A thought on generalized Morrey spaces

Morrey spaces can complement the boundedness properties of operators that Lebesgue spaces can not handle. Morrey spaces which we have been handling are called classical Morrey spaces. However, classical Morrey spaces are not totally enough to describe the boundedness properties. To this end, we need to generalize parameters $p$ and $q$, among others $p$.


Introduction
This note studies the function spaces which Guliyev, Mizuhara and Nakai defined in [28,69,73]. Let 0 < q ≤ p < ∞. We define the classical Morrey space M p q (R n ) to be the set of all measurable functions f for which the quantity |Q(x, r)| is finite, where Q(x, r) = {y ∈ R n : |x 1 − y 1 |, |x 2 − y 2 |, . . . , |x n − y n | ≤ r}. To describe the endpoint case or to describe the intersection space, sometimes it is useful to generalize the parameter p: let us suppose that p comes from the function t n p . So, we envisage the situation where t n p is replaced by a general function ϕ. More precisely, it is known as the Adams theorem that I α maps M p q (R n ) to M s t (R n ) whenever 1 < q ≤ p < ∞ and 1 < t ≤ s < ∞ satisfy 1 Here I α is the fractional integral operator given by (12) for a nonnegative measurable function f : R n → [0, ∞]. However, it is known that we can not take s = ∞.
In fact, I α fails to be bounded from M n/α q (R n ) to L ∞ (R n ) for all 1 ≤ q ≤ n α . To compensate for this failure, we can use generalized Morrey spaces. We will define the generalized Morrey norm by Here ϕ : (0, ∞) → [0, ∞) is a function which does not vanish at least at some point. Another advantage of generalized Morrey spaces is that we can cover many function spaces related to Lebesgue spaces. Although we do not consider the direct applications of generalized Morrey spaces to PDEs, generalized Morrey spaces can be applied to PDEs. Applications to the divergence elliptic differential equations can be found in [59]. Applications to the nondivergence elliptic differential equations can be found in [35,114,123]. Applications to the parabolic differential equations can be found in [112,128]. See [30,42,113] for the parabolic oblique derivative problem. See [61] for applicatitons to Schrödinger equations. We refer to [64] for the application to singular integral equations.
This note is organized as follows: We will collect some preliminary facts in Section 2 In Section 3, we define generalized Morrey spaces and then discuss the structure of generalized Morrey spaces. The conditions on ϕ will be important. So we discuss them quite carefully. Section 4 is a detailed discusssion of the boundedness properties of the operators. Here we handle the Hardy-Littlewood maximal operators, the Riesz potentails, the Riesz transforms and the fractional maximal operators as well as their generalizations. Section 5 is a survey of other related function spaces.
Here we list a series of (somewhat standard) notation we use in this note.
• Let (X, d) be a metric space. We denote by B(x, r) the ball centered at x of radius r. Namely, we write B(x, r) ≡ {y ∈ R n : d(x, y) < r} when x ∈ R n and r > 0. Given a ball B, we denote by c(B) its center and by r(B) its radius. In the Euclidean space R n , we write B(r) instead of B(o, r), where o ≡ (0, 0, . . . , 0).
• A metric measure space is a pair of a metric space (X, d) and a measure µ such that any open set is measurable. • By a "cube" we mean a compact cube in R n whose edges are parallel to the coordinate axes. The metric closed ball defined by ∞ is called a cube.
If a cube has center x and radius r, we denote it by Q(x, r). Namely, we write Q(x, r) ≡ y = (y 1 , y 2 , . . . , y n ) ∈ R n : max j=1,2,...,n |x j − y j | ≤ r when x = (x 1 , x 2 , . . . , x n ) ∈ R n and r > 0. From the definition of Q(x, r), its volume is (2r) n . We write Q(r) instead of Q(o, r). Given a cube Q, we denote by c(Q) the center of Q and by (Q) the sidelength of Q: (Q) = |Q| 1/n , where |Q| denotes the volume of the cube Q. • Given a cube Q and k > 0, k Q means the cube concentric to Q with sidelength k (Q). Given a ball B and k > 0, we denote by k B the ball concentric to B with radius k r(B). • Let A, B ≥ 0. Then A B and B A mean that there exists a constant C > 0 such that A ≤ CB, where C depends only on the parameters of importance. The symbol A ∼ B means that A B and B A happen simultaneously, while A B means that there exists a constant C > 0 such that A = CB.
• When we need to emphasize or keep in mind that the constant C depends on the parameters α, β, γ etc Instead of A B, we write A α,β,γ,... B.

Lebesgue spaces and integral inequalities.
Here we recall Lebesgue spaces and the integral inequalities.
• The Lebesgue norm L p (µ)-(quasi-)norm of a measurable function f is given by e. x ∈ X }, p = ∞.
Here the equivalence relation ∼ is defined by: and below omit this equivalence in defining function spaces.
As usual if (X, B, µ) is the Lebesgue measure, then we write L p (R n ) = L p (µ) and · WL p = · WL p (µ) .
In addition to L p (µ) with 0 < p ≤ ∞, it is convenient to define L 0 (µ): The space L 0 (µ) denotes the set of all measurable functions considered modulo the difference on the set of measure zero.
2.2. Integral operators. We will study the Hardy-Littlewood maximal operators, the Riesz potentails, the Riesz transforms and the fractional maximal operators. So we recall the definition together with the boundedness properties on L p (R n ). For Theorems 2.4, 2.5, 2.9, 2.10, 2.11 we refer to [24,92,116,115].
The mapping M : f → M f is called the Hardy-Littlewood maximal operator.
Here we are interested in the following type of the singular integral operators: Definition 2.6 (Singular integral operator). A singular integral operator is an L 2 (R n )-bounded linear operator T that comes with a function K ∈ C 1 (R n \ {0}) satisfying the following conditions: • (Size condition) For all x ∈ R n , • (Gradient condition) For all x ∈ R n , |∇K(x)| |x| −n−1 .
• Let f be an L 2 (R n )-function. For almost all x / ∈ supp(f ), The function K is called the integral kernel of T .
More generally we consider the following type of singular integral operators: Definition 2.7. A (generalized) Calderón-Zygmund operator is an L 2 (R n )-bounded linear operator T , if it satisfies the following conditions: • There exists a measurable function K such that for all L ∞ (R n )-functions f with compact support we have • The kernel function K satisfies the following estimates.
As a typical example we consider the Riesz transform: Example 2.8. Let j = 1, 2, . . . , n. The singular integral operators, which are represented by the j-th Riesz transform given by, x j − y j |x − y| n+1 f (y)dy, (9) are integral operators with singularity.
Theorem 2.9. Suppose that T is a CZ-operator. Then T is weak-(1, 1) bounded, that is for all f ∈ L 1 (R n ) ∩ L 2 (R n ).
We have the strong boundedness for 1 < p < ∞.
Theorem 2.10. Let 1 < p < ∞ and T be a generalized singular integral operator, which is initially defined on L 2 (R n ). Then T can be extended to L p (R n ) for all 1 < p < ∞ so that T f p p f p (11) for all f ∈ L p (R n ).
As a different type of singular integral operators, we consider the fractional integral operators. Let 0 < α < n. Let I α be the fractional maximal operator given by for a nonnegative measurable function f : R n → [0, ∞].
The following theorem is known as the Hardy-Littlewood-Sobolev theorem. Generalized Morrey spaces can be used to refine this theorem.
• Assume that the parameters 1 < p < n α and 1 q = 1 p − α n . Then we have Let 0 ≤ α < n. The fractional maximal operator M α of order α is defined by for a measurable function f . Note that M α maps M n/α 1 (R n ) boundedly to L ∞ (R n ). As an opposite endpoint case, we have the following boundedness: for all f ∈ L 1 (R n ).
Using the boundedness of M , we can prove the boundedness of M α . Here for the sake of convenience for readers we supply the proof. Theorem 2.13. Let 1 < p < q < ∞ and 0 < α < n satisfy 1 Proof. By the monotone convergence theorem we may assume that f ∈ L ∞ c (R n ). For λ > 0 we let Ω λ ≡ {M α f > λ}. We observe that We choose u > 0 so that n(1 − u) = pα. Then we have Thus, we have Observe that By the Hölder inequality, we obtain As a result, In addition to the linear operators above we sometimes consider the commutator generated by BMO and these operators. Here we recall the BMO (bounded mean oscillation) as follows: for f ∈ L 1 loc (R n ). The "norm" · BMO is called the BMO(R n ) norm. The space BMO(R n ) collects all f ∈ L 1 loc (R n ) such that f BMO is finite. Usually BMO(R n ) is considered modulo the constant functions.
As the following theorem shows, the BMO functions have high local integrability.
Theorem 2.15 (John-Nirenberg inequality). For any λ > 0, a cube Q and a nonconstant BMO(R n )-function b, where D depends only on the dimension.
We will consider the commutators generated by BMO and these operators. For the properties of these commutators we refer to [25]. Let a ∈ BMO(R n ). Then the we can consider the operators the ambiguity which the definition of BMO(R n ) causes.
Thanks to the following theorem, we can extend our results of generalized Morrey spaces by mimicking the proof of the boundedness for the similar operators. However, due to the similarity we do not consider the boundedness and the definition of these operators. Instead, we give some references when needed.
For the proof of Theorem 2.16 we refer to [25].

Generalized Morrey spaces
Likewise we can replace q by a function Φ.
• [73] Define the generalized Morrey space M ϕ q (R n ) to be the set of all measurable functions f such that • [28] Define the weak generalized Morrey space wM ϕ q (R n ) to be the set of all measurable functions f such that Although we torelate the case where ϕ(t) = 0 for some t > 0, it turns out that there is no need to consider such possibility.
Before we go into more details, clarifying remarks may be in order.
Remark 3.2. In [73] Nakai defined the generalized Morrey space M ϕ q (R n ) to be the set of all measurable functions f such that or more generally See also [19,29,33,60,76,120,123]. Here we follow the notation by Sawano, Sugano and Tanaka [100,101,102], for example. See also [71]. See [3,12,86,88,89,109] for another notation of generalized Morrey spaces, for example. Despite the difference of the notation, the idea of defining the predual space depends on [60,109].
Remark 3.3. Some prefer to call ϕ the weight (see [26] for example), while some prefer to call · ×w M ϕ q the weighted generalized Morrey norm (see [41] for example).
We can recover the Lebesgue space L q (R n ) by letting ϕ(t) ≡ t n q for t > 0 as we have mentioned. To compare Morrey spaces with generalized Morrey spaces, we sometimes call Morrey spaces classical Morrey spaces. In addition to the function ϕ(t) = t n p , we consider the following typical functions: • The function ϕ ≡ 1 generates L ∞ (R n ) thanks to the Lebesgue convergence theorem.
Thus, if f is a measurable function such that f M ϕ q < ∞ and if a > 0, then we have f L q (Q(x,r)) = 0 for any cube Q(x, r).
In general, for 0 < q < ∞ and ϕ 1 , ϕ 2 : (0, ∞) → [0, ∞) satisfying ϕ 1 (t 1 ) = 0 and ϕ(t 2 ) = 0 for some • Let 0 < q < ∞ The uniformly locally L q -integrable space L q uloc (R n ) is the set of all measurable functions f for which We can define the weak uniformly locally L qintegrable space wL q uloc (R n ) similarly. For f ∈ L 0 (R n ), the norm is given by The fifth example deserves a name. We define the small Morrey space as follows: is the set of all measurable functions f for which the quantity , the norm is given by Example 3.6. [14, Proposition A] Let x ∈ R n and r > 0. Then In fact, simply observe that The following min(1, q)-triangle inequality holds: Lemma 3.7. Let 0 < q < ∞ and ϕ : (0, ∞) → (0, ∞) be a function. Then Proof. This is similar to classical Morrey spaces: Use the min(1, q)-triangle inequality for the Lebesgue space L q (R n ).
Proposition 3.8. Let 0 < q < ∞, and let ϕ : Proof. The norm inequality follows from Lemma 3.7. The proof of the completeness is a routine, which we omit.
Proposition 3.8 guarantees that the (quasi-)norm of M ϕ q (R n ) is complete. However, it may happen that M ϕ q (R n ) = {0} as is seen from Example 3.10. We check that this extraordinary thing never happens if ϕ satisfies a mild condition. Proposition 3.9. [80, Lemma 2.2(2)] Let 0 < q < ∞, and let ϕ : (0, ∞) → [0, ∞) be a function satisfying ϕ(t 0 ) = 0 for some t 0 > 0. Then the following are equivalent: We may assume that f (0) = 0 and that x = 0 is the Lebesgue point of |f | q . Then since f ∈ L q loc (R n ), by the Lebesgue differential theorem, 1 |Q(r)| Q(r) |f (y)| q dy ∼ 1 for all 0 < r < 1. Here the implicit constants depend on f . Thus, Here the implicit constants depend on f again. Meanwhile Finally, if (c) holds, then χ Q(x,1) ∈ M ϕ q (R n ) for any x ∈ R n . Since M ϕ q (R n ) is a linear space, χ Q(x,m) ∈ M ϕ q (R n ) for any x ∈ R n . Since any function g ∈ L ∞ c (R n ) admits the estimate of the form |g| ≤ N χ Q(x,m) for some m, N ∈ N, we conclude (a).
Here we consider the case where M ϕ q (R n ) is close to M p q (R n ) in a certain sense.
if and only if β 2 ≥ 0. Indeed, according to Proposition 3.9 the case β 2 < 0 must be excluded in We have the following scaling law: Proof. We content ourselves with showing the equality |f | η The nesting property holds like classical Morrey spaces.

3.2.
The class G q . It will turn out demanding to consider all possible functions ϕ. We will single out good functions. We answer the question of what functions are good.
Here we list a series of the functions in G q .
• Let u ∈ R, and let ϕ(t) = t u for t > 0. Then ϕ belongs to G q if and only if 0 ≤ u ≤ n q .
for t > 0. Then ϕ belongs for t ≥ 0. Then ϕ / ∈ G q because ϕ is not increasing.
We start with a simple observation that any function in G q enjoys the doubling property: Proof. The left inequality is a consequence of the fact that ϕ is increasing, while the right inequality follows from the fact that t → t − n q ϕ(t) is decreasing.
The next proposition justifies that we can naturally use the class G q .
Proof. Let f be a measurable function.
We claim that for any cube Q ∈ Q and any positive number t ≤ (Q). In fact, let [a j , a j + (Q)], so that (a 1 , a 2 , . . . , a n ) is the "bottom" corner of Thus, (20) follows. If we let then it is easy to see that ϕ 1 (t) ≤ ϕ(t) for all t > 0 and hence On the other hand, for any r > 0 and ε > 0, we can find r ≥ r such that ϕ * (r) ≤ (1 + ε)r n q ϕ 1 (r )r − n q .
Thus for any cube Q(x, r), In view of Proposition 3.16, it follows that we can always suppose that ϕ ∈ G q .
As the next proposition shows, G q is a good class in addition to the nice property that it naturally arises in generalized Morrey spaces.
Proposition 3.18. Let ϕ ∈ G q with 0 < q < ∞. Then we can find a continuous function ϕ * ∈ G q such that ϕ * is strictly increasing and that ϕ ∼ ϕ * .
Proof. We look for ϕ * in a couple of steps. (t > 0).
Since ϕ is increasing, the function ϕ 0 is increasing. Also, by the fact that ϕ is doubling, we see that ϕ 0 and ϕ are equivalent. Thus, we may assume that ϕ is continuous.
is decreasing. Likewise we can check that ψ is increasing. Thus, ψ ∈ G q . Since ψ + ϕ ϕ, we can assume that ϕ is strictly increasing in (1, ∞) and that ϕ is continuous.
• Finally, we consider where N ≡ n q + 1. Then it is easy to check that ϕ * is equivalent to ϕ since each term is dominated by 2 −k ϕ. Likewise since ϕ ∈ G q , ϕ * ∈ G q . Finally since ϕ is strictly increasing in (1, ∞), the function ϕ(2 k t) is strictly increasing on (2 −k , ∞) for any k ∈ N. Since k ∈ N is arbitrary, ϕ * is strictly increasing.
Thus, we are led to the following definition: Definition 3.19. The class W stands for the set of all continuous functions ϕ : We apply what we have obtained to small Morrey spaces.
Example 3.20. Let 0 < q < ∞. Let us see how we modify ψ in M ψ q (R n ) to obtain the equivalent space M ϕ q (R n ) with ϕ ∈ G q . • Let ϕ(t) ≡ max(t n p , 1) and ψ(t) ≡ t n p χ (0,1] (t) for t > 0. Then with the equivalence of norms m p q (R n ) = M ψ q (R n ) = M ϕ q (R n ). • Let ϕ(t) ≡ max(t a , 1) with a ≥ n q and ψ(t) ≡ t n q χ (0,1] (t) for t > 0. Then with the equivalence of norms L q uloc (R n ) = M ψ q (R n ) = M ϕ q (R n ). Note that ϕ ∈ G q ∩ W but that ψ ∈ G q \ W in both cases.
So far we have shown that we may assume that ϕ ∈ G q . As a result, we may assume that ϕ is doubling. This observation makes the definition of the norm · M ϕ q more flexible. Remark 3.21. In (15), cubes can be replaced with balls; an equivalent norms will be obtained. Precisely use the norm given by to go through the same argument given for the norm defined by means of cubes.
Related to this definition, we give some definitions related to the class G q . Although we show that it is sufficient to limit ourselves to G q , we still feel that this class is too narrow as the function of ϕ(t) = t log(e + t −1 ) shows. So, it is convenient to relax the condition on ϕ. The following definition will serve to this purpose.
The implicit constants in these inequality are called the almost decreasing/increasing constants.
• Let a, b be real parameters with b = 0. Let ϕ a,b (t) = t a (log(e + t)) b for t > 0. Then ϕ a,b is almost increasing for any a > 0. If a < 0, then ϕ a,b is almost decreasing. • The function ϕ(t) = t+sin 2t, t > 0 is almost increasing but not increasing.
• Let 0 < p, q < ∞ and β 1 , β 2 ∈ R. We write We set ϕ(t) = t n p −B (t) (t > 0) as we did in Example 3.10. We observe that ϕ is almost increasing for all p > 0 and β 1 , β 2 ∈ R. We also note that t → ϕ(t)t − n q is almost decreasing if and only if p = q and β 1 ≤ 0 ≤ β 2 or p > q. Note that ϕ is an equivalent to a function ψ ∈ G q if and only if either p = q, β 1 ≤ 0 ≤ β 2 or p > q since we can neglect the effect of "log".
• Let E ⊂ (0, ∞) be a non-Lebegsgue measurable set. Then ϕ = 1 + χ E is almost increasing and almost decreasing. Note that ϕ is not measurable. • A simple but still standard example is as follows: Let m ∈ N and define where l m (t) is given inductively by: for t > 0. The for all 0 < q < ∞ and m ∈ N, ϕ ∈ G q .
As the following lemma shows, we can always replace an almost increasing function with an increasing function.
Having set down the condition of ϕ, we move on to the norm estimate of the function. The lemma below gives an estimate for the norm of Proof. One method is to reexamine Example 3.6. Here we give a direct proof. It is easy to see that To prove the opposite inequality we consider for any cube Q = Q(y, r). When R ≤ r, then since ϕ ∈ G q . When R > r, then again by virtue of the fact that ϕ ∈ G q .
A direct consequence of the above quantitative information is: Here for the second inequality we invoked Lemma 3.25. Now we consider the role of ϕ. We did not tolerate the case where by the Lebesgue differentiation theorem. The next theorem concerns a situation close to this.
Theorem 3.27. [75, Proposition 3.3] Let 0 < q < ∞ and ϕ ∈ G q . Then, the following are equivalent: If these conditions are satisfied, then M ϕ Proof. If inf t>0 ϕ(t) > 0, then by the Lebesgue differentiation theorem we get We also investigate the reverse inclusion to Theorem 3.27.
By combining these two theorems, we obtain the following result.
Corollary 3.29. [75, Proposition 3.3] Let 0 < q < ∞ and ϕ ∈ G q . Then, the following are equivalent: So, if log ϕ grows or decays slowly we can say that M ϕ q (R n ) is close to L ∞ (R n ).
The next example shows that when the support of the functions are torn apart, the norm does not increase even in the case of generalized Morrey spaces.
Example 3.30. Let 0 < q < ∞ and ϕ ∈ G q . Suppose that we have a collection of only to consider cubes contained in 3Q; if a cube Q intersects both Q and R n \ 3Q, then its triple 3Q engulfs Q and the radius of Q is smaller than that of Q . Thus we have We fix a cube Q(x, r) and estimate We let Accordingly we set As for I 1 , we use (23) and the fact that {Q(a j , r j )} ∞ j=1 is disjoint: As for I 2 , we have only to consider only one summand, since Q(a j , 3r j ) engulfs Q(x, r) if j ∈ H 2 : we can deduce I 2 ≤ 1 using (24).
As an application of Example 3.30, we present another example. Denote by [t] the integer part of t ∈ R. For positive sequences Then as k → ∞.
Devide equally a + [0, 1] n into k n cubes to have a non-overlapping collection Furthermore, we devide [0, k −1 ] n equally into m k n cubes to obtain a collection If we set Then . This is achieved by verifying the assumption of Example 3.30. Let is not overlapping for each l = 1, 2, . . . , 3 n . To check this we need to verify (23) and (24). From Lemma 3.25, we have (23). Meanwhile, Thus for each l = 1, 2, . . . , 3 n From these examples, we obtain the following conclusion: Proof. The "if" part is trivial thanks to Lemma 3.12. Let us concentrate on the "only if" part. Assume q 1 < q 2 . We employ Example 3.31 with q = q 1 and ϕ = ϕ 1 . Then As a result, When inf t>0 ϕ 1 (t) > 0, the situation is different.
Example 3.33. When ϕ 1 is constant, we use Theorem 3.28 to have the following characterization. Let 0 < q 1 , q 2 < ∞, and let ϕ 1 = 1 ∈ G q1 and ϕ 2 ∈ G q2 . Then For the later purpose we use the following characterization of * Proof.
Let R > 0 be large enough as to have supp(g) ⊂ B(R). Then the above inequality reads as: We also have the following characterization of the space M Thus By Lemmas 3.34 and 3. 35, we see that Although we can define many other closed subspaces as we did for Morrey spaces, we content ourselves with these three definitions, which we actually use.

Boundedness properties of the operators in generalized Morrey spaces
Having set down the boundedness properties of generalized Morrey spaces, we are now interested in the boundedness properties of the operators.

4.1.
Maximal operator in generalized Morrey spaces and the class Z 0 . After we generalize the parameter p in the space M p q (R n ), we realize that the boundedness of the maximal operator is obtained due to the condition on q ∈ (1, ∞).
for all measurable functions f .
We observe that we did not requere anything other than ϕ ∈ G q as an evidence of the fact that the parametre q play a central role for the boundedness of the Hardy-Littlewood maximal operator. Theorem 4.1 extends the result in [10] from Morrey spaces to generalized Morrey spaces.
Proof. Once we assume ϕ ∈ G q , the proof of this theorem will be an adaptation of the classical case. Fix a cube Q(x, r). We need to prove The proof of (30) follows from the boundedness of the Hardy-Littlewood maximal operator and the fact that ϕ(5r) ϕ(r) for any r > 0. As for (31), we use with Q = Q(x, r). Then by virtue of the fact that ϕ is increasing, we obtain where for the last inequality, we used the nesting property of M ϕ q (R n ), 1 ≤ q < ∞.
Similar to Theorem 4.1, we can prove the following theorem: Once again any function G q will do as long as ϕ ∈ G q . Theorem 4.3 extends the result in [10] from Morrey spaces to generalized Morrey spaces.
Proof. Simply resort to the weak-(1, 1) boundedness of M and modify the proof of Theorem 4.1.
We move on to the vector-valued inequality. We use the following estimate: For all measurable functions f and cubes Q, we have Proof. We write out M [χ R n \5Q f ](y) in full: where R runs over all cubes. In order that χ R (y) f L 1 (R\5Q) be nonzero, we need to have y ∈ R and R \ 5Q = ∅. Thus, R meets both Q and R n \ 5Q. If R ∈ Q is a cube that meets both Q and R n \ 5Q, then (R) ≥ 2 (Q) and 2R ⊃ Q. Thus, (32) follows.
Unlike the usual maximal inequality, we need the integral condition (33).
Then for all Proof. When u = ∞, the result is clear from Theorem 4.1. The proof is essentially the same as the classical case except that we truly use (33). The proof of the estimate of the inner term remains unchanged except in that we need to generalize the parameter p to the function ϕ.
We can handle f j,1 's in a standard manner as before. Going through a similar argument to the classical case and using Lemma 4.4, we will have If we use the definition of the Morrey norm, we obtain Since ϕ ∈ G q , we obtain If we use (33) and ϕ ∈ G q , then we have Inserting this estimate into (35), we obtain the counterpart to the classical case.
Since (33) is an important condition, we are interested in its characterization. In fact, we have the following useful one.
As we have mentioned, we need (33) for the vector-valued maximal inequality. We give an example showing that (33) is absolutely necessary: By no means (33) is artificial as the following proposition shows: holds for all sequences of measurable functions {f j } ∞ j=1 . Then (33) holds.
Proof. Assume to the contrary; for all m ∈ N ∩ [2, ∞), there would exist r m > 0 such that ϕ(2 m r m ) ≤ 2ϕ(r m ) for r = r m . Fix m ∈ N for the time being. Let us As a result, By our assumption, we have where D does not depend on m. This contradicts to the fact that m ∈ N ∩ [2, ∞) is arbitrary.
Proof. Let It remains to let R → ∞.
We change variables to have the following variant: Proof. Set Then our assumption reads as: according to Theorem 4.9. If we express this inequality in terms of ψ, we obtain the desired result.
In the next proposition, we further characterize and apply our key assumption (33).
holds if and only if holds (33), or equivalently, ϕ satisfies (38) for some ε > 0. If one of these conditions is satisfied, then ∞ r ds ϕ(s) u s 1 ϕ(r) u (r > 0) (40) for all 0 < u < ∞, where the implicit constant depends only on u.
We present an example of the functions in M ϕ q (R n ). Let 0 < η < ∞. We define the powered Hardy-Littlewood maximal operator M (η) by Example 4.15. [80, Proposition 2.11] Let 0 < q < ∞ and ϕ ∈ G q . Define We claim that the following are equivalent; (a) f ∈ M ϕ q (R n ), where · ∞ denotes the ∞ -norm, it follows that (a) and (c) are equivalent.
and that for all l ∈ Z if and only if h ∈ M ϕ q (R n ). To verify this, we let f, g be as in Example 4.15. Suppose first h ∈ M ϕ q (R n ). Then holds as well.
Conversely, assume that and hold for all l ∈ Z. Then we have h ∼ f from (43). Thus f ∈ M ϕ q (R n ) by (42), from which it follows that h ∈ M ϕ q (R n ).
We further present some examples of the functions in M ϕ p (R n ). Proof. First note that ϕ ∈ Z − n q is equivalent to 1 r n r 0 ϕ(t) q t n−1 dt ϕ(r) q (r > 0).

4.2.
Singular integral operators on generalized Morrey spaces. Let T be a singular integral operator. To define the function T f for f ∈ M ϕ q (R n ) we follow the same strategy as the one for f ∈ M p q (R n ). To this end, we need to establish the following estimate: We note that 1 ϕ ∈ Z 0 appeared once again.
Proof. Let Q be a fixed cube. Then we need to prove To this end, we decompose f according to 2Q: As for f 2 , we use the size condition of K, the integral kernel of T , to have the local estimate: By the definition of the norm, Lemma 3.12 and (33), we obtain It remains to integrate this pointwise estimate.
To carry out our program of proving the boundedness of the singular integral operators, we need to investigate the predual and its predual.
run over all expressions above.
Example 4.23. Let 1 < q < ∞ and ϕ ∈ G q . Let A be a non-zero L q (R n )-function supported on a cube Q. Then B ≡ ϕ( (Q)) A is a (ϕ, q)-block supported on Q.
Proposition 4.24. Let 1 < q < ∞ and ϕ ∈ G q be such that ϕ(t) t n q for alll t > 0. Then a measurable function f belongs to H ϕ q (R n ) if and only if f admits a decomposition: for some sequence {A j } ∞ j=1 of (ϕ, q)-blocks supported on cubes of volume less than or equal to 1 and {λ j } ∞ j=1 ∈ 1 (R n ) and B ∈ L q (R n ) with unit norm. Furthermore run over all expressions above.
Proof. Let A be a (ϕ, q)-block supported on Q with (Q) 1. Then we can say that A is a (ϕ, q)-block suppported on Q is and only if 2A has the L q (R n )-norm 1 and A is supported on Q. So, in the decomposition in Definition 4.22 any block A j with the cube Q j satisfying |Q j | 1 can be combined into a block supported on "R n ".
The following lemma justifies the definition above.  A is a (ϕ, q)-block, we can find a cube Q such that supp(A) ⊂ Q and that A L q ≤ |Q| − 1 q ϕ( (Q)). By the Hölder inequality, About the definition above, the following proposition is fundamental: Proposition 4.26. [60, Lemma 4.2] Let 1 < q < ∞ and ϕ ∈ G q . Let f ∈ M ϕ q (R n ) and g ∈ H ϕ q (R n ).
Proof. Let ε > 0 be fixed. Then we can decompose g as and each b k is a (ϕ, q)-block. According to Lemma 4.25, we have Since ε > 0 is arbitrary, it follows that For j ∈ Z and k = (k 1 , k 2 , . . . , k n ) ∈ Z n define the set Q jk by A dyadic cube is a set of the form Q jk for some j ∈ Z, k = (k 1 , k 2 , . . . , k n ) ∈ Z n . The set of all dyadic cubes is denoted by D; D = D ≡ {Q jk : j ∈ Z, k ∈ Z n } . For j ∈ Z the set of dyadic cubes of the j-th generation is given by It is easy to see that H ϕ q (R n ) is a normed space. Similar to the classical case, we can prove the following decomposition theorem: Proof. We suppose that f H ϕ q < 1. It suffices to find a decomposition where λ(Q) is a non-negative number with Q∈D λ(Q) < 3 n .
First, decompose f as and each b k is a (ϕ, q)-block. We will divide K into the disjoint sets K(Q) ⊂ N, so that supp(b k ) ⊂ 3Q and |Q k | ≥ |Q| whenever k ∈ K(Q). We achieve this as follows: Let D = {Q (j) } ∞ j=1 be an emumeration of D. For each k ∈ K, we write for each k. We set K(Q (j) ) ≡ {k ∈ K : j k = j}.
We set .
We now rewrite f as Since each b k is a (ϕ, q)-block, we obtain which implies that b(Q) is a (ϕ, q)-block supported in 3Q. These complete the proof.
Example 4.28. Let 1 < q < ∞, ϕ ∈ G q , and let f ∈ H ϕ q (R n ). Assume that inf t>0 t − n q ϕ(t) > 0, so that Proposition 4.24 is applicable. Then f can be decomposed as where λ(Q) is a non-negative number with Corollary 4.29. Let 1 < q < ∞ and ϕ ∈ G q . Then every function in H ϕ q (R n ) is locally integrable.
Proof. Simply combine Lemma 3.12, Proposition 4.26 and the fact that χ Q ∈ M ϕ q (R n ) for any cube Q.
Proposition 4.30. Let 1 < q < ∞ and ϕ ∈ G q . Assume in addition that ϕ satisfies (33). Suppose that f and f k , (k = 1, 2, . . .), are nonnegative, that each Proof. By Theorem 4.27 f k can be decomposed as and b k (Q) is a (ϕ, q)-block supported in Q and Using (48), (49) and the weak-compactness of the Lebesgue space L q (Q) we now apply a diagonalization argument and, hence, we can select an increasing sequence {k j } ∞ j=1 of integers that satisfies the following: lim where b(Q) is a (ϕ, q)-block supported in Q. We set Then, by the Fatou theorem and (48), which implies f 0 ∈ H ϕ q (R n ).
We will verify that lim j→∞ Q0 for all Q 0 ∈ D. Once (53) is established, we will see that f = f 0 and hence f ∈ H ϕ q (R n ) by virtue of the Lebesgue differentiation theorem because at least we know that f 0 locally in L q (R n ). By the definition of f kj , we have for any cube Q containing Q 0 . We distinguish two cases. If Thus, we are in the position of using the Lebesgue convergence theorem based on Example 4.12. Thus we obtain (53). If lim t→∞ ϕ(t)t − n q > 0, then we go through a similar argument using Example 4.28 to obtain (53).
Since f k ↑ f a.e., we must have by (53) for all Q 0 ∈ D. This yields f = f 0 a.e., by the Lebesgue differentiation theorem, and, hence, f ∈ H ϕ q (R n ). Since we have verified f ∈ H ϕ q (R n ), it follows that This completes the proof of the theorem.
The proof of the following theorem is completely the same as the classical case once Proposition 4.30 is proved.
• The dual of H ϕ q (R n ) is M ϕ q (R n ). More precisely, we have the following mappings: -Any f ∈ M ϕ q (R n ) defines a continuous functional L f by: and and Proof.
• (a) This is a consequence of Proposition 4.26.
For the sake of the simplicity we denote L q (Q j ) by the set of L q (R n )-functions supported on Q j . The functional g → L(g) is well defined and bounded on L q (Q j ). Thus by the duality L p -L p there exists f j such that for all g ∈ L q (Q j ). By the uniqueness of this theorem we can find an L q loc (R n ) function f such that f | Qj = f j a.e. for any j.
We will prove f ∈ M p q (R n ). For this purpose, we take an arbitrary Q and estimate; For a fixed cube Q and a fixed function f we set Then we can write Notice that the function |Q| As a result we have I ≤ L * . This is the desired result. (c) Carefully reexamine the proof of (a) and (b). The proof of (c) is already included in them. • The heart of the matters is to prove (b); (a) and (c) are obtained similarly to (1). Let L : M ϕ q (R n ) → C be a bounded linear mapping. Since ϕ −1 ∈ Z 0 , we have ϕ(t) t 1/P , 0 < t ≤ 1 for some P > 1. For such P > 0, As a consequence, for each k ∈ N the mapping f ∈ L P (R n ) → L (χ [−k,k] n f ) ∈ C is a bounded linear mapping that can be realized by some f k ∈ L P (R n ). Furthermore, f k = f k+1 χ [−k,k] n from the L P (R n )-L P (R n )-duality. So, f = lim Using f k = f k+1 χ [−k,k] n and Proposition 4.30, we see that f ∈ H ϕ q (R n ) and that , it follows that L = L f , as required.
We have the following conclusion.
Theorem 4.32. [73, Theorem 2] Let 1 < q < ∞ and let ϕ ∈ G q satisfy 1 ϕ ∈ Z 0 . Then any singular integral operator, which is initially defined for L ∞ c (R n )functions, can be naturally extended to a bounded linear operator on M ϕ q (R n ). More precisely, T has the following properties: • For all f ∈ L ∞ c (R n ), T f ∈ M ϕ q (R n ) and satisfies T f M ϕ q f M ϕ q . In particular T extends to a bounded linear operator on M ϕ q (R n ). • The adjoint operator T * is bounded on H ϕ q (R n ). • The adjoint T * * of T * is a bounded linear operator on M ϕ q (R n ).
Theorem 4.32 extends the result by Peetre in [84] from Morrey spaces to generalized Morrey spaces. The proof is based on [87].
Proof. It suffices to prove the first assertion, since the remaining assertions are general facts on functional analysis.
Then there exists R > 0 such that f is supported on Q(R). We can decompose . We argue as follows: Let Q be a cube. We need to show We decompose this estimate according to 3Q: Since T is known to be L q (R n )bounded, Meanwhile, using the size condition of T , we have Thus We did not use (33) for the proof of boundedness of the Hardy-Littlewood maximal operator. However for the proof of boundedness of the singular integral operators, (33) is absolutely necessary as the following proposition shows: Since V is a cone, we have We use this estimate and the boundedness of M on M ϕ q (R n ) to obtain By using the inequality log m |R 1 f m (x)| for x ∈ V ∩ B(r m ) and the boundedness of This implies log m ≤ D where D is independent of m, contradictory to the fact that m ≥ 3 is arbitrary. Hence, there exists some m 0 ∈ N such that ϕ (2 m0 r) > 2ϕ(r). Thus the integral condition (33) holds.
We disprove that T can not be exteded to a bounded linear operator on m p q (R n ). Then and fail. In fact, let ϕ(t) = min(t We end this section with extension to the vector-valued inequality. Theorem 4.35. Let 1 < q < ∞, 1 < u < ∞ and ϕ ∈ G q . Let also T be a singular integral operartor. Assume in addition that (33) holds. Then for all Proof. Similar to Theorem 4.5. We also use a well-known inequality: Remark 4.36. One can consider the singular integral operators having rough kernel. Let Ω : S n−1 ≡ {x ∈ R n : |x| = 1} → C be a measurable function having enough integrability and having integral zero. Then define T Ω by See [5] for example.
and obtained a factorization theorem as an application of the M ϕ q (R n )-boundednes of the commutators.

Generalized fractional integral operators in generalized Morrey spaces.
To consider the operator like (1 − ∆) −1 , we are oriented to considering for any suitable function f on R n , where ρ : (0, ∞) → (0, ∞) is a suitable measurable function. Generalized Morrey spaces allow us to consider more general fractional integral operators.
Let us discuss what condition we need in order to guarantee that I ρ enjoys some boundedness property. As we did in [18, p. 761], we always assume that ρ satisfies the "Dini condition" for I ρ .
so that I ρ χ Q (x) is finite for any cube Q and x ∈ R n . In addition, we also assume that ρ satisfies the "growth condition": there exist constants C > 0 and 0 < 2k 1 < k 2 < ∞ such that as was proposed by Perez [85]. Condition (64) is weaker than the usual doubling condition: there exists a constant D > 0 such that whenever r > 0 and s > 0 satisfy r ≤ 2s ≤ 4r.  • ρ(t) = t α , which generates I α , satisfies the doubling condition.
• ρ(t) = t α e −t satisfies the growth condition but fails the doubling condition.
To check that our example is not so artificial we consider the following kernel.
where G s is given by: The function G s is called the Bessel kernel.
The above examples are natural in some sense but somewhat artificial because the second example and the third one do not appear naturally in the context of other areas of mathematics. Here we present some other examples related to partial differential equations.
Example 4.49. Note that the solution to (1 − ∆)f = g, where f is an unknown function and g is a give function is given by: Although it is impossible to find I ρ χ B(x,r) (y), y ∈ R n , we still have a partial but important estimate. Proof. Take x ∈ B(R/2). We write the integral in full: A geometric observation shows that B(x, R/2) ⊆ B(R). Hence, we have Note that we only use the spherical coordinates to obtain the last integral.
In the case of the radially symmetric functions, we can calculate I ρ g R (x) for x small.
the inequality Proof. We prove the right-hand inequality, the left-hand inequality being similar.
A geometric observation shows that |x − y| ∼ |y| for all x ∈ B R 3 and y ∈ R n \ B 2R 3 . Since θ satisfies (66), then It remains to write the most right-hand side in terms of the spherical coordinates.
For covenience, writeρ Sometimes, we are interested in the case where matters are reduced to the classical fractional integral operators. Now we present three different criteria for the boundedness of I ρ . We prove the following three theorems on the boundedness of I ρ on generalized Morrey spaces.
Note that the left-hand side of (68) equals dt.
Proof of Theorem 4.53(Necessity). Assume that I ρ is a bounded linear operator from M ϕ p (R n ) to M ψ 1 (R n ). Let r > 0. By Lemma 4.50 and the doubling property of ψ, we obtaiñ Since ψ ∈ G 1 and I ρ is assumed bounded from M ϕ p (R n ) to M ψ 1 (R n ), it follows that Since χ B(r) M ϕ p ∼ ϕ(r), we concludẽ ρ(r) ϕ(r) ψ(r) .
Thus Theorem 4.53 is proved.
Proof of Theorem 4.53(Sufficiency) . For a ball B(z, r), we let f 1 ≡ f χ B(z,2r) and f 2 ≡ f − f 1 . Then a geometric observation shows B(z, r) ⊂ B(y, 3r) for all y ∈ B(z, 2r). Hence by the Fubini theorem and the normalization, By the use of the definition of the Morrey norm, (68) and the doubling condition of ψ, we obtain Thus the estimate for f 1 is valid. As for f 2 , we let x ∈ B(z, r). Then we have and decomposing the right-hand side dyadically we obtain If we use (68) once again and the doubling condition on ψ, then we obtain |I ρ f 2 (x)| ψ(r) −1 . Thus the estimate for f 2 is valid as well.
. This calculation shows we cannot delete max(1, log r −1 ) and that (1 − ∆) − s 2 does not map M p q (R n ) to L ∞ (R n ) when n p = s and 1 < q ≤ p < ∞.  p − α n with 1 < p < s < ∞ and 0 < α < n. Then I α does If we consider the truncated fractional maximal operator i α given by |y| n−α dy, Hence it follows from Theorem 4.53 that I ρ f M ψ We give a result, which improves Example 4.54.
We move on to the Adams type estimate. • The operator for all r > 0. If ϕ ∈ Z − n p , then (69) is necessary for the boundedness of I ρ from M ϕ p (R n ) to M ϕ p/q q (R n ).
If we use (70) Recall that we are assuming ϕ ∈ Z − n p . Now we invoke Lemma 4.17 to conclude Thus necessity is proven.
As we have mentioned, we want an estimate of Hedberg-type. We may ask ourselves whether inf can be removed, that is, we may assume sup t>0 ϕ(t) = ∞.
However, it can happen that sup t>0 ϕ(t) < ∞ as example below shows.
Lemma 4.59. [18, Lemma 3.1] Let 1 ≤ p < q < ∞, and let ϕ ∈ G p ∩W satisfy (69). Let ρ : (0, ∞) → (0, ∞) be a measurable function satisfying (64). If a measurable function f satsfies f M ϕ p = 1, then Once this estimate is satisfied, we can conclude the proof of Theorem 4.57 as follows: We choose an arbitrary ball B = B(z, r). If we integrate Lemma 4.59, then we have If we multiply both sides by ϕ(r) p , then we have by virtue of the boundedness of the maximal operator M on M ϕ p (R n ). The ball B being arbitrary, we obtain the desired result.
Proof. Recall that k 1 and k 2 appeared in the condition (64) on ρ. Let We have for given x ∈ R n and r > 0. Let Σ I and Σ II be the first and second summations above. Now we invoke the overlapping property: As a result, we have Thus thanks to (70) We use ρ ϕ ∈ Z 0 or (69) now. If we use (69), then we have By the doubling property of ϕ, we obtain Σ II 1 Hence, for all r > 0. First . Then, the conclusion is immediate from (73).
for all r > 0. Since r > 0 and x ∈ R n are arbitrary, it follows that M f (x) ≤ sup r>0 1 ϕ(r) . We can thus find R > 0 such that M f (x) = 2ϕ(R) and, with this R, we can obtain the desired estimate.
In order that I ρ be bounded from M ϕ p (R n ) to M ϕ p/q q (R n ), we must havẽ ρ(r) ϕ(r) 1 ϕ(r) p q according to Theorem 4.53 with ψ = ϕ p/q . We note that if ρ(r) = r α , with 0 < α < n, then I ρ = I α is the classical fractional integral operator, also known as the Riesz potential, which is bounded from L p (R n ) to L q (R n ) if and only if 1 p − 1 q = α n , where 1 < p, q < ∞. The necessary part is usually proved by using the scaling arguments.
Theorem 4.57 characterizes the kernel function ρ for which I ρ is bounded from L p (R n ) to L q (R n ) for 1 < p < q < ∞. We have the following result: Corollary 4.60. [18, Corollary 1.5] Let 1 < p < q < ∞. The operator I ρ is bounded from L p (R n ) to L q (R n ) if and only if ρ(r) r n p − n q for all r > 0.
For ρ(r) = r α , Corollary 4.60 further reads that the operator I ρ is bounded from L p (R n ) to L q (R n ) if and only if α = n p − n q , where 1 < p < q < ∞. With Theorems 4.53-4.57 we can characterize the function ρ for which I ρ is bounded from one Morrey space to another.
The next corollary generalizes the previous characterization in Corollary 4.60.
We show by examples that two statements in Theorem 4.57 are of independent interest. As before we write This function is used to describe the "log"-growth and "log"-decay properties. Also, we fix p and q so that 1 < p < q < ∞. The key properties we are interested in are summarized in the following Let λ < 0 satisfy 0 < p q − 1 λ < n and − n p < λ. Take µ 1 , µ 2 ∈ R arbitrarily. Set β i ≡ p q − 1 µ i for i = 1, 2. Define ϕ(r) ≡ r −λ −µ1−,µ2 (r) and ρ(r) = ϕ(r) 1− p q for r > 0. Then this pair (ρ, ϕ) fulfills the assumptions ρ ϕ ∈ Z 0 and ϕ ∈ Z − n p in Theorem 4.57. Indeed, for r > 0 we havẽ Example 4.63 is an endpoint case of the above example.
Based upon these preliminary results and Lemma 4.51, we will prove Theorems 4.57-4.53.
We remark that (69) includes (70). We prove an estimate. Once we prove Lemma 4.59 below, we can obtain the boundedness of I ρ from M ϕ p (R n ) to M ϕ p/q q (R n ) as we will see below. Here we use the fact that the Hardy-Littlewood maximal operator M is bounded on M ϕ p (R n ), if p > 1 and ϕ is almost decreasing; see Theorem 4.1.
We end this section with comparison of our results with the existing results. We move on to the case of Spanne type.   [17,49,50,70,117,118] for the study of the boundedness of I ρ .
• See [94] for a different type of generalization of the form:

Generalized fractional maximal operators in generalized Morrey spaces.
We discuss the boundedness property of the generalized fractional maximal operator, defined by: where f ∈ L 1 loc (R n ) and ρ is a suitable function from (0, ∞) to [0, ∞).
• If we let ρ(t) = min(t α , 1), then we obtain the local fractional maximal operator m α ; M ρ = m α , where So, we are interested in when M ρ is bounded from M ϕ q (R n ) to M ψ t (R n ). We start with the following necessary condition: Proof. Let R > 0 be fixed. We utilize the pointwise estimate ρ(R)χ B(R) ≤ M ρ χ B(2R) , and the doubling condition of ϕ to obtain Our first result completely characterizes the boundedness of M ρ on generalized Orlicz-Morrey spaces.
The proof hinges on the following Hedberg inequality: Lemma 4.76. [51, (15)] Let 0 < a < 1 < q < ∞. Let ϕ ∈ G q . Then for any Once Lemma 4.76 is proved, we have only to resort to the scaling law (Lemma 3.11) and the boundedness of M on M ϕ q (R n ).
Proof of Lemma 4.76. Remark that both ϕ is bijective. Let R > 0. By using the definition of M , we obtain Thus, it follows that Since R > 0 being arbitrary, we obtain (75).

Proof.
• Using (74), we may assume that ρ = ϕ 1−a . Then since ϕ is a doubling function and 0 < a < 1, we have Thus, Since we can verify (69), it remains to resort to the boundedness of I ϕ 1−a from M ϕ q (R n ) to M ϕ a a −1 q (R n ).
• We let Then we have the boundedness of M ρ on M ϕ 1 (R n ). Hence, by Theorem 4.75, we conclude that the inequality (74) holds.
Finally, under the assumption that ρ ∼ ϕ/ψ, we prove that inequality (76) holds if and only if ϕ satisfies (33). To do this, it is enough to show that (33) follows from (76). Now, assume that the integral condition (33) fails. Then, for any m ∈ N, there exists r m > 0 such that ϕ(2 m r m ) ≤ 2ϕ(r m ).
Since θ ∈ G n and ρ ∼ ϕ/ψ = ϕ/θ(ϕ), ρ(r) ρ(s) for all r ≤ s. Due to this fact and the inequality M ρ f j ρ(2 j r m )χ B(rm) , we have We combine the inequalities (77) and (78) with the boundedness of M ρ from wM ϕ q ( u ) to M ψ 1 ( u ) to obtain m ≤ D where D is independent of m, contradictory to the fact that m ∈ N is arbitrary. Thus the integral condition (33)

5.
Overview of other types of generalizations 5.1. Morrey spaces for general Radon measures and Morrey spaces over metric measure spaces. In addition to the genealization of p into functions, one can replace the Lebesgue measure by general Radon measures. Here we work on a metric measure space (X, d, µ). We refer to [125] for an exhaustive account of the analysis on metric measure spaces. Let k > 0 and 0 < q ≤ p < ∞. We define the Morrey space M p q (k, µ) as Here B(µ) stands for the set of all balls having positve µ-measure. In the Euclidean space R n , Q(µ) stands for the set of all cubes having positve µ-measure. Clearly we have L p (µ) = M p p (k, µ), and by applying Hölder's inequality to (79) we have f M p q 1 (k,µ) ≥ f M p q 2 (k,µ) for all p ≥ q 1 ≥ q 2 > 0 and k ≥ 1. Thus the following inclusions hold: Proposition 5.2. Let (X, d, µ) be the Euclidean space R n with the Euclidean distance and the Lebesgue measure. Then for all M p q (k, µ) = M p q (2, µ) p ≥ q > 0 and k > 1.
We do not recall its proof whose proof hinges on a geometric structure of R n ; see [104,Proposition 1.1]. It can happen that M p q (1, µ) is a proper subset M p q (2, µ) in Proposition 5.2, as was shown in [99]. Instead of the norm above, we can use See [72]. See also [95]. We can also define the generalized Morrey spaces with Radon measures. We work on a metric measure space (X, d, µ). Let ϕ : (0, ∞) → (0, ∞) be a function. Then define ϕ(µ(B(x, kr))) 1 µ(B(x, kr)) B(x,r) |f (z)| q dµ(z) See [45,91] for generalized Morrey spaces with general Radon measures and [90] for weak Morrey spaces with general Radon measures. See [56,53,93,96,97] for this direction of approaches. We refer to [126] for Morrey spaces with gradient in this type of setting. We can consider some concrete cases. See [20,34,39] for generalized Morrey spaces on Heisenberg group, [48] for parabolic genealized Morrey spaces and [13,40] for anisotropic genealized Morrey spaces. See [16,32] and [121] for generalized Morrey spaces on Carnot groups and for generalized Morrey spaces for the p-adic fields.

5.2.
Local generalized Morrey spaces. Motivated by the works [28,43], define the generalized local Morrey space LM ϕ q (R n ) to be the set of all measurable functions f such that The space LM ϕ q (R n ) is sometimes referred to as the B σ -spaces. See [78,106] for B σ -spaces.

5.3.
Generalized Orlicz-Morrey spaces. Instead of p, we can generalize q using Young functions. In the definition below we exclude the case where ϕ(t) = ∞ for some t ∈ (0, ∞).
There are three types of generalizations. We start with the Orlicz-Morrey spaces of the first kind. To this end we start with the defintion of the (ϕ, Φ)-average over Q.
Definition 5.4 ((ϕ, Φ)-average). Let ϕ : (0, ∞) → (0, ∞) be a function and Φ : (0, ∞) → (0, ∞) a Young function. For a cube Q and f ∈ L 0 (Q), define the (ϕ, Φ)-average over Q by: In [75] Nakai defined the generalized Orlicz-Morrey space of the first kind via the (ϕ, Φ)-average. space L ϕ Φ (R n ), the generalized Orlicz-Morrey space of the first kind, is defined to be the set of all measurable functions f for which the norm f L ϕ Φ is finite. Likewise the function space WL ϕ Φ (R n ), the weak generalized Orlicz-Morrey space of the first kind, is defined to be the set of all measurable functions f for which the norm f WL ϕ Φ = sup λ>0 λ χ (λ,∞] (|f |) L ϕ Φ is finite. We move on to Orlicz-Morrey spaces of the second kind. To define Orlicz-Morrey spaces of the second kind, we need another notion of the average. Definition 5.6. Let Φ : [0, ∞) → (0, ∞) a Young function. For a cube Q, define its Φ-average over Q by: With this new definition of the average in mind, Sawano, Sugano and Tanaka define generalized Orlicz-Morrey spaces of the second kind [103].
The function space WM ϕ Φ (R n ) is defined to be the weak generalized Orlicz-Morrey space of the second kind as the set of all measurable functions f for which the norm f WM ϕ Φ is finite.
Finally, Deringoz, Samko and Guliyev define generalized Orlicz-Morrey space of the third kind as follows: Definition 5.8. The generalized Orlicz-Morrey space Z ϕ Φ (R n ) of the third kind is defined as the set of all measurable functions f for which the norm is finite.
We do not go into the details of these function spaces; here we content ourselves with mentioning that the first kind and the second kind are different and that the second kind and the third kind are different according to [22]. See [11,36,52,98,102,119] for more about these function spaces.

5.4.
Generalization of ϕ to the function depending also on x. One can consider the case where ϕ depends on x not only on r.

5.5.
Martingale Morrey spaces. One can use martingales to generalize Morrey spaces [77,81]. Let (Ω, Σ, P ) be a probability space, and F = {F n } n≥0 a nondecreasing sequence of sub-σ-algebras of Σ such that Σ = σ( n F n ). For the sake of simplicity, let F −1 = F 0 . The set B ∈ F n is called atom, more precisely (F n , P )-atom, if any A ⊂ B, A ∈ F n , satisfies P (A) = P (B) or P (A) = 0. Denote by A(F n ) the set of all atoms in F n .
The expectation operator and the conditional expectation operators relative to F n are denoted by E and E n , respectively. It is known as the Doob theorem that, if p ∈ (1, ∞), then any L p -bounded martingale converges in L p . Moreover, if p ∈ [1, ∞), then, for any f ∈ L p , its corresponding martingale {f n } ∞ n=1 with f n = E n f is an L p -bounded martingale and converges to f in L p (see for example [82]). For this reason a function f ∈ L 1 and the corresponding martingale {f n } ∞ n=1 will be denoted by the same symbol f .
Let M be the set of all martingales such that f 0 = 0. For p ∈ [1, ∞], let L 0 p (Ω, F, P ) be the set of all f ∈ L p (Ω, F, P ) such that E 0 f = 0. For any f ∈ L 0 p (Ω, F, P ), its corresponding martingale (f n ) with f n = E n f is an L p -bounded martingale in M. For this reason we regard as L 0 p (Ω, F, P ) ⊂ M. Let B = {B n } n≥0 be sub-families of F = {F n } n≥0 with B n ⊂ F n for each n ≥ 0. We denote by B ⊂ F this relation of B and F.
In this paper we always postulate the following condition on B: There exists a countable subset B ⊂ B 0 such that P B∈B B = 1.
We first define generalized martingale Morrey-Campanato spaces with respect to B as the following: Replacing sup in the Morrey norm by other norms. Instead of taking the supremum, Fueto considered to take the L p -norm in [21]. Although Feuto worked in the weighted setting, we describe it in the unweighted setting.
Definition 5.10. Let 0 < q ≤ p < ∞ and 0 < r ≤ ∞. One defines M p q,r (R n ) to be the set of all f ∈ L q loc (R n ) for which f M p q,r = |Q νm | See [6,67] for applications to partial differential equations.

5.7.
Grandification of the parameter q. In addition to generalization of the parameter p, we can also grandify the parameter q; for f ∈ L 0 (R n ), we define f M p q),θ ≡ sup The space M p q),θ (R n ) collects all f ∈ L 0 (R n ) for which f M p q),θ is finite. See [58,68,71] for more details and variants. 5.8. The case of the variable expoenent. By a variable exponent we mean any measurable function from R n to a subset of (−∞, ∞]. We define · L p(·) which is called the variable Lebesgue norm or the Nakano-Luxenburg norm. be a measurable function. Then define the variable exponent Lebesgue space L p(·) (R n ) with variable exponents by where ρ p (f ) ≡ χ p −1 (0,∞) |f | p(·) 1 + f L ∞ (p −1 (∞)) . Moreover, for f ∈ L p(·) (R n ) one defines the variable Lebesgue norm by Using this techinique, we can generalize the exponent q. See [37,65,66] for example.