Boundedness of the commutator of the intrinsic square function in variable exponent spaces. (English) Zbl 1404.46027
Summary: In this paper, we show that the commutator of the intrinsic square function with BMO symbols is bounded on the variable exponent Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^n)\) applying a generalization of the classical Rubio de Francia extrapolation. As a consequence we further establish its boundedness on the variable exponent Morrey spaces \(\mathcal{M}_{p(\cdot), u}\), Morrey-Herz spaces \(M\dot{K}_{q, p(\cdot)}^{\alpha(\cdot), \lambda}({\mathbb { R}}^n)\) and Herz-type Hardy spaces \(H\dot{K}_{p(\cdot)}^{\alpha(\cdot), q}({\mathbb {R}}^n)\), where the exponents \(\alpha(\cdot)\) and \(p(\cdot)\) are variable. Observe that, even when \(\alpha(\cdot)\equiv \alpha\) is constant, the corresponding main results are completely new.
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 42B35 | Function spaces arising in harmonic analysis |
| 42B25 | Maximal functions, Littlewood-Paley theory |
References:
| [1] | , the variable exponent Lebesgue and some other function spaces arising in analysis such as Besov spaces and Triebel-Lizorkin spaces etc. have been intensively studied in the recent years, see [4, 6, 29, 31, 35, 42, 43] and their references. These spaces are of interest in their own right, and also have applications to fluid dynamics [30], image restoration [2] and PDE with non-standard growth conditions [15]. In many applications, a crucial step has been to prove that the classical operators, such as maximal operators, singular integrals and their commutators, are bounded in variable exponent Lebesgue spaces Lp(·)(Rn). As shown in [5], one of the important methods used for extending the boundedness results from the weighted Lebesgue spaces Lp(ω) to variable Lp(·)(Rn) spaces is the method of Rudio de Francia extrapolation. In Section 2 we show that the commutator [b, Sβ] is bounded on variable Lp(·)(Rn) spaces applying a generalization of the THE COMMUTATOR OF THE INTRINSIC SQUARE FUNCTION941 classical Rubio de Francia extrapolation, which generalizes the corresponding statement in [33, Theorem 3.1] to the variable exponent case. The variable exponent Morrey spaces Mp(·),uare introduced and studied in Ho [16]. Simultaneously, he has given some sufficient conditions on u for the boundedness of fractional integrals and fractional maximal operator on such spaces. In 2017, Ho [18] established the boundedness of the vector-valued intrinsic square function on variable Morrey spaces Mp(·),u, where u ∈ Wp(·) (see Definition 9 below) is a Morrey weight function for Lp(·)(Rn). Hence, it is natural to ask whether the variable exponent Morrey spaces estimates for the commutator [b, Sβ] are still true if b ∈ BMO(Rn)? The main result in Section 3 is to give an affirmative answer to this question. The variable exponent Herz spacesK˙p(·)α,q(Rn) and Morrey-Herz spaces α,λ M ˙Kq,p(·)(Rn) were recently studied by Izuki [20,21], and he also obtained some basic lemmas on generalization of the BMO norms. Under natural regularity assumptions on the exponent α and p, either at the origin or at infinity, Almeida and Direhem in [1] established the boundedness of a wide class of sublinear operators including maximal, potential and Calder´on-Zygmund operators on the α(·),q variable exponent Herz spaces ˙Kp(·)(Rn), where the exponent α is variable as well. Lu and Zhu [27] further generalized some results in [1] and obtained the boundedness for such sublinear operators and their commutators on the Morrey-Herz spaces M ˙Kq,p(·)α(·),λ(Rn) with variable exponent α(·) and p(·). Motivated by the above results, in Section 4 we consider the boundedness properties α(·),λ of the commutator [b, Sβ] on the variable Morrey-Herz spaces M ˙Kq,p(·)(Rn). It is worth pointing out that even in the particular case α(·) ≡ α is constant, the main results are also new. There are several versions of Herz-type Hardy spaces with variable exponents, see [9, 10, 37]. The variable Herz type Hardy spaces H ˙Kp(·)α(·),q(Rn), as well as their atomic decomposition characterizations, have been studied by Dong and Xu in [9]. Using these decompositions, they proved some boundedness results for singular integral operators on these spaces. Following the idea of [9], in Section 5 we further obtain the boundedness of the commutator [b, Sβ] α(·),q from the variable Herz type Hardy spaces H ˙Kp(·)(Rn) to the variable Herz α(·),q spaces ˙Kp(·)(Rn). In general, we define B := B(x, r) = {y ∈ Rn: |x − y| < r}. fBdenotes the integral average of f on B, namely, fB= |B|−1RBf (x)dx. p0(·) means the conjugate exponent defined by 1/p(·) + 1/p0(·) = 1. By S(Rn) and S0(Rn) we denote the Schwartz class and the space of tempered distributions, respectively. For x ∈ R, we denote by [x] the largest integer less than or equal to x. The symbol C stands for a positive constant, which may vary from line to line. The expression f . g means that f ≤ Cg, and f ≈ g means f . g . f . 942L. WANG 2. Preliminaries and lemmas We first recall some basic lemmas and definitions on the variable exponent Lebesgue spaces, see [3, 8] for more information. Given a measurable set E ⊂ Rn, and a measurable function p(·) : E → [1, ∞), let Lp(·)(E) denote the space of all measurable functions f on E such that Z Ip(·)(f ) :=|f (x)|p(x)dx < ∞. E This is a Banach space with respect to the Luxemburg-Nakano norm kf kLp(·)(E)= inf{µ > 0 : Ip(·)(f /µ) ≤ 1}. p(·) The space Lloc(E) is defined by Lp(·)loc(E) = {f : f ∈ Lp(·)(K) for all compact subsets K ⊂ E}. For brevity, we denote p−E:= ess infx∈Ep(x), p+E:= ess supx∈Ep(x), p−:= p−, p Rn+:= p+Rn. Using this notation, we define P(E) := {p(·) : E → [1, ∞) : p−E> 1 and p+E< ∞} and B(E) := {p(·) ∈ P(E) : M is bounded on Lp(·)(E)}, where M is the Hardy-Littlewood maximal operator defined by Z M f (x) = supr−n|f (y)|dy. r>0B(x,r)∩E When p(·) ∈ P(Rn), f ∈ Lp(·)(Rn) and g ∈ Lp0(·)(Rn), the generalized H¨older inequality holds in the form Z (5)|f (x)g(x)|dx ≤ rpkf kLp(·)(Rn)kgkLp0 (·)(Rn), Rn with rp= 1 + 1/p−− 1/p+, see [22, Theorem 2.1]. Diening’s characterization of variable Lp(·)(Rn) spaces on which the maximal operator is bounded has the following important consequence, see [7, Theorem 8.1]. Lemma 1. Let p(·) ∈ P(Rn). Then the following conditions are equivalent: (a) p(·) ∈ B(Rn). (b) p0(·) ∈ B(Rn). (c) p(·)/q ∈ B(Rn) for some 1 < q < p−. (d) (p(·)/q)0∈ B(Rn) for some 1 < q < p−. THE COMMUTATOR OF THE INTRINSIC SQUARE FUNCTION943 Let ω be a weight function on Rn, that is, ω is real-valued, non-negative and locally integrable. For 1 < p < ∞, we say that ω is an Apweight if 1Z1Zp−1 supω(x)dxω(x)1−p0dx< ∞. B|B|B|B|B The following is a direct generalization of the classical Rubio de Francia extrapolation theorem, see [5, Corollary 1.11]. Lemma 2. Given a family F and an open set Ω ⊂ Rn, assume that for some 1 < p0< ∞, and for every ω ∈ Ap0, ZZ f (x)p0ω(x)dx ≤ C0g(x)p0ω(x)dx,(f, g) ∈ F . ΩΩ Let p(·) ∈ P(Rn) be such that there exists 1 < p1< p−with (p(·)/p1)0∈ B(Ω). Then for all (f, g) ∈ F such that f ∈ Lp(·)(Ω) kf kLp(·)(Ω)≤ CkgkLp(·)(Ω). In [33, Theorem 3.1] it was shown that for all 1 < p < ∞ and all ω ∈ Ap, ZZ |[b, Sβ]f (x)|pω(x)dx ≤ C|f (x)|pω(x)dx. RnRn Thus, we can apply Lemmas 1 and 2 to get the following. Theorem 3. Let p(·) ∈ B(Rn). Then we have for all f ∈ Lp(·)(Rn) and b ∈ BMO(Rn), k[b, Sβ]f kLp(·)(Rn)≤ Ckf kLp(·)(Rn). The following Lemmas 4, 5 and 7 are due to Izuki [20] (see also Diening [7]), and Lemma 8 comes from Almeida and Drihem [1, Lemma 2.1]. Lemma 4. Let p(·) ∈ B(Rn). Then we have |B|−1kχBkLp(·)(Rn)kχBkLp0 (·)(Rn)≤ C. Lemma 5. Let p(·) ∈ B(Rn). Then we have for all measurable subsets E ⊂ B, ≤ C,Lp0 (·)(Rn)≤ C |E|δ2, kχBkLp(·)(Rn)|B|kχBkLp0 (·)(Rn)|B| where δ1, δ2are constants with 0 < δ1, δ2< 1. Remark 6. We would like to stress that everywhere below, the constants δ1 and δ2are always the same as in Lemma 5. Lemma 7. If p(·) ∈ B(Rn), b ∈ BMO(Rn), k > j (k, j ∈ N), then we have 1 supk(b − bB)χBkLp(·)(Rn)≈ kbk∗, B⊂RnkχBkLp(·)(Rn) k(b − bBj)χBkkLp(·)(Rn)≤ C(k − j)kbk∗kχBkkLp(·)(Rn), where Bk= B(0, 2k) and χBkis the characteristic function of Bkfor k ∈ Z. 944L. WANG A function α(·) : Rn→ R is called log-H¨older continuous at the origin (or has a log decay at the origin), if there exists a constant Clog> 0 such that Clog |α(x) − α(0)| ≤,x ∈ Rn. log(e + 1/|x|) If, for some α∞∈ R and Clog> 0, there holds Clog |α(x) − α∞| ≤,x ∈ Rn, log(e + |x|) then α(·) is called log-H¨older continuous at infinity (or has a log decay at the infinity). Lemma 8. Let r1, r2> 0 and α(·) ∈ L∞(Rn) be log-H¨older continuous both at origin and at infinity. Then we have 1α+,0 < r≤ r r221/2, r1α(x)≤ Crα(y)2×1,r1/2 < r2≤ 2r1, r1α−,r r22> 2r1, for any x ∈ B(0, r1)\B(0, r1/2) and y ∈ B(0, r2)\B(0, r2/2). 3. Boundedness on variable exponent Morrey spaces We first recall the following definitions given by Ho in [16]. Definition 9. Let p(·) : Rn→ (1, ∞) with 1 < p−≤ p+< ∞. A Lebesgue measurable function u(x, r) : Rn+1+→ R+is said to be a Morrey weight function for Lp(·)(Rn) if, for any x ∈ Rnand r > 0, ∞kχk XB(x,r)Lp(·)(Rn) (6)u(x, 2j+1r) < Cu(x, r). kχB(x,2j+1r)kLp(·)(Rn) j=0 We denote the class of Morrey weight functions by Wp(·). Definition 10. Let p(·) ∈ B(Rn) and u ∈ Wp(·).The variable exponent Morrey space Mp(·),uis the collection of all Lebesgue measurable functions f satisfying 1 kf kMp(·),u=supkf χB(x,R)kLp(·)(Rn)< ∞. x∈Rn,R>0u(x, R) We note that condition (6) is introduced by Ho in [16] to generalize the well known Hardy-Littlewood-Sobolev theorem to the case of variable exponent Morrey spaces on unbounded domains, and is also used to show the FeffermanStein vector-valued maximal inequalities for weighted Morrey spaces, see [17]. For any p(·) ∈ B(Rn), let Kp(·)denote the supremum of those q > 1 such that p(·)/q ∈ B(Rn) and Ep(·)be the conjugate of Kp0(·). The following result can be seen as a special case of the general result in [17] for Banach function spaces. THE COMMUTATOR OF THE INTRINSIC SQUARE FUNCTION945 Proposition 11. Let p(·) ∈ B(Rn). For any 1 < q < Kp(·)and 1 < τ < Kp0(·), there exist constants C1, C2> 0 such that for any x ∈ Rnand r > 0, kχB(x,2jr)kLp(·)(Rn) C12jn(1−1τ)≤≤ Cjn kχB(x,r)kLp(·)(Rn)22q, ∀j ∈ N. Now, let us state the main result in this section. Theorem 12. Suppose 0 < β ≤ 1 and p(·) ∈ B(Rn). If there exists a constant C > 0 such that for any x ∈ Rnand r > 0, u fulfills ∞kχ XB(x,r)kLp(·)(Rn) (7)(j + 1)u(x, 2j+1r) < Cu(x, r), kχB(x,2j+1r)kLp(·)(Rn) j=0 then we have for all f ∈ Mp(·),uand b ∈ BMO(Rn), k[b, Sβ](f )kMp(·),u≤ Ckbk∗kf kMp(·),u. Remark 13. Note that if u ≡ 1, then the variable exponent Morrey spaces Mp(·),ureduce to the variable exponent Lebesgue spaces Lp(·)(Rn). Therefore, the above theorem is a generalization of Theorem 3. Remark 14. Condition (7) is satisfied by a number of Lebesgue measurable functions. For instance, if for any 0 ≤ γ < 1/Ep(·), a weight function u satisfies u(x, 2r) ≤ 2nγu(x, r) for any x ∈ Rnand r > 0, then (7) holds. In fact, for any γ < 1/Ep(·), there always exists a τ < 1/Kp0(·)such that γ < 1 − 1/τ < 1 − 1/Kp0(·)= 1/Ep(·). An application of Proposition 11 gives XB(x,r)kLp(·)(Rn)u(x, 2j+1r)X∞ (j + 1)< C(j + 1)2jn(τ1+γ−1)< C. kχB(x,2j+1r)kLp(·)(Rn)u(x, r) j=0j=0 To be more precise, the weight function u(x, r) = rγ(x), 0 ≤ γ(x) ≤ γ+< 1/Ep(·), satisfies condition (7). In addition, it is easy to check that condition (7) together with Proposition 11 yields u(x, 2r) ≤ Cu(x, r) for any x ∈ Rnand r > 0. Proof of Theorem 12. Let f ∈ Mp(·),u. For any x0∈ Rnand r > 0, we decom0,2r)and h =P∞j=1f χB(x0,2j+1r)\B(x0,2jr). Then we have 1 kχB(x u(x0, r)0,r)[b, Sβ](f )kLp(·)(Rn) 11 ≤kχB(x0,r)[b, Sβ](g)kLp(·)(Rn)+kχB(x0,r)[b, Sβ](h)kLp(·)(Rn) u(x0, r)u(x0, r) := U + V. For U , using u(x0, 2r) ≤ Cu(x0, r) and the Lp(·)(Rn)-boundedness of [b, Sβ], we obtain 1 U ≤ Ckbk∗kf χB(x0,2r)kLp(·)(Rn) u(x0, 2r) 946L. WANG 1 ≤ Ckbk∗supkf χB(x0,R)kLp(·)(Rn) x0∈Rn,R>0u(x0, R) ≤ Ckbk∗kf kM. p(·),u We now turn to estimate V . For any 0 < β ≤ 1, φ ∈ Cβand (y, t) ∈ Γ(x), we have Z (8)(b(x) − b(z))φt(y − z)h(z)dz Rn ∞Z ≤ Ct−nX|b(x) − b(z)||f (z)|dz, j=1RfjT{z:|y−z|≤t} where fRj:= B(x0, 2j+1r) B(x0, 2jr). Note that if x ∈ B(x0, r), (y, t) ∈ Γ(x) and z ∈ fRjT{z : |y − z| ≤ t}, then 111 (9)t ≥(|x − y| + |y − z|) ≥|x − z| ≥(|z − x0| − |x − x0|) > 2j−2r. 222 Thus, from (8), (9) and the Minkowski inequality, we get |[b, Sβ](h)(x)| ZZZ2dydt12 =sup(b(x) − b(z))φt(y − z)h(z)dz Γ(x)φ∈CβRntn+1 Z∞Z∞Z1 ≤ Ct−nX|b(x) − b(z)||f (z)|dz|2dydt2 2j−2r|x−y|<tj=1RfjT{z:|y−z|≤t}tn+1 ∞ ≤ CX1Z|b(x) − b(z)||f (z)|dz (2j+1r)nB(x j=10,2j+1r) ∞ X1Z ≤ C|b(x) − bB(x0,r)||f (z)|dz (2j+1r)nB(x j=10,2j+1r) ∞ X1Z + C|b(z) − bB(x0,r)||f (z)|dz (2j+1r)nB(x j=10,2j+1r) := V1+ V2. For V1, the generalized H¨older inequality implies that ∞ X1 V1≤ C|b(x) − bB(x0,r)|kf χB(x0,2j+1r)kLp(·)(Rn)kχB(x0,2j+1r)kLp0 (·)(Rn). (2j+1r)n j=1 For V2, noting that |bB(x0,2j+1r)− bB(x0,r)| ≤ C(j + 1)kbk∗(see [32, Page 206]), we apply Lemma 7 with B = B(x0, 2j+1r) and obtain ∞ X1Z V2≤ C|bB(x0,2j+1r)− bB(x0,r)||f (z)|dz (2j+1r)nB(x j=10,2j+1r) THE COMMUTATOR OF THE INTRINSIC SQUARE FUNCTION947 ∞ X1Z + C|b(z) − bB(x0,2j+1r)||f (z)|dz (2j+1r)nB(x j=10,2j+1r) ∞ ≤ Ckbk∗X(j + 1)1kf χ (2j+1r)nB(x0,2j+1r)kLp(·)(Rn)kχB(x0,2j+1r)kLp0 (·)(Rn) j=1 ∞ X1 + Ckf χB(x0,2j+1r)kLp(·)(Rn)k(b − bB(x0,2j+1r))χB(x0,2j+1r)kLp0 (·)(Rn) (2j+1r)n j=1 ∞ ≤ Ckbk∗X(j + 1)1kf χ (2j+1r)nB(x0,2j+1r)kLp(·)(Rn)kχB(x0,2j+1r)kLp0 (·)(Rn). j=1 Combining the estimates of V1and V2, by Lemma 4, we get kχB(x0,r)[b, Sβ](h)kLp(·)(Rn) ∞ ≤ Ckbk∗X(j + 1)1kχ (2j+1r)nB(x0,r)kLp(·)(Rn)kf χB(x0,2j+1r)kLp(·)(Rn) j=1 kχB(x0,2j+1r)kLp0 (·)(Rn) ∞kχ ≤ Ckbk∗X(j + 1)B(x0,r)kLp(·)(Rn)u(x j=10,2j+1r)kLp(·)(Rn)0, 2j+1r) 1 ×supkf χB(x0,R)kLp(·)(Rn) x0∈Rn,R>0u(x0, R) ∞kχk ≤ Ckbk∗kf kMp(·),u(j + 1)B(x0,r)Lp(·)(Rn)u(x kχB(x0,2j+1r)kLp(·)(Rn)0, 2j+1r). j=1 Thus, in view of the condition (7), we arrive at the estimate 1 V =kχB(x u(x0, r)0,r)[b, Sβ](h)kLp(·)(Rn) ∞kχk ≤ Ckbk∗kf kMp(·),uX(j + 1)B(x0,r)Lp(·)(Rn)u(x0, 2j+1r) j=10,2j+1r)kLp(·)(Rn)u(x0, r) ≤ Ckbk∗kf kM. p(·),u This completes the proof of Theorem 12. 4. Boundedness on variable exponent Morrey-Herz spaces Let Bl= {x ∈ Rn: |x| ≤ 2l}, Rl= Bl\Bl−1and χl= χRlbe the characteristic function of the set Rlfor l ∈ Z. Definition 15. Let 0 < q ≤ ∞, p(·) ∈ P(Rn) and α(·) : Rn→ R with α(·) ∈ L∞(Rn). The homogeneous Herz space ˙Kp(·)α(·),q(Rn) is defined as the 948L. WANG p(·) class of all f ∈ Lloc(Rn{0 |
| [2] | A. Almeida and D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl. 394 (2012), no. 2, 781-795. · Zbl 1250.42077 |
| [3] | Y. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383-1406. · Zbl 1102.49010 |
| [4] | D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Applied and Numerical Harmonic Analysis, Birkh¨auser, Springer, Heidelberg, 2013. · Zbl 1268.46002 |
| [5] | D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, The maximal function on variable Lpspaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 223-238. · Zbl 1037.42023 |
| [6] | D. Cruz-Uribe, SFO, A. Fiorenza, J. Martell, and C. P´erez, The boundedness of classical operators on variable Lpspaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239-264. · Zbl 1100.42012 |
| [7] | L. Diening, Maximal function on generalized Lebesgue spaces Lp(·), Math. Inequal. Appl. 7 (2004), no. 2, 245-253. · Zbl 1071.42014 |
| [8] | , Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), no. 8, 657-700. · Zbl 1096.46013 |
| [9] | L. Diening, P. Harjulehto, P. H¨ast¨o, and M. R ˙uˇziˇcka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. · Zbl 1222.46002 |
| [10] | B. Dong and J. Xu, Variable exponent Herz type Hardy spaces and their applications, Anal. Theory Appl. 31 (2015), no. 4, 321-353. · Zbl 1349.42049 |
| [11] | D. Drihem and F. Seghiri, Notes on the Herz-type Hardy spaces of variable smoothness and integrability, Math. Inequal. Appl. 19 (2016), no. 1, 145-165. · Zbl 1334.42047 |
| [12] | C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. · Zbl 0222.26019 |
| [13] | , Hpspaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193. THE COMMUTATOR OF THE INTRINSIC SQUARE FUNCTION961 · Zbl 0257.46078 |
| [14] | V. Guliyev, M. Omarova, and Y. Sawano, Boundedness of intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces, Banach J. Math. Anal. 9 (2015), no. 2, 44-62. · Zbl 1311.42049 |
| [15] | Y. Han, M.-Y. Lee, and C.-C. Lin, Atomic decomposition and boundedness of operators on weighted Hardy spaces, Canad. Math. Bull. 55 (2012), no. 2, 303-314. · Zbl 1271.42030 |
| [16] | P. Harjulehto, P. H¨ast¨o, ´U. V. Lˆe, and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), no. 12, 4551-4574. · Zbl 1188.35072 |
| [17] | K.-P. Ho, The fractional integral operators on Morrey spaces with variable exponent on unbounded domains, Math. Inequal. Appl. 16 (2013), no. 2, 363-373. · Zbl 1260.42009 |
| [18] | , Atomic decompositions of weighted Hardy-Morrey spaces, Hokkaido Math. J. 42 (2013), no. 1, 131-157. · Zbl 1269.42010 |
| [19] | , Intrinsic square functions on Morrey and block spaces with variable exponents, Bull. Malays. Math. Sci. Soc. 40 (2017), no. 3, 995-1010. · Zbl 1372.42014 |
| [20] | Y. Hu and Y. Wang, The commutators of intrinsic square functions on weighted Herz spaces, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 4, 1421-1437. · Zbl 1356.42012 |
| [21] | M. Izuki, Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent, Rend. Circ. Mat. Palermo (2) 59 (2010), no. 3, 461-472. · Zbl 1206.42011 |
| [22] | , Fractional integrals on Herz-Morrey spaces with variable exponent, Hiroshima Math. J. 40 (2010), no. 3, 343-355. · Zbl 1217.42034 |
| [23] | O. Kov´aˇcik and J. R´akosn´ık, On spaces Lp(x)and Wk,p(x), Czechoslovak Math. J. 41 (1991), 592-618. · Zbl 0784.46029 |
| [24] | A. K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Adv. Math. 226 (2011), no. 5, 3912-3926. · Zbl 1226.42010 |
| [25] | Y. Liang and D. Yang, Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3225-3256. · Zbl 1318.42027 |
| [26] | S. Lu and L. Xu, Boundedness of rough singular integral operators on the homogeneous Morrey-Herz spaces, Hokkaido Math. J. 34 (2005), no. 2, 299-314. · Zbl 1081.42012 |
| [27] | S. Lu, D. Yang, and G. Hu, Herz Type Spaces and Their Applications, Science Press, Beijing, 2008. |
| [28] | Y. Lu and Y. P. Zhu, Boundedness of some sublinear operators and commutators on Morrey-Herz spaces with variable exponents, Czechoslovak Math. J. 64(139) (2014), no. 4, 969-987. · Zbl 1349.42044 |
| [29] | E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665-3748. · Zbl 1244.42012 |
| [30] | A. Nekvinda, Hardy-Littlewood maximal operator on Lp(x)(R), Math. Inequal. Appl. 7 (2004), no. 2, 255-265. · Zbl 1059.42016 |
| [31] | M. R ˙uˇziˇcka, Electrorheological Fluids: modeling and mathematical theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000. · Zbl 0962.76001 |
| [32] | J. Tan, Z. Liu, and J. Zhao, On some multilinear commutators in variable Lebesgue spaces, J. Math. Inequal. 11 (2017), no. 3, 715-734. · Zbl 1375.42033 |
| [33] | A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Pure and Applied Mathematics, 123, Academic Press, Inc., Orlando, FL, 1986. · Zbl 0621.42001 |
| [34] | H. Wang, Intrinsic square functions on the weighted Morrey spaces, J. Math. Anal. Appl. 396 (2012), no. 1, 302-314. · Zbl 1252.42021 |
| [35] | , Endpoint estimates for commutators of intrinsic square functions in Morrey type spaces, Math. Inequal. Appl. 18 (2015), no. 3, 801-826. · Zbl 1331.42024 |
| [36] | , Commutators of singular integral operator on Herz-type Hardy spaces with variable exponent, J. Korean Math. Soc. 54 (2017), no. 3, 713-732. · Zbl 1366.42018 |
| [37] | H. Wang and H. Liu, The intrinsic square function characterizations of weighted Hardy spaces, Illinois J. Math. 56 (2012), no. 2, 367-381. · Zbl 1284.42063 |
| [38] | H. Wang and Z. Liu, The Herz-type Hardy spaces with variable exponent and their applications, Taiwanese J. Math. 16 (2012), no. 4, 1363-1389. 962L. WANG · Zbl 1260.42012 |
| [39] | M. Wilson, The intrinsic square function, Rev. Mat. Iberoam. 23 (2007), no. 3, 771-791. · Zbl 1213.42072 |
| [40] | , Weighted Littlewood-Paley Theory and Exponential-Square integrability, Lecture Notes in Mathematics, 1924, Springer, Berlin, 2008. |
| [41] | X. Yan, D. Yang, W. Yuan, and C. Zhuo, Variable weak Hardy spaces and their applications, J. Funct. Anal. 271 (2016), no. 10, 2822-2887. · Zbl 1440.42100 |
| [42] | D. Yang, C. Zhuo, and E. Nakai, Characterizations of variable exponent Hardy spaces via Riesz transforms, Rev. Mat. Complut. 29 (2016), no. 2, 245-270. · Zbl 1337.42025 |
| [43] | D. Yang, C. Zhuo, and W. Yuan, Triebel-Lizorkin type spaces with variable exponents, Banach J. Math. Anal. 9 (2015), no. 4, 146-202. · Zbl 1339.46040 |
| [44] | , Besov-type spaces with variable smoothness and integrability, J. Funct. Anal. 269 (2015), no. 6, 1840-1898. · Zbl 1342.46038 |
| [45] | C. Zhuo, Y. Sawano, and D. Yang, Hardy spaces with variable exponents on RD-spaces and applications, Dissertationes Math. (Rozprawy Mat.) 520 (2016), 74 pp. · Zbl 1376.42033 |
| [46] | C. Zhuo |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
