp = Cw (B) \\b\\{.

To prove the requested equivalence we also need to have the right inequality, that is easily obtained using Holder inequality, then we get (32). Note that (31) follows from (32) in the case w = 1.

The following lemma was proved in [24].

Lemma 17. Let b be afunction in BMO(R"). Let also 1 < p < ot, x e r", and r1,r2 > 0. Then

(k)\l™ №)-b*"2> \Pdy

< C\1 +

where C > 0 is independent of f, x, r1 and r2.

The following lemma was proved in [17].

Lemma 18. (i) Let w e ATO and b be afunction in BMO(R"). Let also 1 < p < >x>, x e r", and r1 ,r2 > 0. Then

< C(1 +

where C > 0 is independent of f, x, r1 and r2.

(ii) Let w e A p and b be afunction in BMO(R"). Let also

1 < p < ot, x e r", and r1,r2 > 0. Then

1 f I,-TT f \h(y)- hB(X.

J1-p (B(x,ri)) JB(x,rl) \

X2),w\

x w(y)1 p dy

< C (1 +

where C > 0 is independent of f, x, r1 and r2.

4. Proof of the Theorems

Proof of Theorem 5. Let p e (1, ot). For arbitrary x0 e r" and r > 0,set B = B(x0, r). Write f = f1 + f2 with f1 = fx2B

and f2 = fXt(2B). Hence

\\Tlf\\L^(B) < WAl^(B) + WTbf2\\L^By (38)

From the boundedness of T- in Lp(w) (see Lemma 11) it follows that:

H^ML^ <WTif1\L,„ #ILll/1lL, = rfcll

\\LPtW(2By

For the term \\T-/2\\L (B), without loss of generality, we

can assume m = 2. Thus, the operator T-jf2 can be divided into four parts

T-J2 (X) = {h (x) - (h)Bw) (b>2 (X) - (b2\w)

x\ K{x,y)f2 {y)dy

+ f K(x,y)(b1 (y)-(b1)Bw)

x{h (y)-(h)B,w)f2 (y)dy -(h m-(*1)b,w)

xf K(x,y)(b2 (y)-(b2)B,w)f2 (y)dy

-(h M-(h)B,w)

xf K(x,y)(b1 (y)-(b1)Bw)f2 (y)dy

= h (x) + h (x) + Is (x) + I4 (x) . For x e B we have

\Trf2 (*) \ < \I1 (*)\ + \I2 (*)\ + \I3 (*) \ + \l4 (X)|

< \b1 (x)-(b1)BJ^b2 (x)-(b2)BJ

J (2B) \xo -y\

\.(2B)\h1 ^Mbw

x\b2 (y)-(b2\w dy

+ \h (x) - (k)BJ\ X\.(2B)\b2 V-Mbw

\f(y)[

xo -y[

(x) - (b2)

\H2B)\b (y)-(biK

\f(y)\t Ix0 -

n%! \b, (y)-(b)

H2B) X -yf

\P \Hp

x \f (y)\ dy ) w (x) dx)

\h (x) - (h)BJ

\b2(y)-(b2)

t(2B) X -y\n x w (x) dx)

( \ \b2 (x) - (b2),

(y)-(h)B,u

\f(y)\dy

H2B) \xo - yf

\f(y)\dy

x w (x) dx)

n%! \bj (x)-(b,)B,w I B^jH2B) |xo -y\n

\P \l'P

x \f (y)\ dy ) w (x) dx)

= I1 + I2 + h + h.

Let us estimate I

1 V 7 J (2B) \xo -y\n

~~w(B)l'P ^ft \b> (y) - (bi)e,w \

\f(y)\dy

L£ dy

w(BB)"' \ \ nib(y)-(bX*

J2r J2r<\x0-y\<t j=i

x\f(y)\dy-di

w(Bf \ \ n\bt (y)-(bt)B,w |

J2r )B(x0,t)1j^1 1 1

x\f(y)\dy-d1 ■

Applying Holder's inequality and by Lemma 18, we get

I1 < w(B)1/p

xT n(\ \b, (y)-(b,)BJP'w(y)1-2p'dy

J2r j=1\JE(x0,t)

x ||f IiiPj^(B(x0,i))

<11 \blw(B)1/p

x\2r (l + ln r ) W'W \LP' (B(xoA)IIJ \LeJB(xo,t))tn+1

t\2\ -up\ \ dt

ln2 (e+-

Lt,,,(B(xo ,t))

xw(B(x0,t))-1lp

Let us estimate I

I2 = (\ \b1 (x) - (b1)B,w\Pw(x)dx)

( \b2 (y)-(b2)BJ x, ,

x \ ^(2B) \x0 -yf if(y)idy 1lw(B)1lp \i{2B)\b2 (y)-(b2\w\\f(y)\

Hb1lw(B)1/p

xl \b2

\b1lw(B)1/p

x\ \ \b2 (y)-(b2)B,w \\f(y)\dy

J2r JB(xo,t)

f+T ■

Applying Holder's inequality and by Lemma 18, we get

h <\\b1\^tw(B)1/P

xT (I \b (y)-(b2)B,w f w(y)1-i dy)

J2r \JB(x0,t) )

\\Lp JB(xo,t)) tn+1

IÎINL«^)11' f (l +^

+ m- )\\w-1li\\ ,, „

r)\\ \\Lp, (B(x0,t))

\\Lpw(B(x0,t)) tn+1

rn\ w(B)i!i

fln2 {e + !r)Wf\\L^^))w(в(x0 .t))-1li -

In the same way, we shall get the result of I3

I3 <\\blW(B)1li [ n {e+r)\\f\K,Mx0,t))

xW(B(x0,t))-1li

In order to estimate I4 note that

f n\bt (x) - (bi)Bw\iw(x)dx)

(2B) \x0 -y\n 7 n(f \bt (x) - (bi)E w\2iw(x)dx)

\f(y)\

\f(y)\

(2B) |xo - y\

n-1li\\

< 1 \\t\\LptW(B(x0,^))\\W \\l, (B(x0,t)) fn+1

i MLpM)r(.BMy

]t(2B) \x0 -y\" By Lemma 18, we get

h ¿Kv(B)1lP L.^r^Tndy- (49)

" "* )'J(2B) \Xq -yj

Applying Holder's inequality, we get

f \f(y)\

Thus, by (50)

I4 < m w(B)1li f \\* J;

\\Lp^Wxo^(B(xo,t))-1li -r-

Summing up I1 and I4,forall p e [1, œ) we get

x f ln (e+-

< m^B)1^ 2

r ) \U \\Lpm(B(x0,t))

xw(B(x0,t))-1li -.

On the other hand,

\\lpJ2B) - lBl \\f\\Lp,J2B) f

<|B| f

2r f*1

LpJB(x0,t)) tn+1

V ' \\ \\LP> (B)hr

\\lpw (B(x0,t))^+1

(47) < w(B)1li \

\\LPiW(B(xQ,t)) \\ ^ \\L , (B(x0,t)) fn+1

-1li\\

< [w]^w(B)1li

)W(B(xo,t)Y1l^.

\\LtiW(B(x0,t)y v v // t

Finally,

\\Tlf\LjB) ¿WfkMB) + Kw(B)1li

x\ lnm (e+-

\\Lr>.,„(B(xl

,t)) (54)

xW(B(x0,t))-1lp

and the statement of Theorem 5 follows by (53).

Proof of Theorem 6. Let p = 1. To deal with this result, we split f as above by f = f1 + f2, which yields

\\Tif\\wL 1w(B) ^ \\Tlfl\\wLhw(B) + \\Tlf2\\wLhw(B)' (55) From the boundedness of T' from L 0 (w) to WL

1,w (see

Lemma 13) it follows that:

\\TifA WL j (B) - \\Tlfu

<\\b\\Jf1\k„ = \\b\\

\\L^m(2B) '

For the last term \\Tif2^^wl (B), without loss of generality, we still assume m = 2. By homogeneity it is enough to assume

X/2 = Hfcjl* = \\b21|, = 1 and hence, we only need to prove that

w({xeB:^[Tlf2 (x)| > 1})

<w(B)\2r ^^(BM)w(B(X0,t)T for all B = B(x0, r). In fact, by Lemma 12, we get w(xeB:^Tlf2 (*)|>1)

< sup-w (x e r" : lTt f, (%)| > t)

t>o 0(1/t) K 1 hJlK n !

1 ^ (58)

< su^^TTTw (x e B : (f2) (x) > t) t>0 0(1/t)

= (xeB:M(0 (f2)) (x) > t) ,

t>0 0(1/t)

where O(i) = t lnm(e + t). We use the Fefferman-Stein maximal inequality

$(t)dx<-\ \f(x)\M$(x)dx, (59)

JtaR»:MfM>t) t Jr»

»:Mf(x)>t) t

for any functions f and $ > 0. This yields w({xeB:M(O (f2))(x)>t})

Xb (x)w(x)dx ( )

t J{x£R":^(f2)(x)>t) (60)

^f ®(f2)(x)M(wXB)(x). t Jr»

w(xeB:liTl^(x)l > 1)

< sup^T^ (xeB:M(0 (f2)) (x) > t) t>0 0 (1/t)

< su^T-t^ \ 0 (f2) (x) M (wxb) (x) .

t>0 t0 (1/t) Jr»

Proof of Theorem 7. By Theorem 5 and Theorem 14 we have for p > 1

jMj, (w) #11, sup V2(x,r)-1

x\ lnm (e+-

WLpJB(x,t))

xw(B(x,t))-1lp j

= jjfcjj sup f2(x,r) 1

*xeR»,r>0

X i0 Xvr (e+7r) WJ WLpJB(x,^)) xw(B(x,t-1))-1/P |

= jjfcjj* sup f2(x,r-1) r—

*xeR»,r>0 r

x\0 lnm\e+t)wj Wh.M-1))

xw(ß(x,t 1))

-Up dt

< jjfcjj sup (p1 (X, r ^

xeR»,r>0

xW(B(x,r-1))-1/PjjfjjL^)) = jjfcjj sup (p1 (x,r)-1w(B (x,r))-1/p

* xeR»,r>0

(62) □

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for careful reading the paper and useful comments. The research of Vagif S. Guliyev was supported by the grant of Ahi Evran University Scientific Research Projects (PYO.FEN.4003.13.003) and (PYO.FEN.4003-2.13.007).

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