Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents

The target is potential theory in connection with Morrey spaces on general metric measure spaces. The present paper is oritented to investigating Sobolev’s inequality, Trudinger exponential integrability and continuity for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents. A counterexample shows that our results are reasonable. In addition to the example above, what is new about this paper is that everything can be developed once the underlying measure does not charge any point.

Morrey spaces date back to the work of Morrey [25] in 1938. His observation has become a useful tool for partial differential equations and with this tool we can study the existence and regularity of solutions of partial differential equations. Nowadays, his technique turned out to be a wide theory of function spaces called Morrey spaces. The (original) space M p u (R d ) with 1 ≤ u ≤ p < ∞ is a normed space whose norm is given by In the present paper, we are oriented to Sobolev's inequality for Riesz potentials of functions in Morrey spaces of variable exponents in the non-doubling setting, which will extend the results in [3,15,21,22,24,35]. We also establish Trudinger exponential integrability for Riesz potentials of functions in Morrey spaces of variable exponents in the non-doubling setting, as extensions of our earlier papers [21,22,34]. Further, we discuss continuity of Riesz potentials of variable order, which extends [7,12,21,23].
The assumption (1.1) will be necessary for the definition of U α(·),τ f in order that the integral is not infinite.
Here and in what follows we tacitly assume that f = 0 outside G. Observe that this naturally extends the Riesz potential operator when (X, d) is the d-dimensional Euclidean space and μ = dx.
and for all cubes Q.
In view of the integral kernel of (1 − ) −α/2 (see [37]) and the Adams theorem, we have is bounded as long as The operator norm of Hence we can say that Proposition 1.3 substitutes (1.5). We refer to [36] for a counterexample showing that (1.5) is no longer true for α = d p . Meanwhile, the function q(·) can be used to describe the Hardy-Littlewood maximal operator control in very subtle settings. To describe the situation, we place ourselves in the setting of the Euclidean space R d . We denote again by B(x, r ) the open ball centered at x ∈ R d and of radius r . For a locally integrable function f on R d , we consider the Hardy-Littlewood maximal function For the fundamental properties of the Hardy-Littlewood maximal function, see Duo andikotxea [5] and Stein [37]. It is known as Stein's theorem that there exists a universal constant C > 0 such that for all functions f supported on a ball B with radius 1.
Remark that, if X = R d , the parameter κ is not essential as long as κ > 1 as Proposition 1.4 below shows: Proposition 1.4 Let κ 1 , κ 2 > 1 and X = R d be the Euclidean space. Suppose that G is a bounded open set. Assume in addition that ν and β satisfy the log-Hölder continuity and the loglog-Hölder continuity, respectively. Then the spaces L ,ν,β;κ 1 (G) and L ,ν,β;κ 2 (G) coincide as sets and their norms are equivalent.
We shall prove Proposition 1.4 in Sect. 3. In the present paper, we consider a generalized and modified Hardy-Littlewood maximal function defined by for a locally integrable function f on G.
In the present paper, we shall also show that most of the results known as the limiting cases can be carried over to the non-doubling measure spaces. It counts that we take an attentive care of the parameters κ appearing in (1.4). For example, unlike the doubling measure spaces, we need delicate geometric observations [see the Proof of Lemma 4.2 and (6.17), for example]. Because we need careful geometric observations, we need to set everything up from the start. Section 4 is our actual starting point.
We organize the remaining part of the present paper as follows: In Sect. 2, we intend to justify that the modification is necessary in the non-doubling setting by proving Theorem 1.1. To construct a counterexample, we shall refine the one in [32]. In Sect. 3, we see some more examples of this metric measure setting.
From Sect. 4, we are going to construct a general theory. Section 4 is devoted to the study of the modified centered Hardy-Littlewood maximal operator M 16 .
We are going to obtain Sobolev's inequality for Riesz potentials U α(·),32 f of functions in L ,ν,β;2 (G) in Sect. 5. To this end, we apply Hedberg's trick [14] by the use of the boundedness of the Hardy-Littlewood maximal function M 16 adapted to our setting. Our result (see Theorem 4.1 below) is given in Sect. 5, which extends the results in [15,21,22,24,35].
In Sect. 7, we discuss the continuity of Riesz potentials of variable order, as an extension of [7,12,21,23]. For related results, see [8,19,20]. More precisely, in Sect. 7 we discuss the continuity of Riesz potentials U α(·),4 f , which can be considered as generalized variable smoothness. It seems of interest in other fields of mathematics such as PDEs that we investigate the continuity of functions according to each points. Indeed, the fundamental solution of − u = f on R d is continuous except on the origin. Therefore, we are interested in tools with which to investigate continuity differently according to the points. In view of the continuity we postulate on variable exponents, we can say that this is achieved to some extent.
Finally we explain some notations used in the present paper. The function χ E denotes the characteristic function of E. Throughout the present paper, let C denote various constants independent of the variables in question.

Proof of Theorem 1.1
In Sect. 2, we prove Theorem 1.1. Here will be a series of definitions which are valid only in Sects. 2.1-2.2.

The space we work on
First, we define a set X on which we work. Definition 2.1 (Definition of X ) Define a set X as follows: (For the graph of A 0 we refer to the footnote 1.) (For the graph of X 0 we refer to the footnote 1.)

Remark 2.2
Here and below, we adopt the following rules in Sect. 2: 1. The letter z without subindex denotes the point in C. 2. Points in X are written in the bold letters such as x, y, z. 1 3. Symbols such as x j , y j , z j and so on are complex numbers and they denote the jth component of elements in X .
The point is that we give a "singular" metric on X . The precise definition is as follows:  1 We draw graphs of A 0 and X 0 . Note that A 0 is an annulus and X 0 is the union and X is a countable product of X 0 .
3. Let x = {x j } ∞ j=1 and y = {y j } ∞ j=1 be points in X . Then define the distance d(x, y) of x and y by

One defines a sphere S k by
At this moment, about the definition of the natural number N 0 , it is only the fact that the function N : (0, ∞) → R is a decreasing function that counts for the moment.
Before we start a (long) proof, let us outline it. Roughly speaking, sphere testing suffices. Based upon the metric space (X, d) given above, we shall show that (X, d) is in fact a separable metric measure space and that there does exist a Borel measure μ such that L p,4,ν (μ) and L p,2,ν (μ) do not coincide as sets: More precisely, we shall show that Choose an increasing sequence {k(n)} ∞ n=1 such that then F L p,2,ν (μ) ≥ 2 n and F L p,4,ν (μ) ≤ 1 for all n ∈ N. This implies F ∈ L p,4,ν (μ) \ L p,2,ν (μ). The remainder of this subsection is devoted to some preparatory observation on this metric measure space and in the next subsection we get the conclusion. Note that and that Thus, d X = 2. Let us check that d is a metric function and that χ S k is μ-measurable.

Lemma 2.4 In Definition 2.3, d is a metric function, that is,
Furthermore, the d-topology is exactly the product topology of X .
Since any d-open set is open with respect to the product topology, the product topology is not weaker than the d-topology. However, we can express X as a union of d-open balls as long as X is given by with someÑ ∈ N, (z 1 , z 2 , · · · , zÑ ) ∈ X 0Ñ and (r 1 , r 2 , · · · , rÑ ) ∈ (0, ∞)Ñ . Therefore, two topologies coincide. Lemma 2.5 Let r > 0 and x = (x 1 , x 2 , · · · ) ∈ X. Then (2.6) Proof From the definition of the open ball, we have Since N : (0, ∞) → R is left-continuous and assumes its value in Z, if δ is slightly less than r , N (δ) = N (r ). Together with the monotonicity of the most-right hand side of the above equality, we conclude The measure is given by way of product: Definition 2.6 (Definition of the measure) Let H 1 denote the 1 dimensional Hausdorff measure.

One defines a function
As for the measures μ and μ 0 , we have the following relations.
Proof From the definition of μ 0 , we see This observation yields the lower bound for μ 0 ∞ j=k A j . It remains to obtain the upper bound for μ 0 ∞ j=k A j . First of all, let us assume that k = 0. Note that μ 0 (A 0 ) = 2πγ . Hence, when k = 0, from (2.8), we deduce Note that γ ∈ (0, 1) and that Thus, (2.7) was proved.
Proof This is an easy consequence of (2.6).
Proof In view of (2.6), we have and hence, from (2.7) and Lemma 2.5, we deduce 2πγ . (2.11) Meanwhile, from the right inequality of (2.7), we have For this case, it is not so hard to see that this sequence is almost decreasing. The next lemma is an easy consequence of a simple geometric observation and the inequality sin −1 r ≥ r for r ∈ 0, π 2 .

Lemma 2.11
For all x ∈ X and r > 0, we have In view of (2.10), we have For the definition of (z, r ) see Definition 2.1. Now that N (r ) ≥ N (10r ) and μ 0 is a probability measure, it suffices to prove The left circle is x 2 + y 2 = 1 and the right circle is (x − 1) 2 + y 2 = 1/4. Let us observe that the length of the set {x 2 + y 2 = 1, (x − 1) 2 + y 2 < r 2 } grows linearly when r is small enough.
We specify the natural number N 0 in Definition 2.3 by for some a ∈ N large enough. As long as b ∈ [1,9] and k is an odd multiple of a, log N 0 b ·3 −k and log N 0 3 −k have the same integer part since we have (2.19).
The next lemma concerns the norm estimates of χ S k .

Lemma 2.13
Let k be an odd multiple of a. Then, equivalence Proof The lower bound of χ k L p,2,ν (μ) is a consequence of Lemma 2.12: It is easy, from Lemma 2.12, to see that By using sup, we have Actually, we distinguish two cases by setting Let us suppose r ≤ 3 6−k . Then we have Let us suppose r > 3 6−k instead. Then we have from Lemma 2.11. Thus, by choosing an integer m ≤ k so that 3 6−m < r ≤ 3 7−m , by virtue of Lemma 2.9 with (l, k) replaced by (m − 1, k − 1), we obtain be a ball such that B(x, r ) meets S k at a point y, that is, y ∈ S k ∩ B(x, r ). We distinguish three cases assuming that k is an odd multiple of a. Case 1 Assume first that 3 −k+2 /10 ≤ r ≤ 3 −k+6 . Then d(x, y) < r implies .

Meanwhile let us set
Since k is an odd multiple of a, we have N ( Recall that y ∈ S k . A geometric observation shows that Hence, from (2.21), we deduce (2.28) Case 2 If 3 −k+2 /10 ≥ r , then we use which follows from a geometric observation. Now we go into the structure of the measure μ; if we insert the definition of the measure μ, then we obtain . Now we consider a transform given by z ∈ A k → 3 k z ∈ A k and we deduce .

Remarks and examples
3.1 Proof of Proposition 1.4 We follow the idea in [10], [ Before we start the proof of Proposition 1.4, we need some preparatory observations. By symmetry, we can assume that κ 1 > κ 2 . Next, since 0 < ν − ≤ ν + < ∞ and −∞ < β − ≤ β + < ∞, we can find a constant K independent of x such that for all r > 0. Based upon these observations, we prove Proposition 1.4. We need to compare the following two conditions (3.2) and (3.3): By virtue of (3.6) and the convexity of (y, ·), (3.3) holds with λ 2 = K m 0 N λ 1 .

Other examples of non-doubling metric measure spaces
The doubling condition had been playing a key role in harmonic analysis. However, nondoubling measure spaces occur very naturally in many branches of mathematics. The typical examples we envisage are the following ones: x n ) ∈ R n : x 2 1 + x 2 2 + · · · + x 2 n < 1} be the unit ball in R n . Equip B(1) with a metric given by Then (B(1), g) is called the space with constant curvature −1 and if we denote by μ the induced measure, then μ(B(x, r )) grows exponentially.

Example 3.2 Equip the Euclidean space
Then (R n , | · 1 − · 2 |, μ) is called the Gauss measure space and the operator is a self-adjoint operator on L 2 (μ). Recently, the first author, Liguang Liu and Dachun Yang considered Morrey spaces in [16]. Let us set In [

An estimate of the modified centered Hardy-Littlewood maximal operator M 16
In Sect. 4 we work on a bounded open set G and we write d G for the diameter of G.
For a locally integrable function f on G, recall that in (1.6) we defined the centered and generalized Hardy-Littlewood maximal operator by In what follows, as we did in Sect. 1, if f is a function on G, then we assume that f = 0 outside G.
As a starting point of the present paper, we shall prove the following estimate of the centered Hardy-Littlewood maximal operator M 16 . For the case q = 0, see Kokilashvili-Meskhi [15]. As a consequence of Theorem 4.1 the centered Hardy-Littlewood maximal operator M 16 is bounded from L ,ν,β,G;2 (G) to L ,ν,β,G;4 (G).
To prove Theorem 4.1, we need several lemmas. Let us begin with the following result, which concerns an estimate for the case p(x) ≡ p 0 and q(x) ≡ 0 (cf. [

Lemma 4.2
Assume that p(·) and ν(·) satisfy p(·) ≡ p 0 > 1 and ν − > 0, respectively. Let f be a μ-measurable function on G satisfying for all x ∈ G and 0 < r < d G . Then there exists a constant C > 0 such that 1 μ (B(z, 4r )) for all z ∈ G and 0 < r < d G , where the constant C is independent of f satisfying (4.3).
Proof Let f satisfy (4.3), and fix z ∈ G and 0 < r < d G . Write A 0 ≡ B(z, 2r ) and Let us set
The estimate for I 1 is now valid.
Integrating the above estimate over B(z, r ), we obtain (B(z, r )).
It is significant that ν(·) and β(·) do not have to be continuous. The next lemma concerns an estimate for x such that | f (x)| is large. For convenience of the readers, we supply its proof the following key inequality (4.5) which is similar to the one dealt in [27].
for each x ∈ G. Define g(y) ≡ f (y) p(y) (log(e + f (y))) q(y) for y ∈ X . Then there exists a constant C > 0, independent of f , such that for all x ∈ G.
Proof Let x ∈ G and r > 0. We let (4.6) To prove (4.5), it suffices to show that 1 μ (B(x, 16r )) for all x ∈ G and 0 < r < d G with the constant C independent of x and r . Indeed, once (4.7) is proved, if we insert (4.6) to (4.7) and consider the supremos over all admissible x and r , then we will have In view of the definition of g and the fact that the inverse function of t → t P (log t) Q with P > 1 and Q ∈ R is equivalent to the function t → t 1/P (log t) −Q/P , it follows that (4.8) implies the desired conclusion (4.5). So let us prove (4.7).
Keeping Lemmas 4.2 and 4.3 in mind, we prove Theorem 4.1.
We plan to prove Theorem 5.1 by using three auxiliary estimates, keeping in mind the original proof of Hedberg [14].
The next lemma concerns an estimate inside balls.
Proof The proof is similar to the one in [24,Lemma 2.3]. Assuming that μ does not charge a point {x}, we have If we use (1.2), then we see the geometric series of the most right-hand side converges and, with a constant C independent of x, we have Thus, Lemma 5.3 is proved.
We get information outside a fixed ball by using Lemma 5.4 below.

Lemma 5.4 Let f be a non-negative μ-measurable function on G such that
for x ∈ G and small δ > 0.
Proof Let d G be the diameter of G as before and let j 0 be the smallest integer such that 2 j 0 δ ≥ d G . By Lemma 5.2 and our convention that f is 0 outside G, we have α(x)). Then η > 0 by (5.1). Since the functions log α, q, β are all bounded, p − > 1 and the function ν is positive, we obtain So we need to consider the integral Recalling that η > 0, we have which completes the proof.
With the aid of Theorem 4.1, Lemma 5.3 and Lemma 5.4, we can apply Hedberg's trick (see [14]) to obtain a Sobolev type inequality for Riesz potentials, as an extension of Adams [ for all δ > 0. Here, we optimize the above estimate by letting and we have Then from (5.3) we find for all x ∈ G. It follows from Theorem 4.1 that 1 μ (B(z, 4r )) for all z ∈ G and 0 < r < d G , as required.

Trudinger exponential integrability
Based on what we have culminated in the present paper, we shall obtain Trudinger exponential integrabilities for U α(·),9 f . We seek to discuss the exponential integrability in Sect. 6, assuming that for x ∈ G and r > 0, where we choose a normalization constant c 0 so that inf x∈G (x, 2) = 2. Note that sup x∈G, r ≥2 .
If we combine Theorem 6.1 and Remark 6.2, then we obtain the following result, which was called the Trudinger inequality. Corollary 6.3 Let G be bounded. Suppose ν − > 0 and (6.1) holds. Let ε be a μ-measurable function on G such that Then there exist constants c 1 , c 2 > 0 such that (6.7) (2) in case inf x∈G (q(x) + β(x))/ p(x) ≥ 1, 1 μ (B(z, 4r )) To prove Theorem 6.1, we use the following lemmas. The first lemma can be proved with minor changes of the proof of Lemma 5.4. We begin with investigating the functions from outside the balls. Lemma 6.5 Suppose that ν − > 0 and (6.1) holds. Then there exists a constant C > 0 such that G\B(x,δ) for all x ∈ G, 0 < δ < d G and non-negative μ-measurable functions f satisfying f L ,ν,β;2 (G) ≤ 1.

Proof
Since we can reexamine and modify the proof of Lemma 5.4. Indeed, we need to estimate I(δ), where I(δ) is given by (5.8). Assuming (6.1), we have The proof of Lemma 6.5 is thus complete.
To prove Theorem 6.1, we need another lemma. By generalizing the integral kernel, we are going to prove Theorem 6.1, as is seen from the beginning of the proof. This is where the number "9" comes into play in Theorem 6.1 [see (6.17) below]. Lemma 6.6 Let ε : G → (0, ∞) be a μ-measurable function satisfying (6.2) and let z be a fixed point in G. Also we write ρ(z, r ) ≡ r ε(z) (log(e + 1/r )) (q(z)+β(z))/ p(z) .
To prove Theorem 7.1, we need Lemmas 7.2 and 7.3. Lemmas 7.2 and 7.3 concern an estimate inside the ball and that outside the ball, respectively.