00 Journal of Inequalities and Applications
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A note on rough singular integrals in Triebel-Lizorkin spaces and Besov spaces
Feng Liu, Huoxiong Wu and Daiqing Zhang*
Correspondence: zhangdaiqing2011@163.com Schoolof MathematicalSciences, Xiamen University, Xiamen, 361005, China
Abstract
This paper is concerned with the singular integral operators along polynomial curves. The boundedness for such operators on Triebel-Lizorkin spaces and Besov spaces is established, provided the kernels satisfy rather weak size conditions both on the unit sphere and in the radial direction. Moreover, the corresponding results for the singular integrals associated to the compound curves formed by polynomial with certain smooth functions are also given. MSC: 42B20; 42B25
Keywords: singular integrals; rough kernels; Triebel-Lizorkin spaces; Besov spaces
ft Spri
1 Introduction
Let Rn, n > 2, be the «-dimensional Euclidean space and Sn-1 denote the unit sphere in Rn equipped with the induced Lebesgue measure da. Let ft e L1(Sn-1) be a homogeneous function of degree zero and satisfy
I ft(u) da(u) = 0. (1.1)
For a suitable function h defined on R+ = {t e R: t > 0} and a polynomial PN with PN(0) = 0, where N is the degree of PN, we define the singular integral operators Th&pN along polynomial curves in Rn by
TWN (f)(x) := p.v. f f (x - Pn (iyi)y) ^(y)hn(lyl) dy. (1.2)
Jr." |y|
For PN(t) = t, we denote by Th. Fefferman [1] first proved that Th is bounded
on Lp(Rn) for 1 < p < to provided that ft satisfies a Lipschitz condition of positive order on Sn-1 and h e Lto(R). Subsequently, Namazi [2] improved Fefferman's result to the case ft e Lq(Sn-1). Later on, Duoandikoetxea and Francia [3] showed that Th is of type (p,p) for 1 <p < to provided that ft e Lq(Sn-1) and h e A2(R+), where Ay(R+), y >0, denotes the set of all measurable functions h on R+ satisfying the condition
/ pR \i/V
+)= sup i R1 \h(t)\v dt) < to.
R>0 \ Jo /
ringer
It is easy to check that ATO(R+) = Lto(R+) C Ay2(R+) C Ay^R+) for 0 < y < Y2 < to. In 1997, Fan and Pan [4] extended the result of [3] to the singular integrals along polyno-
©2013 Liu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
mial mappings provided that n e H1(Sn-1) and h e AY(R+) for y >1 with \1/p - 1/2\ < min{l/2,1/y'}, where H1(Sn-1) denotes the Hardy spaces on the unit sphere (see [5, 6]). In 2009, Fan and Sato [7] showed that Th is bounded on Lp(Rn) for some j > max{y ',2} with |1/p - 1/2\ < min{1/y',1/2} - 1/j, provided that h e Ay (R+) for y >1, and n satisfies the following size condition:
sup ii |n(0)n(u')\(log+ tl) da(0)da(u') < to. (1.3)
^ 'eSn-1J JSn-1xSn-1 \ \(0 - u) •
For the sake of simplicity, we denote
FFj (Sn-1) := {n e L1(Sn-1): n satisfies (1.3)}, Vj > 0.
On the other hand, for h(t) = 1, Fan etal. [8] showed that Th>n,pN is bounded on Lp(Rn) for 2j/(2j -1) <p < 2j provided j > 1 and n e Fj(Sn-1), where
Fj(Sn-1) := jn e L1(Sn-1): sup^J^y')^log da(/) < , Vj > 0.
Moreover, see [9,10] for the corresponding results of the singular integrals in the mixed homogeneity setting.
Remark 1.1 It should be pointed out that the functions class Fj(Sn-1) was originally introduced in Walsh's paper [11] and developed by Grafakos and Stefanov [12] in the study of Lp-boundedness of singular integrals with rough kernels. It follows from [12] that Fj1(Sn-1) C Fj2(Sn-1) for 0 < j2 < j1, and U?>1 Lq(Sn-1) C Fj (Sn-1) for any j >0, moreover,
f| Fj (Sn-1) £ H1 (Sn-1) £ y F j (Sn-1). (1.4)
j>1 j>1
We also remark that condition (1.3) was originally introduced by Fan and Sato in more general form in [7]. In addition, it follows from [7, Lemma 1] that
Fj (S1) c Fp (S1), for j > 0. (1.5)
In this paper, we consider the boundedness of Th>npN on the Triebel-Lizorkin spaces and the Besov spaces, which contain many important function spaces, such as Lebesgue spaces, Hardy spaces, Sobolev spaces and Lipschitz spaces. Let us recall some notations. The homogeneous Triebel-Lizorkin spaces FOp,q(Rn) and homogeneous Besov spaces Bpaq (Rn) are defined, respectively, by
F5q(Rn) := f e S'(Rn): \f \\m
J22-iaq\^i *f \q
< TO } (1.6)
LP(Rn) J
i/(Mn) := f e S'(Rn): \f H^) = (£2-iaq\№ *f \\ V))^ < , (1.7)
where a e R, 0 <p, q <to (p = to), S'(Rn) denotes the tempered distribution class on Rn, ) = <(2if) for i e Z and < e CcTO(Rn) satisfies the conditions: 0 < <(x) < 1; supp(<) c {x: 1/2 < |x| < 2}; <(x) > c > 0 if 3/5 < |x| < 5/3. It is well known that
Fp,2(Rn) = Lp(Rn) (1.8)
for any 1 <p < TO,see [13-15], etc. for more properties of Fp,q(Rn) and i?a,q(Rn).
For h(t) = 1, the operator Th is the classical Calderon-Zygmund singular integral operator denoted by T. In 2002, Chen etal. [16] proved that T is bounded on Fp,q(Rn) provided ft e Lr(Sn-1) for some r >1. Subsequently, Chen and Zhang [17] improved the result of [16] to the case ft e Fp (Sn-1) for some p >2. Furthermore, in 2008, Chen and Ding [18] showed that Th is bounded on Fp,q(Rn) for 1 <p, q < to and a e R if ft e . ^S"-1) and h e Lto(R+). In 2010, Chen et al. [19] extended the result of [18] to the singular integrals along polynomial mappings provided that h e AY(R+) for y >1 with max{|1/p - 1/21, |1/q - 1/21} < min{1/2, 1/y '}.
In light of aforementioned facts, a natural question is the following.
Question Is Th>Q>PN bounded on Fp,q(Rn) if ft e Fp (Sn-1) and h e AY (R+) for some y > 1?
In this paper, we will give an affirmative answer to this question. Our main results can be formulated as follows.
Theorem 1.1 Let Th,ft^N be as in (1.2) and h e AY (R+) for some y >1. Suppose that ft e Fp (Sn-1) for some j > max{2, y '} and satisfies (1.1). Then for a e R and max{|1/p -1/21, |1/q -1/21} < min{1/2,1/Y'} - 1/j, there exists a constant C >0 such that
|| Th,ft,PN f) |_Fp,q(RK) < C\\f ll_Fp,q(Rn),
where C = Cn,p>q>h,aN,Y ,p is independent of the coefficients ofPN.
Theorem 1.2 Let Th,ft^N be as in (1.2) and h e AY (R+) for some y >1. Suppose that ft e Fp (Sn-1) for some j > max{2, y '} and satisfies (1.1). Then for a e R, 1 < q < to and |1/p -1/21 < min{1/2, 1/y'} - 1/j, there exists a constant C >0 such that
lTh,ft,PN ^¿«(rk) < C\\f \JBg'q(R»),
where C = Cn,p>q>h,aN,Y ,p is independent of the coefficients ofPN.
By (1.5) and Theorems 1.1-1.2, we get the following results immediately.
Theorem 1.3 Let Th,ft^N be as in (1.2) and h e AY (R+) for some y >1. Suppose that ft e Fp (S1) for some j > max{2, y '} and satisfies (1.1). Then for a e R and max{|1/p -1/2|, |1/q -1/21} < min{1/2, 1/y'} - 1/j, there exists a constant C >0 such that
lTh,ft,PN (f) |_Fp,q(R2) < CWf WjFp,q(R2), where C = Cpqih,a,N,Y ,p is independent of the coefficients ofPN.
Theorem 1.4 Let Th,npN be as in (1.2) and h e AY (R+) for some y > 1. Suppose that n e Fj (S1) for some j > max{2, y '} and satisfies (1.1). Thenfor a e R, 1 < q < to and \1/p —1/2 \ < min{1/2, 1/y'} - 1/j, there exists a constant C >0 such that
\Th,n,PN (f)liia'q(R2) < CWf \iig'q(R2),
where C = Cp>q>h,a,N,y j is independent of the coefficients ofPN.
Remark 1.2 Obviously, by (1.5) and (1.8), our results can be regarded as the generalization of the results in [8] or [7], even in the special case h(t) = 1 or PN (t) = t. Moreover, by (1.4)-(1.5), our results are also distinct from the ones in [18,19].
Furthermore, by Theorems 1.1-1.4, and a switched method followed from [20], we can establish the following more general results.
Theorem 1.5 Let h e Ay (R+) for some y >1 and n e Fj(Sn-1) for some j > max{2, y'} with satisfying (1.1). Suppose that y is a nonnegative (or nonpositive) and monotonic C1 function on (0, to) such that r(t) := y(ty with \r(t)\ < C, where C is a positive constant which depends only on y. Then
(i) for a e R and max{\1/p -1/2\, \1/q —1/2\} < min{1/2, 1/y'} - 1/j, there exists a constant C >0 such that
\Th,PN,y(f)\pp,q(rn) < CWf WFp,q(rn), where
ThPN.y(f)(x) := p.v. i f (x - Pn(y(\y\))y') n(y)hn(\y\) dy. Jmn \y\
(ii) for a e R, 1 < q < to and \1/p -1/2 \ < min{1/2, 1/y '} - 1/j, there exists a constant C >0 such that
\Th,PN ,y (f)\,ig'q(R") < CWf WBaaq (R«).
The constant C = Cn>p>q>h,a,yN,Y, j is independent of the coefficients of PN.
Theorem 1.6 Let y, h and ,y be as in Theorem 1.5. Suppose that n e Fj (S1) for some j > max{2, y'} with satisfying (1.1). Then
(i) for a e R and max{\1/p -1/2i, \1/q —1/2\} < min{1/2, 1/y'} - 1/j, there exists a constant C >0 such that
\Th,PN,y(f)\fp,q(r2) < CWf llf-a,q(R2);
(ii) for a e R, 1<q < to and \1/p -1/2 \ < min{1/2, 1/y '} - 1/j, there exists a constant C >0 such that
\Th,PN,y(f)\iiP-q(r2) < CWf WBpa'q(r2). The constant C = Cp,q>h,a,yN,y, j is independent of the coefficients of PN.
Remark 1.3 Under the assumptions on y in Theorem 1.5, the following facts are obvious (see [20]):
(i) limt^0 y(t) = 0 and limt^TO |y (t) | = to if y is nonnegative and increasing, or nonpositive and decreasing;
(ii) limt^0 |y(t)| = to and limt^TO y(t) = 0 if y is nonnegative and decreasing, or nonpositive and increasing.
Moreover, the inhomogeneous versions of Triebel-Lizorkin spaces and Besov spaces, which are denoted by Fp,q(Rn) and Bpaq(Rn), respectively, are obtained by adding the term W$ *f \\Lp(R«) to the right-hand side of (1.6) or (1.7) with ^ieZ replaced by ^i>1, where $ e S(Rn), supp($) c{f : |f | < 2}, $(x) > c > 0 if |x|< 5/3. The following properties are well known (see [13,14], for example):
Fp,q(Rn) - Fp,q(Rn) n Lp(Rn) and
\f WFp,q(R") - \\f wfp,q(r") + \\f WLP(R") (a > 0);
Bpaq(Rn) - Bpaq(Rn) n Lp(Rn) and Wf WBS'q(R") - Wf WBaq(R") + Wf WlP(R") (a > 0). Hence, by (1.8)-(1.10) and Theorems 1.5-1.6, we get the following conclusion immediately.
Corollary 1.7 Under the same conditions of Theorems 1.5 and 1.6 with a >0, the operator Th,PN,y is bounded on Fp,q(Rn) and BPaq(Rn), respectively.
The paper is organized as follows. After recalling and establishing some auxiliary lemmas in Section 2, we give the proofs of our main results in Section 3. It should be pointed out that the methods employed in this paper follow from a combination of ideas and arguments in [3,19, 20].
Throughout the paper, we let p' denote the conjugate index of p, which satisfies 1/p + 1/p' = 1. The letter C or c, sometimes with certain parameters, will stand for positive constants not necessarily the same one at each occurrence, but are independent of the essential variables.
2 Auxiliary lemmas
For given polynomial Pn(t) = EN=ibiti, we let Px(t) = bt for X e {1,2,...,N} and P0(t) = 0 for all t e R. Without loss of generality, we may assume that bX = 0 for X e {1,2, ...,N} (or there exist some positive integers 0 < l1 < l2 < ••• < ld < N such that PN (t) = Yfi=i bi;tli with bh = 0 for all i e{1,2,..., d}). Let k e Z and Dk = {y e Rn : 2k < |y| < 2k+1}. For X e{1,2, ...,N} and f e Rn, we define the measures {ak,X}keZ by
Ok-M)= f e-2niPx rnsf^ym dy. JDk |y|
It is clear that
Th,ftPN (f) = Yl OkN *f. (2.1)
We have the following estimates.
Lemma 2.1 Let h e Ay (R+) for some y >1 and ^ e FFj(Sn for some j > 0. For X e {1,2,..., N}, k e Z and * e Rn, there exists a constant C >0 such that (i)
|o~X(*)-5u-i(f)| < C|2(k+1)Xbx*|1/X; (2.2)
|3u(£)| < C(log|2(k+1)X^xf |)-i/P, /or |2(k+1)Xbx*| > 1, (2.3)
where y = max{2, y '}. The constant C is independent of the coefficients ofPX.
Proof By the change of the variables, we have
r 2k+1 r dt
|akX(*)| = a(y')(e-2niPx(t)y'^ -e-2niPx-1(t)y'*)da(y')h(t)d-
J2k JSn-1 t
< CH^HL1(s-1)HhHAy(R+)|2(k+1)Xbx*|. (2.4) On the other hand, it is easy to check that
|5kX(f)|< CmLHsn-1)\\h\\Ay(R+). (2.5) Interpolating between (2.4) and (2.5) implies (2.2). Next, we prove (2.3). Let
Hk(^,^0) =ik
,y, ^ = J e-2niPx(W-e)-çdt.
By Van der Coupt lemma, there exists a constant C >0, which is independent of the coefficients of PX and k such that
|Hk(*,y',0)| < Cmin{1, |2(k+1)Xbx* • (y' - 0)|-1/X}.
For |2(k+1)XbX* | > 1, since t/(log t)j is increasing in (ej, to), we have
|H (*./0)|< C(log2ejXly • y -0)|-1)j (()
\Hk{-*,y,^| < (log |2(k+«xbxf |)i , ()
where n = * /№ |. Let y be as in Lemma 2.1, by the change of the variables and Holder's inequality, we have
C 2k+1
|aX(*)|= I I n(y')e-2niPx(t)y'• * da(y')h(t)
J2k JSn-1
< C\\h\\Ay(R+//2 f n(y')e-2niPx(t)y' * da(y')
\j2k JSn-1
y' dt ^1/y'
< C(lk(*))w, (.7)
Ik (H )=/ i ft(y')e-2n ^(t)y' • ? da (y')
J2k JSn-1
Note that
Ik (£ )=/ i ft(y')e-2n P (t)y'' *da (y')
J2k JSn-1
= f if iP^t)(y'-0)- ? da (/) da(0)-
= if Hk , y', 0 )ft(0) da (y;) da (0).
J J Sn-1xSn-1
Combining (2.6)-(2.7) with the fact that ft e FFP (Sn-1), we get (2.3). This proves Lemma 2.1.
Lemma 2.2 [19, Theorem 1.4] Let d > 2 and P = (Pi,..., Pd) with P,- being real-valued polynomials on Rn. For 1<p, q < to, the operator Mp given by
Mp(f )(x) = sup 1 i f (x - P(y))|dy
r>0 r J |y|<r
satisfies the following Lp(lq, Md ) inequality
< Cp>q
where Cp,q is independent of the coefficients ofPjfor all 1 < j < d.
Lemma 2.3 [21, Proposition 2.3] Let 0 < M < N and H : MM ^ MM, G : MN ^ MN be two nonsingular linear transformations. Let {ak }keZ be a lacunary sequence of positive numbers satisfying infkeZ ak+1/ak > a >1. Let ) e S(RM) and $k(f ) = akM/ak). Define the transformations J and Xk by
J (f)(x) = f (Gt (Ht ® idKN-M )x)
Xk(f)(x) = J-1(($k ® 5rN-m) * Jf)(x).
Here, we use 5r« to denote theDirac delta function on Rn, J-1 denote the inverse transform of J and Gl denote the transpose ofG. We have the following inequalities:
Efel*k (fj)( o|:
q/2\ 1/q
LP(Rn )
LP(Rn )
for arbitrary functions {f} e Lp(lq, RN ) and 1 <p, q < to;
/ \ q/2\ 1/q
ElElwol2) )
jez kez 77
/ \ q/2\ 1/q
E EM)f )
jeZ keZ 77
LP(RN )
for arbitrary functions {gkj}kj e Lp(lq(l2), RN ) and 1 <p, q < to.
^ (RN )
Lemma 2.4 For any X e{1,2, ...,N} and arbitrary functions {gk,j}k,j e Lp(lq(l2), Rn), there exists a constant C > 0, which is independent of the coefficients ofPX such that
E(E|ak,X * gkj'Î
q/2\ 1/q
lp(R")
jeZ keZ
q/2 1/q
(2.10)
Lp(R")
for max{|1/p -1/2|, |1/q - 1/2|} < min{1/2,1/y'}.
Proof Since \\h\\A (R+) < C\\h\\A2(R+) when y > 2, we may assume that 1 < y < 2. By duality, it suffices to prove (2.10) for 2 < p, q < 2y/(2- y). Given functions f } with \\fj}\\L(f>ny(£(q/2)' RK) < 1. It follows from the similar argument as in getting (7.7) in [4] that
/ Kx *gkj(x)]\fj(x)dx < C \gkj(x)|2Mpx fj)(x)dx,
jr" jr"
(2.11)
f 2k+1 f dt Mpx (f)(x)= jx + Px (t)y') imyO | da (?) |h(t)|2-Y dt.
J2k Jsn-1 t
By Holder's inequality, we have
p / />2 ; dA 2
Mpx (fj)(x) < yhy2A;Y(R +) Jsn_\J2k f(x + Px(t)?) |Y /2 y J myO l da (y')
< cf I^(y')|f sup -f f (x + Px(t)y')|Y'/2 dt)IV da (y').
JS"-1 \r>0 r J|t|<r /
By Lemma 2.2 and Minkowski's inequality, we have for y'/2 < u, v < to,
Px (fj)
Lu(Rn)
Thus, by (2.11)-(2.12), we get
(2.12)
Lu(Rn)
£(£|ak,x *gkj\'
q/2\ 1/q
Lp(Rn)
sup / E E\ak,x *gk^(x)|2f (x)dx
IÎPIl(p/2)'(l(q/2)',K«)<^Rn jeZ keZ
C sup / E E^gk,j(x)\2Mpx (fj)(x) dx
IK/i}!.^/..^,^. <1./Rn ^
IIfj}IIL(p/2)' (l(q/2)' ,r") R" jeZ keZ
j}llLp/2)' (¿(q/2)',EK) :
Px fj)
Lu (Rn)
/ \ q/2s 1/q 2
Lfogk/) )
E(Eigk,ji;
q/2\ 1/q
LP(Rn)
LP(Rn )
where we take u = (p/2)' and v = (q/2)'. This completes the proof of Lemma 2.4.
Lemma 2.5 [20, Lemma 2.1] Let r, y be as in Theorem 1.5. Suppose that h e Ay (M+) for some y >1, then we have h(y-1)r(y-1) e Ay(M+).
Lemma 2.6 Let T^pN,y be given as in Theorem 1.5. Then
(i) if y is nonnegative and increasing, TKpn ,y (f) = ^-1^-1 №pN f);
(ii) if y is nonnegative and decreasing, TKpn ,y (f) = -Th(y-1)r(y-1),s,pN (f)
(iii) if y is nonpositive and decreasing, Th^N ,y (f) = ^-1^-1),^ pN (f);
(iv) if y is nonpositive and increasing, ThpN ,y f) = -Th(y-1)r(y-1),âpN f), where Û(y) = ft(-y).
Proof We can get easily this lemma by Remark 1.3 and the similar arguments as in getting [20, Lemma 2.3]. The details are omitted. □
3 Proofs of main results
For a function $ e C0°°(M) such that $(t) = 1 for |t| < 1/2 and $(t) = 0 for |t| > 1. Let ty(t) = $(t2), and define the measures {rk,x} by
)U ty(|2(k+1)jb;f |) -^fê)[] ty (|2(k+1)jj I)
j=X+1 j=x
for k e Z and X e{1,2, ...,N}, where we use convention ]~[je0 aj = 1. It is easy to check that
®k,N = E Tk,X.
In addition, by Lemma 2.1, we can obtain the following estimates (see also in [4, (7.39)])
ICa^)|< C|2(k+1)AbAf|1/A; (3.3)
ICa^)| < C(log|2(k+1)AbAf for |2(k+1)AbAf | > 1,
where y = max{2, y'}. Now, we are in a position to prove our main results.
Proof of Theorem 1.1 It follows from (2.1) and (3.2) that
ThftPNf) = E E TkA *f := £Ba(f).
X=1 keZ
By (3.5), to prove Theorem 1.1, it suffices to prove that for any X e{1,2, ...,N},
lBx(f)lFp,q{rn) < C|f hp*(r") (3.6)
for max{|1/p -1/21,11/q —1/21} < min{1/2,1/ y'} - 1/j and a e R, where C = Cn>h>p,q,a,X,Ytp is independent of the coefficients of PX for X e{1,2, ...,N}. For X e{1,2, ...,N}, we choose a Schwartz function T e S(R+) such that
0 < T(t) < 1, supp(T) c [2-X,2X], J2 Tk(t)2 = 1,
where Tk (t) = T (2kXt). Define the operator Sk by
Sf (f ):= Tk (|bxf |)f (f ). Let ®k(f ) = Tk(|bxf |). It is clear that ®k e S(Rn) and
Skf (x) = ®k * f (x). Observe that we can write
Bxf) = £ T^ Sj+kSj+kf) = £ £ S+k (Tkx * Sj+kf):= E Bf (3.7)
keZ V'eZ ' jez keZ jeZ
Invoking the Littlewood-Paley theory and Plancherel's theorem, we get
lKf)HL2(R«) < cW \tkx(M)\2f(m)|2m,
kez Ej+k
Ej+k = {M e Rn : 2-(j+k+1)X < |6xM I < 2-(/'+k-1)x}. This together with (3.3)-(3.4) yields
|BX(f) | l2 (Rn) < CCj\f \ \ L2 (Rn), where
j-»9, j < -1; 2-j|, j >-1,
and y = max{2, y'}. In other words (by (1.8)),
||BXf)||F2,2(Rn) < CCj\\f (Rn). (3.8) Next, we will show that
||BXf)||fg,q(Rn) < C\f \Fp,q(Rn) (3.9)
for max{|1/p -1/21, |1/q -1/21} < min{1/2, Uy'}, a e R, j e Z and k e {1,2,...,N}. To prove (3.9), it suffices to prove that
EKfc)|'
LP(Rn)
(3.10)
LP(Rn)
for max{|1/p -1/21, |1/q -1/2|} < min{1/2,1/y'} and {gi} e LP(tq,Rn), where C is independent of the coefficients of Pk. In fact, (3.10) implies (3.9), that is,
k (f -MlFpq(Rn)
j22-iaqi% * Bk(f )|'
£K(2-ia*f)|4
Lp(Rn)
Lp(Rn)
^2-iaq№ *f |'
= Clf \\FP,q (Rn),
Lp(Rn)
which leads to (3.9). Now, we return to the proof of (3.10). Using Lemmas 2.3-2.4, the definition of Tk,k and the similar argument in getting [19, Proposition 2.3], one can check that
E(E|Tk,k *gk,i\'
ieZ ^keZ
q/2\ 1/q
Lp(Rn)
E E|gk,i|:
ieZ keZ
q/2 1/q
(3.11)
Lp(Rn)
for maxj|1/p - 1/2|, |1/q - 1/2|| < minjl/2, Uy'}. Using Lemma 2.3 again, for 1 <p,q < to and arbitrary functions {g;}ieZ e Lp(lq, R"), we have
ieZ ^keZ
q/2 1/q
LP (Rn)
(3.12)
LP(Rn )
By duality and (3.11)-(3.12), we have
EBkte)!'
LP(Rn) U{fi}U
lp' (tq' M-n)<
/ (Tk,k * Sj+kgi)(x)fi(x) dx
^rn iez kez
< c sup
'}llLp' (tq1 ,Rn)<
EÎElsf
ieZ keZ
4/2\ 1/q1
E(E|Tk,k * Sj+kgA
q/2\ 1/q
ieZ keZ
q/2\ 1/q
lp(rn)
ieZ keZ
lp(rn)
LP (Rn)
This proves (3.10). Interpolating between (3.8) and (3.9) (see [14, 22]), for max{|1/p -1/21,11/q —1/21} < min{1/2,1/y '} — 1/8 and a e R, we can obtain e e (0,1) such that efi/y > 1 and
K(f)k*m < cce\f h^wy (3.13)
which together with (3.7) implies (3.6) and completes the proof of Theorem 1.1. □
Proof of Theorem 1.2 The proof of Theorem 1.2 is to copy the arguments in proving [19, Theorem 1.2]. By Theorem 1.1 and (1.8), for |1/p — 1/2| < min{1/2,1/ y'} — 1/8, there exists a constant C >0 such that
II Th ,Q,PN f ) II LP(rn) < c\f \\LP (R«). (3.14)
Then for |1/p —1/21 < min{1/2,1/ y '} —1/8, 1 < q < to and a e R,we have
/ \ 1 \Th,n,PN (/,)|JgP.q(R«) = i £2—'* Th,npN f ^IV) )
= (£ IïWn (2—*f) W)'
< c(E2—iaq\\^i *f \\Lp(R«))
■ieZ
= c\f \\jja,q (R«).
Theorem 1.2 is proved. □
Proofs of Theorems 1.5-1.6 Using Lemmas 2.5-2.6 and Theorems 1.1-1.2, we get Theorem 1.5. Theorem 1.6 follows from Theorem 1.5 and Remark 1.1. □
Competing interests
The authors declare that they have no competing interests. Authors' contributions
Three authors worked jointly in drafting and approving the finalmanuscript. Acknowledgements
The authors would like to thanks the referees for their carefulreading and invaluable comments. This work was supported by the NNSF of China (11071200) and the NSF of Fujian Province of China (No. 2010J01013).
Received: 12 April 2013 Accepted:11 September 2013 Published: 07 Nov 2013
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10.1186/1029-242X-2013-492
Cite this article as: Liu et al.: A note on rough singular integrals in Triebel-Lizorkin spaces and Besov spaces. Journal of Inequalities and Applications 2013, 2013:492
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