Weak Type Inequalities for Some Integral Operators on Generalized Nonhomogeneous Morrey Spaces

We prove weak type inequalities for some integral operators, especially generalized fractional integral operators, on generalized Morrey spaces of nonhomogeneous type.The inequality for generalized fractional integral operators is proved by using two different techniques: one uses the Chebyshev inequality and some inequalities involving the modified Hardy-Littlewood maximal operator and the other uses a Hedberg type inequality and weak type inequalities for the modified Hardy-Littlewoodmaximal operator. Our results generalize the weak type inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces and extend to some singular integral operators. In addition, we also prove the boundedness of generalized fractional integral operators on generalized non-homogeneous Orlicz-Morrey spaces.


Introduction
In this paper, we prove that the theory of generalized Morrey spaces can be staged on the nondoubling setting on R  , so that we assume that  is a positive Borel measure on R  satisfying the growth condition; that is, there exist  ∈ [0, ] and   > 0 such that  ( (, )) ≤     (1) for any ball (, ) centered at  ∈ R  with radius  > 0 (see [1][2][3][4]).For 0 <  ≤  and a measurable function  : (0, ∞) → (0, ∞), we define the generalized fractional integral operator   by
These two conditions imply that  satisfies the doubling condition.Note that if () =  −/ , then  , () =   () is the nonhomogeneous Lebesgue space.
The study of the boundedness of the fractional integral operator   on generalized Morrey spaces was initiated in [15,Theorem 3].The following theorem presents the weak type inequalities for   on generalized nonhomogeneous Morrey spaces.
The proof of Theorem 1 employs some inequalities involving the modified Hardy-Littlewood maximal operator   (see [8]), which is defined for any locally integrable function  by and the Chebyshev inequality which is presented in the following theorem.
Theorem 3 (see [20]).Let  be a measurable subset of R  .If  is an integrable function on , then, for every  > 0, one has One of the reasons why we are fascinated with the generalized fractional integral operators is that these operators appear naturally in the context of differential equations; see [21,Section 6.4] for a nice explanation in connection with the holomorphic calculus of operators and see [22, (4.3)] and [23,Lemma 2.5] for a detailed account that (1 − Δ) −/2 with  > 0 satisfies the requirement of  in the present paper.In addition, investigating generalized Morrey spaces is not a mere quest to the abstract theory; it arises naturally in the context of Sobolev embedding.In [24], the following proposition is proved.Proposition 4 (see [24,Theorem 5.1]).Let 1 <  < ∞ and 0 <  < .Then, there exists a positive constant  , such that holds for all  ∈  , (R  ) with (1 − Δ) /2  ∈  , (R  ) and for all balls , where  , is the abbreviation of  , with () =  − .
Later Proposition 4 is strengthened by [25,Example 5].An example in [24] as well as the necessary and sufficient condition obtained in [25,Theorem 1.3] implicitly shows that the log factor above is absolutely necessary.
In this paper, we will prove the weak type inequalities for   which is a generalization of Theorem 1.In Section 2, we will prove the weak type inequalities for   by using the Chebyshev inequality and some inequalities involving operator   .In Section 3, we will prove a Hedberg type inequality on generalized nonhomogeneous Morrey space by adapting the proof of a Hedberg type inequality on homogeneous setting in [25].Through the weak type inequalities for   , we then prove the weak type inequalities for   on generalized nonhomogeneous Morrey spaces.In Section 4, we extend our results to the singular integral operators defined in [1].Finally, in Section 5, we prove the boundedness of   on generalized nonhomogeneous Orlicz-Morrey spaces.See [26][27][28] for related results.
Throughout the paper,  denotes a positive constant which is independent of the function  and the variable  and may have different values from line to line.We also denote by   ( ∈ N) the fixed constants that satisfy certain conditions.

Weak Type Inequalities for 𝐼 𝜌 via the Chebyshev Inequality
Now, we give an inequality which is used in the proof of the weak type inequalities for   in the following lemma.
By letting  ≡ 1 or  ≡  (,) , we have the following.( Remark 7.These two inequalities will be used later to prove one of our main theorems.The next lemma presents an inequality involving the modified Hardy-Littlewood maximal operator   .This inequality is an important part of the proof of the weak type inequalities for   in [16,19].See [8] for similar results. Lemma 8 (see [16]).
Proof.Let (, ) be any ball in R  .For every  ∈ (, ) and  > 0, let Let By the dyadic decomposition of R  \ (, ) and the growth condition of , we have ()      () .
Remark 10.Note that () =   , where 0 <  <  satisfies the condition of Theorem 9 and, for this , we obtain the weak type inequalities for   in Theorem 1.

Weak Type Inequalities for 𝐼 𝜌 via a Hedberg Type Inequality and Weak Type Inequalities for 𝑀 𝑛
In this section, we will prove weak type inequalities for   using a different technique, namely, via a Hedberg type inequality and weak type inequalities for   .It turns out that some hypotheses can be removed.The Hedberg type inequality is presented in the following proposition.
Sihwaningrum et al. [19] proved the weak type inequalities for   on generalized nonhomogeneous Morrey space by assuming that   satisfies the integral condition; that is, ∫ ∞  (()  /) ≤ ()  for every  > 0. In [19], the weak type inequalities for   are also proved by using the weak type inequalities for   .In this paper, we remove the integral condition of   in the hypothesis of our proposition below.See [32,Theorem 2.3] and [33,Theorem 2.3] for such attempts.
With Propositions 11 and 12, we are now ready to prove the weak type inequalities for   on generalized nonhomogeneous Morrey spaces.

𝛾 )
. (59) By summing the two previous estimates, we get the desired inequality.
Remarks 1. (i) Note that the hypotheses ∫ ∞  (()  /) ≤ ()  in Theorem 9 are not included in Theorem 13, since we can prove the weak type inequalities for   without this condition.
(ii) The conditions on , namely, inf >0 () = 0 and sup >0 () = ∞, are not included in the hypotheses in Theorem 13.However, we have to use the weak type inequalities for   on generalized nonhomogeneous Morrey spaces and a Hedberg type inequality for   in the proof of Theorem 13.

Boundedness of Singular Integral Operators
Proposition 12 carries over to the singular integral operator whose definition is given in [1].Recall that the singular integral operator  is a bounded linear operator on  2 () for which there exists a function  that satisfies three properties listed below.As for this singular integral operator , the following result is due to Nazarov, Treil, and Volberg.Proposition 14 (see [1,2]).The singular operator  is bounded on   () for 1 <  < ∞.Moreover, there exists a constant  > 0 such that for every  > 0.
Proof.The proof is a modification of that of Proposition 12.
We decompose  =  1 +  2 as before.The treatment of  1 is the same as that in Proposition 12 but by using the weak type inequality for  in Proposition 14.We need to take care of  2 .By the condition (4.a), Hölder's inequality, and the growth condition of , we have If we use our integrability assumption, then we have a pointwise estimate: So, we are done.
Remark 16.If we define the generalized weak Morrey space of nonhomogeneous type  , () to be the set of all measurable functions  such that          , () ( { ∈  (, ) :      ()     > }) then the inequality (64) amounts to the boundedness of  from  , () to  , ().Similarly, our previous results can be translated into this language.In the following section, we will use these notations for convenience.

Generalized Nonhomogeneous Orlicz-Morrey Spaces
Our results above can be carried over to generalized nonhomogeneous Orlicz-Morrey spaces.We first formulate our main results and then prove them later in Sections 5.1-5.3.
We define the generalized weak Orlicz-Morrey spaces of nonhomogeneous type as follows.For a Young function Φ, the generalized weak Orlicz-Morrey space of nonhomogeneous type  Φ, () =  Φ, (R  , ) is the set of all measurable functions  for which the norm In view of the growth condition, we may suppose that || assumes its value in {0} ∪ [1, ∞).Since Φ is a Young function, we have Therefore, Φ As a result,  ∈ In view of the doubling property, we are done with the maximal operator.
As for the singular integral operator, we combine the above proof and that of Theorem 15.We mimic the argument above for  1 , while we use estimate (66) obtained in the proof of Theorem 15.We omit the further details.

Proof of Theorem 20.
We start with the proof of a Hedberg type inequality.Let  > 0.Then, as in (37), we have Here, for the last inequality, we used Theorem 19.
for any ball (, ) and -measurable function .