Φ-Admissible Sublinear Singular Operators and Generalized Orlicz-Morrey Spaces

where 0 ≤ λ ≤ n, 1 ≤ p < ∞. Here and everywhere in the sequel B(x, r) stands for the ball in R of radius r centered at x. Let |B(x, r)| be the Lebesgue measure of the ball B(x, r) and |B(x, r)| = V n r , where V n = |B(0, 1)|. M p,λ (R) was an expansion of L p (R) in the sense that M p,0 (R) = L p (R). We also denote by WM p,λ ≡ WM p,λ (R) the weak Morrey space of all functions f ∈ WLloc Φ (R) for which

We also denote by  , ≡  , (R  ) the weak Morrey space of all functions  ∈  loc Φ (R  ) for which          , = sup ∈R  ,>0 −/          ((,)) < ∞, where   ((, )) denotes the weak   -space (for  loc Φ (R  ) see Definition 4).Morrey found that many properties of solutions to PDE can be attributed to the boundedness of some operators on Morrey spaces.Maximal functions and singular integrals play a key role in harmonic analysis since maximal functions could control crucial quantitative information concerning the given functions, despite their larger size, while singular integrals, Hilbert transform as its prototype, nowadays intimately connected with PDE, operator theory and other fields.
Let  ∈  loc 1 (R  ).The Hardy-Littlewood (H-L) maximal function of  is defined by The Calderón-Zygmund (C-Z) singular integral operator is defined by ( It is well known that the maximal and singular integral operators play an important role in harmonic analysis (see [2,3]).
We find it convenient to define the generalized Orlicz-Morrey spaces in the following form (see Definition 3 for the notion of Young functions).Definition 1.Let (, ) be a positive measurable function on R  × (0, ∞) and Φ a Young function.We define the generalized Orlicz-Morrey space  Φ, (R  ) as the space of all functions  ∈ for some ball  which contains , with proving the absolutely convergence of the integral in the second term and the independence of the choice of the ball  (see [12,13] for example).Also,  ∞ comp (R  ) is dense in Orlicz spaces  Φ (R  ) if and only if Φ satisfies the Δ 2 condition.
The main purpose of this paper is to find sufficient conditions on general Young function Φ and the functions  1 ,  2 ensuring that the sublinear operators generated by singular integral operators are of weak or strong type from generalized Orlicz-Morrey spaces  Φ, 1 (R  ) into  Φ, 2 (R  ).Note that the Orlicz-Morrey spaces were introduced and studied by Nakai in [12,14].Also the boundedness of the operators of harmonic analysis on Orlicz-Morrey spaces see also, [9,10,[13][14][15][16][17].
By  ≲ , we mean that  ≤  with some positive constant  independent of appropriate quantities.If  ≲  and  ≲ , we write  ≈  and say that  and  are equivalent.
Recall that a function Φ is said to be quasiconvex if there exist a convex function  and a constant  > 0 such that Let Y be the set of all Young functions Φ such that 0 < Φ () < +∞ for 0 <  < +∞.
Definition 4 (Orlicz space).For a Young function Φ, the set is called Orlicz space.The space  loc Φ (R  ) endowed with the natural topology is defined as the set of all functions  such that   ∈  Φ (R  ) for all balls  ⊂ R  .
For Young functions Φ and Ψ, we write Φ ∼ Ψ if there exists a constant  ≥ 1 such that For a Young function Φ and 0 ≤  ≤ +∞, let If Φ ∈ Y, then Φ −1 is the usual inverse function of Φ.We note that for some  > 1.
For a Young function Φ, the complementary function Φ() is defined by The complementary function Φ is also a Young function and It is known that for  ∈ R  and  > 0; (2)  is bounded in Φ(R).
In the case Φ() =   the Φ-admissible singular operator will be called the -admissible singular operator.
In the case Φ() =   the weak Φ-admissible singular operator will be called weak -admissible singular operator.
Necessary and sufficient conditions on Φ for the boundedness of  in Orlicz spaces  Φ (R  ) have been obtained in [ [20, page 15]).With this remark taken into account, the known boundedness statement runs as follows.
The following theorem was in fact proved in [11].
Theorem 8 (see [11]).Let Φ be any Young function.Then the maximal operator  is bounded from Sufficient conditions on Φ for the boundedness of the singular integral operator  in Orlicz spaces  Φ (R  ) are known; see [20,Theorem 1.4.3] and [17,Theorem 3.3].The following theorem was proved in [21].
Remark 10.Note that, from Theorems 8 and 9 we get the any Young function Φ the maximal operator  and for the Young function Φ ∈ Δ 2 the singular integral operator  are the weak Φ-admissible singular operator.Also for the Young function Φ ∈ ∇ 2 the maximal operator  and for the Young function Φ ∈ Δ 2 ∩ ∇ 2 the singular integral operator  are the Φadmissible singular operator.
Definition 11 (generalized Orlicz-Morrey space).Let (, ) be a positive measurable function on R  × (0, ∞) and Φ be any Young function.We denote by  Φ, (R  ) the generalized Orlicz-Morrey space, the space of all functions  ∈  loc Φ (R  ) with finite quasinorm Also, by  Φ, (R  ) we denote the weak generalized Orlicz-Morrey space of all functions  ∈  loc Φ (R  ) for which According to this definition, we recover the Orlicz space  Φ and weak Orlicz space  Φ under the choice  ≡ 1: Also according to this definition, we recover the generalized Morrey space  , and weak generalized Morrey space  , under the choice Φ() =   : The following statement, containing Guliyev results obtained in [22][23][24], was proved in [25] (see also [26]).
Theorem 12. Let 1 ≤  < ∞, and ( 1 ,  2 ) satisfies the condition where  does not depend on  and .Then, for 1 <  < ∞ a -admissible sublinear singular operator  is bounded from  , 1 (R  ) to  , 2 (R  ) and for 1 ≤  < ∞ a weak -admissible sublinear singular operator  is bounded from We will use the following statement on the boundedness of the weighted Hardy operator: where  is a weight.
Theorem 13.Let V 1 , V 2 and  be weights on (0, ∞) and V 1 () be bounded outside a neighborhood of the origin.The inequality holds for some  > 0 for all non-negative and non-decreasing  on (0, ∞) if and only if Moreover, the value  =  is the best constant for (28).

Φ-Admissible Sublinear Singular Operator in the Spaces 𝑀 Φ,𝜑
In this section, sufficient conditions on  for the boundedness of the Φ-admissible sublinear singular operator  in generalized Orlicz-Morrey spaces  Φ, (R  ) are obtained.
Proof.Let Φ be any Young function and the operator  be a Φ-admissible sublinear singular operator.With the notation 2 = ( 0 , 2), we represent  as Thus, On the other hand, by (20) we get and then Thus, Let the operator  be a weak Φ-admissible sublinear singular operator.Then, by the weak boundedness of  on Orlicz space and (39) it follows that Then by (39) and (44), we get inequality (31).