In this paper, we are focused on the decomposition of generalized Orlicz-Morrey spaces of the third kind and its applications. The generalized Orlicz–Morrey space Mφ,Φ(Rn) of the third kind is defined as the set of all measurable functionsf for which norms. 4], the decomposition of functions requires the vector-valued inequality for the maximal Hardy–Littlewood operator, and the synthesis of functions requires duality.
If φ (t )=Φ−1(t−n), then Mφ,Φ(Rn)=LΦ(Rn), which is beyond the scope of generalized Orlicz–Morrey spaces of the second kind defined in [54] for example, constructed in [14]; see Definition 9.1 for its definition. Our strategy for the proof of Theorems 1.1 and 1.2 is as follows: The proof of Theorem 1.1 depends on duality. Meanwhile, to prove Theorem 1.2, we will transform the generalized Orlicz–Morrey space Mφ,Φ(Rn) of the third order into the generalized Hardy–Orlicz–Morrey space HMφ,Φ(Rn) of the third order, as we do in (5.1).
By "cube" we mean a compact cube whose edges are parallel to the coordinate axes, namely the metric sphere defined by ∞ is called a cube. As an auxiliary step, in Section 3 we will prove the boundedness of the Hardy–Littlewood maximal operator on generalized Orlicz–Morrey spaces of the third order. The converse theorems 1.1 and 1.2 for Hardy spaces with variable exponents and Hardy–Orlicz spaces are proved in [40] and [41], respectively.
We apply Theorems 1.1 and 1.2 to prove that the singular integral operators are bounded on generalized Orlicz-Morrey spaces of the third kind in section 6. A function is said to belong to Bφ,Ψ(Rn) if there is a series λ= {λj}∞j=1∈1(N) and a set{Aj}∞j=1of(φ, Ψ )-blocks of the third kind exists such that. This means that we can assume that f is expressed as a finite linear combination of (φ, Ψ )-blocks of the third kind or even that fitself is a (φ, Ψ )-block of the third kind.
In this section we aim to recall a boundedness criterion for the Hardy-Littlewood operatorM defined in (1.19) and to extend the Fefferman-Stein vector-valued inequality from Lp(q,Rn) to Mφ,Φ(q,Rn). The vector-valued generalized Orlicz–Morrey spaceMφ,Φ(q,Rn) of the third kind is the set of all sequencesF= {fj}∞j=1of measurable functions for which. We say thatf∈HMφ,Φ(Rn), the generalized Hardy–Orlicz–Morrey space of the third kind if and only if iff∈S(Rn) and it satisfies sup.
If ∈S(Rn)andMf∈Mφ,Φ(Rn), then a pointwise estimate is sup. 5.5) allows us to invoke Proposition 5.1; we first get. We shall prove the Olsen inequality on generalized Orlicz-Morrey spaces of the third kind.
Boundedness of the Fractional Integral Operator
Olsen’s Inequality Revisited
By the Fefferman–Stein inequality for generalized Morrey spaces of the third kind; see Theorem 3.5, we can remove the maximum operator and we get. 8 More general form of the boundedness of the maximal operator In this section we look at the case when φ also depends on x. In this case, the generalized Orlicz–Morrey spaceMφ,Φ(Rn) of the third kind is defined as the set of all measurable functionsf for which the norm.
Likewise, the weak generalized Orlicz-Morrey space WMφ,Φ(Rn) of the third kind is defined as the set of all measurable functions for which the norm. We can use Corollary8.4 to consider generalized Besov–Orlicz–Morrey spaces of the third kind, as we did in. Here and below, by a "weight" we mean a measurable function which is finite and positive almost everywhere.
On the other hand, the estimate for MF2 is valid based on an estimate corresponding to Lemma3.4. By the weak-type vector valued maximal inequality for Orlicz spaces (see [30]) we have MF1(·). On the other hand, the estimate for MF2 is again valid based on an estimate corresponding to Lemma3.4.
Suppose we are given a function YoungΦandφ1, φ2∈GΦ such that a uniform estimate overx∈Rnandr >0;. Note that [69, Lemma 2.13] is called the modular inequality, while Corollary 8.11 is the vector-valued norm inequality. If we apply these results, then our results carry over to the case where Φ depends on x.
Comparison of Many Generalized Orlicz–Morrey Spaces
According to the examples in [14], we can say that the LandM˜ scales are different and that M˜ and Mare are different.
Comparison of the Assumptions of the Theorems
The spaces Lφ,Φ(Rn),M˜φ,Φ(Rn) and Mφ,Φ(Rn) are defined by Nakai in [37] (with Φ independent of x), by Sawano, Sugano and Tanaka in [54] ( with Φindependent ofx) and by Deringoz, Guliyev and Samko in [6, Definition 2.3], respectively. Using Corollary 8.11 and the main results in [33], the smooth atomic decomposition for Orlicz-Morrey spaces of the third kind can be obtained. However, the cost to be paid is the magnitude of the moment condition number N in the words of [69, Definition 5.4].
Guliyev was partially supported by the grant of the Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF and by the grant of the Presidium of the Azerbaijan National Academy of Science 2015. The authors are grateful to Professor Mitsuo Izuki of Okayama University for his careful reading of the manuscript. Akbulut, A., Guliyev, V.S., Mustafayev, R.: On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces.
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