Sobolev Embeddings for Generalized Riesz Potentials of Functions in Morrey Spaces L ( 1 , φ ) ( G ) over Nondoubling Measure Spaces

Our aim in this paper is to deal with the Sobolev embeddings for generalized Riesz potentials of functions in Morrey spaces over nondoubling measure spaces.


Introduction
In this paper, we show that many endpoint results about the Adams theorem still hold in the nondoubling setting and that the integral kernel can be generalized to a large extent.In [1], in the setting of the Lebesgue measure, for 0 <  < , recall that Adams considered and proved the boundedness of the fractional integral operator   given by The operator   is also called the fractional integral operator or the Riesz potential.We denote by (, ) the ball { ∈ R  : | − | < } with center  and of radius  > 0, and by |(, )| its Lebesgue measure, that is, |(, )| =     , where   is the volume of the unit ball in R  .Let  be a bounded open subset of R  .We denote its diameter by   ; = sup {      −      : ,  ∈ } .
For  ∈  1 (), we define the integral mean over (, ) by Let 1 ≤  < ∞.If  is a positive function on the interval (0, ∞) satisfying the doubling condition (see (23)), then we define the Morrey space  (,) () to be the family of all  ∈   loc () for which there is a positive constant  such that − ∫

Journal of Function Spaces and Applications
Much about the case  > 1 is known.Recall that the Adams theorem about the boundedness of fractional integral operators [1,Theorem 3.1] provided the parameters , ,  satisfy See also research papers [2][3][4][6][7][8][9][10][11][12][13][14][15][16]] and a survey [5].Meanwhile, only a few results are known for the case  = 1.Trudinger [17, Theorem 1] proved that if  ∈  1,1 () =  1 () then exp(| 1 |) ∈  1 () for some constant  > 0; this implies that the operator  1 is bounded from  1,1 () to exp( 1 )().See also Serrin [18] for an alternative proof.Recently, the boundedness of Riesz potentials from  (1,) () to Orlicz-Morrey spaces was shown in [19].This result extends [20,21].One of the reasons why the case when  = 1 is difficult is the failure of the boundedness of the Hardy-Littlewood maximal operator .In connection with this failure, we do not have Littlewood-Paley characterization.Due to these two difficulties, the case when  = 1 is hard to analyze.However, from the point of PDEs, we are faced with analyzing the quantity lim ↓0 (sup in connection of the Kato condition, where  is the potential operator of the operator −Δ + .See [22, Section 2], for example.Consequently, despite the difficulty arising from harmonic analysis, the case when  = 1 occurs naturally.As another evidence that the case when  = 1 is of importance, we recall that the space for any measurable function , where where {  } ∞ =−∞ is a Littlewood-Paley patch.By choosing a smooth function  ∈  ∞ (R  ) such that  (0,4)\(0,2) ≤  ≤  (0,8)\(0,1) , recall that we can define the th Littlewood-Paley patch by for  ∈ S  (R  ).Note that ( 13) is a direct consequence of the translation invariance of the space  1, (R  ).But this loss caused by ( 13) is quite big.Note that fails.See the appendix for a proof.When  > 1, an approach using the Littlewood-Paley patch is taken effectively [27].Indeed, for all  ∈  , (R  ).However, for the case when  = 1, due to the fact that the estimate ( 13) is essential when we consider the Littlewood-Paley patch, we prefer to avoid the Littlewood-Paley patch.See [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] for a huge amount of culmination of this approach.Instead of using the Littlewood-Paley patch, we still have a good approach for the case when  = 1.Just make a closer look at the integral kernel.Our method being simple enough, there is no need to stick to the geometric structure of R  .Our result relies completely only upon the positivity of the integral kernel.So, here and below, we work on a separable metric space  equipped with a nonnegative Radon measure , where we do not postulate any other condition on .By (, ), we denote the open ball centered at  of radius  > 0. While, given a point  1 and  2 in R  , we write | 1 −  2 | for the distance of the points  1 and  2 , and we write (, ) for the distance of the points  and  in .We assume that ({}) = 0 and that 0 < ((, )) < ∞ for  ∈  and  > 0 for simplicity.In the present paper, we do not postulate on  the "so-called" doubling condition.Recall that a Radon measure  is said to be doubling, if there exists a constant  > 0 such that for all  ∈ supp()(= ) and  > 0. Otherwise  is said to be nondoubling.In connection with the 5-covering lemma, the doubling condition had been a key condition in harmonic analysis.
Our aim in this paper is to show that, for the case  = 1, the operator   and its generalization   are bounded from Morrey spaces  (1,) to Orlicz-Morrey spaces, or, to generalized Hölder spaces, whose definitions will be given in the next section, in the nondoubling setting.Our result extends the results in [17][18][19][20][21].The definition of   is the following: let  be a function from (0, ∞) to itself and satisfy for all sufficiently small  > 0. We do not have to postulate the doubling condition on .See Remark 3 for an example which fails the doubling condition.We define where  ∈  1 ().Instead of using we discuss   defined above.This modification will be necessary in Lemma 9 for example.An example in [44, Section 2] shows that  †  is less likely to be bounded in general, although there does not exist a proof.We refer to [45] for an attempt of definining fractional integral operators by using the underlying measure .
Note that ( 18) is necessary in order that the image by   of  (,) , the indicator functions of the balls, belongs to  , () at least when  is the Lebesgue measure.Indeed, if for any sufficiently small  > 0.Then, for  ∈ (, /2) such that (, ) ⊂ , we have by using the spherical coordinate.We organize the remaining part of the present paper as follows.In Section 2, we set up some notations.Section 3 is devoted to stating our main results fully based on the notations in Section 2. Some auxiliary lemmas are collected in Section 4. Finally, theorems in the present paper are proven in Section 5.

Notation and Terminologies
Let G be the set of all continuous functions from (0, ∞) to itself with the doubling condition, that is, there exists a constant   ≥ 1 such that We call the smallest number   satisfying (23) the doubling constant of .Note that in view of [46, page 445] and [47, (1.2)], the doubling condition on  is a natural one.For  ∈ G, we define the Morrey space  (1,) () as follows: with the norm Then, a routine argument shows that  (1,) () is a Banach space.Due to the fact that R  is a geometrically doubling space, we can prove that for all  > 1. See [48, Proposition 1.1] for a technique used to prove this inequality.Note here that if  1 ,  2 ∈ G and  1 / 2 is bounded above on (0,   ), then in particular, if there exists a constant  ≥ 1 such that  −1  1 () ≤  2 () ≤  1 () for all  > 0, then with equivalent norms.A ball testing shows the following.
Here and below, we write  ≲  to indicate that there exists a constant  independent of Morrey functions such that  ≤ .The symbol  ∼  stands for  ≲  ≲ .
Let us consider the family Y of all continuous, increasing, convex, and bijective functions from [0, ∞) to itself.For Φ ∈ Y, the Orlicz space  Φ () is defined by where If Φ 1 , Φ 2 ∈ Y are equivalent in the sense that there exists a constant  ≥ 1 with for all  > 0, then we see easily that with equivalent norms.If for large  > 0, then  Φ () will be denoted by exp (  ) () , exp exp (  ) () , For Φ ∈ Y and  ∈ G, the Orlicz-Morrey space  (Φ,) () is defined by where          (Φ,) () (see [50,51]).Then, again it is routine to prove that ‖ ⋅ ‖  (Φ,) () is a norm and that  (Φ,) () is a Banach space.Note that the space  Φ is a special case of Orlicz-Morrey spaces when  = .
For  ∈ G such that  is bounded, the generalized Hölder space is defined by where Then, ‖‖ Λ  () is a norm modulo constants and thereby Λ  () is a Banach space.Since  is bounded, every  ∈ Λ  () is bounded.If () → 0 as  ↓ 0, then every  ∈ Λ  () is continuous.For details, we refer to [52].

Main Results
In this section, we state our main theorems, whose proofs are given in Section 5. Throughout this paper, let  be a bounded open set in  and denote by   , the doubling constant of  ∈ G.
(1) Here it is not significant for us to choose 16; it counts that any number will do as long as it is small enough.
We now state a result for Orlicz-Morrey spaces.

Proofs of the Theorems
We are now ready to prove our theorems.
for ,  ∈ .On the other hand, we have by (52)

)
Let  ∈ G. Assume the following condition on .
is a constant depending only on   ,   ,  1 ,  2 , and .