Here we discuss the meaning of the assumptions of Theorems1.1and1.2. In [25], we have proved the following theorems:
Theorem 9.3 [25, Theorem 1] Suppose that the parametersp, q, s, tsatisfy 1< q≤p <∞, 1< t≤s <∞, q < t, p < s.
Assume that{Qj}∞j=1⊂Q,{aj}∞j=1⊂Mst(Rn)and{λj}∞j=1⊂ [0,∞)fulfill ajMs
t ≤ |Qj|1s, supp(aj)⊂Qj,
∞ j=1
λjχQj
Mp
q
<∞. (9.1) Thenf≡∞
j=1λjaj converges inS(Rn)∩Lqloc(Rn)and satisfies fMpq
∞ j=1
λjχQj
Mp
q
. (9.2)
Theorem 9.4 [25, Theorem 2] Suppose that the real parametersp, q, Lsatisfy 1< q≤p <∞, L∈N0.
Letf ∈Mpq(Rn). Then there exists a triplet{Qj}∞j=1⊂Q,{aj}∞j=1⊂L∞(Rn)∩PL⊥(Rn) and{λj}∞j=1⊂ [0,∞)such thatf=∞
j=1λjaj inS(Rn)∩Lqloc(Rn)and that, for allv >0
|aj| ≤χQj, {λjχQj}∞j=1
Mp
q(v)fMpq. (9.3) Assumption (1.6) and (1.7) correspond to conditionsp < sandq < t in Theorem9.3, respectively. Assumption (1.13) corresponds top <∞.
According to the counterexample in [52, Proposition 4.1], we know that we can not relax the assumption q < t; if this were true for q=t, then this would contradict to the counterexample in [52, Proposition 4.1].
However, it is not known that we can relax the assumptionp < s.
Remark 9.5 According to the best knowledge of the authors, it seems that there are three decompositions for Morrey spaces.
1. In 2005, Kruglyak and Kuznetsov considered the Calderón–Zygmund decomposition [31].
2. In 2007, the “so called” smooth decomposition is obtained [50]. The key idea is to de- velop the idea obtained in [23,34,61,62]. This decomposition is investigated very inten- sively in [24,32,33,45,47,48]. Later, in [51], by using this atomic decomposition, the above scale turned out to be the one defined by Yang and Yuan [64,65]. See [66–68,70]
for more for this new scale. We refer to [71] for an exhaustive account of these function spaces as well as the results on this decomposition. In particular, remark that Yang, Yuan and Zhuo obtained the smooth decomposition for Musielak–Orlicz spaces in [69]. Using Corollary8.11and the main results in [33], one can obtain the smooth atomic decompo- sition for the Orlicz–Morrey spaces of the third kind. However, the cost that must be paid is the size of the numberN of the moment condition in words of [69, Definition 5.4].
With the results in this paper and the ones in [33],Nmust be large enough. To overcome this disadvantage, one needs another approach, which is the future work.
3. Probably, [27] is the first work on the non-smooth decomposition of functions Hardy–
Morrey space. The paper [25] complements the case when Mφ,Φ(Rn) is the classical Morrey spaceMpq(Rn).
Acknowledgement The research of V. Guliyev was partially supported by the grant of Science Develop- ment Foundation under the President of the Republic of Azerbaijan, Grant EIF-2013-9(15)-46/10/1 and by the grant of Presidium Azerbaijan National Academy of Science 2015. This paper is written during the stay of Y. Sawano in Ahi Evran University. Y. Sawano is thankful to Ahi Evran University for this support of the stay there. Y. Sawano is thankful to Professor Jie Xiao for his pointing out that (7.18) is correct under some restricted conditions. The authors are thankful to Professor Mitsuo Izuki at Okayama University for his careful reading of the manuscript.
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