On Reflexivity and Other Geometric Properties of Morrey Spaces

Abstract

We propose necessary and sufficient conditions for the local Morrey space to have one of the following properties: the absolutely continuous norm, the Fatou property, the reflexivity. We propose conditions for the global Morrey space $GM_ {l, X}^\tau$ to have the Fatou property. We give an example of global Morrey spaces $GM_{l,X}^ \tau$, which do have not an absolutely continuous norm, and spaces l and X have an absolutely continuous norm.

1 Introduction

It is well known that to study specific properties of operators in analysis, for example, the Hilbert operator, the Hardy—Littlewood maximal function operator, and so on, it is very important to choose correctly spaces in which one can describe various properties of these operators. Spaces $M_{\lambda , L^p}$, introduced by Morrey [1], and their generalizations [2, 3] play an important role in harmonic analysis and in the study of partial differential equations. For the use of spaces in harmonic analysis, properties of these spaces play an important role. In this paper, we propose necessary and sufficient conditions for the local Morrey space with one of following properties: the absolutely continuous norm, the Fatou property, the reflexivity. Parameters of Morrey spaces $M^{\tau }_{l, X}$ are actually two ideal spaces: the function space X and the sequence space l. It is in terms of these spaces that the criteria of absolutely continuous norms, a presence of the Fatou property, and the reflexivity for local Morrey spaces will be given. Note that an important role in describing the criterion of the reflexivity is played by the theorem on the representation of the dual space to the local Morrey space, obtained in the paper of the author [4]. We propose the criterion for the presence of the Fatou property of global Morrey spaces $GM^{\tau }_{l, X}$. We demonstrate an example of global Morrey space $GM^{\tau }_{l, X}$ showing that even if both ideal spaces l, X have an absolutely continuous norm, the space $GM^{\tau }_{l, X}$ does not have this property. This example shows that the absolute continuity of the norm $G{M_{l, X} ^ \tau }$ depends on some other properties of X and l.

Note that a presence of the Fatou property for classical Morrey spaces $M_{\lambda , L^p} (\mathbb {R}^ n)$, ( $0 < \lambda < \frac {n}{p}$), can be extracted from results of [5], and an absence of the absolutely continuous norm for of $M_{\lambda , L^p} (\mathbb {R}^ n)$ can be obtained from Theorem 5 of [6]. The theorems given in this paper contain both of these results.

We note that results thus obtained are also of interest for classical Morrey spaces.

2 Preliminaries

Let μ is the Lebesgue measure in $\mathbb {R}^ n$, let S(μ,  Ω) is a space of all measurable functions $x: \Omega \to \mathbb {R}$ and let χ(D) stand for characteristic function of D. Along with Lebesgue spaces L ^p, p ∈ [1, ∞] ideal and symmetric spaces X are often used in harmonic analysis. Recall their definitions (see, for example [7,8,9]).

Banach space X of measurable functions on Ω is said to be an ideal if it follows from the condition x ∈ X, the measurability of y and the validity of the inequality |y(t)|≤|x(t)| for almost all t ∈ Ω that y ∈ X and ∥y|X∥≤∥x|X∥ (the symbol ∥x|X∥ denotes the norm of the element x in the space X).

Let v ∈ S(μ), v > 0 a. e. (v is a weight). We denote by the symbol X _v a new ideal space in which the norm is given by the equation ∥x|X _v∥ = ∥x ⋅ v|X∥. When X = L ^p, our definition of weighted space differs somewhat from the standard one: when X = L ^p the weight is usually included in the measure.

For every ideal space X, the dual ideal space X ^′ is well defined: it consists of functionals, continuous on X and representable in the integral form, whose norm is defined by the equation

$$\displaystyle \begin{aligned}\| g | X ^{\prime} \| = \sup \{\int_{\Omega} g (t) x (t) dt: \| x | X \| \le 1 \}.\end{aligned}$$

It is easy to verify the equality (X _v)^′ = (X ^′)_1∕v.

When $x: \Omega \to \mathbb {R}$ we denote by λ(f, γ), (γ > 0) the distribution function of x, namely, λ(x, γ) = μ{t ∈ Ω : |x(t)|≤ γ}, and by x ^∗ the rearrangement of x in nonincreasing order. An ideal space X is said to be symmetric if it follows from the condition x ∈ X, the measurability of y and the validity of the inequality λ(y, γ) ≤ λ(x, γ) for all $\gamma \in \mathbb {R}_+$ that y ∈ X and ∥y|X∥≤∥x|X∥. Examples of symmetric spaces are Orlicz, Lorentz, and Marcinkiewicz spaces. Details can be found in [8, 9].

Along with function spaces we need ideal spaces of sequences. Let e ⁱ = {…, 0, 1, 0, …}, ($i \in \mathbb {Z}$, the unit stands in the i-th place) be the standard basis in the space of two-side sequences. We denote by the symbol l an ideal space of sequences $x = \sum _{i=- \infty } ^ {\infty } x_i e ^ i$ ($x_i \in \mathbb {R}$) with the norm ∥x|l∥.

Definitions of Lebesgue, Orlicz, Lorenz, and Marcinkiewicz symmetric sequences spaces and the weighted sequence l _ν similar to definitions of the corresponding function spaces.

All properties listed above for function spaces are preserved for sequence spaces. For details concerning the theory of sequence spaces, see [10].

Let’s go to definitions of basic properties that we will explore in Morrey spaces.

Definition 2.1

(See, for example, [7, 10]). An ideal space of functions X ⊂ S(μ,  Ω) has an absolutely continuous norm (X has the A—property ) if for every x ∈ X following two conditions are satisfied:

$$\displaystyle \begin{aligned}\lim_{\delta \to 0} \sup_{\{D: \mu (D) \le \delta \}} \| x \chi (D) | X \| = 0, \ \ \ \lim_ {R \to \infty} \| x \chi (\Omega \setminus B (R, 0) | X \| = 0. \end{aligned}$$

A discrete ideal space l has an absolutely continuous norm if for each x ∈ l the following two conditions are satisfied:

$$\displaystyle \begin{aligned}\lim_ {k \to \infty} \| \varSigma ^ {- k}_{- \infty} e ^ i x_i | l \| = 0, \ \ \ \lim_ {k \to \infty} \| \varSigma_ {k} ^ {\infty} e ^ i x_i | l \| = 0. \end{aligned}$$

It is well known that spaces $L ^{p} _{\Omega }$ ($l ^{p} _{\omega }$) for p ∈ [1, ∞) have the A—property, and $L ^{\infty } _{\omega }$ ($L ^{\infty } _{\omega }$) have not the A—property.

Definition 2.2

Say (see, for example, [7,8,9]) that an ideal space X ⊂ S(μ,  Ω) has the Fatou property if from 0 ≤ x _n  ↑ x; x _n ∈ X and sup_n∥x _n|X∥ < ∞ it follows that x ∈ X and ∥x|X∥ =sup_n∥x _n|X∥.

The main property of spaces with the Fatou property is that the equality X ^′′ = X holds and the norms in these spaces coincide, i.e.

$$\displaystyle \begin{aligned}\sup \{\int _{\Omega} x (t) f (t) dt: \| f | X ^{\prime} \| \le 1 \} = \| x | X \|. \end{aligned}$$

It is well known that Lebesgue spaces $l ^{p} _{\omega }$, ($l ^{p} _{\omega }$) for p ∈ [1, ∞] have the Fatou property, and the space c ⁰ have not the Fatou property. Other examples of ideal spaces with the Fatou property are of Orlicz, Lorentz, and Marcinkiewicz spaces. Details can be found in [9].

The classical Morrey space $M_{\lambda , L ^ p}$, $(\lambda \in \mathbb {R})$ (see [1]), consists of all functions $f \in L ^ {1, loc} (\mathbb {R} ^ n)$ for which the following norm is finite:

$$\displaystyle \begin{aligned}\| f | M_{\lambda, L_p} \| = \sup_ {x \in \mathbb{R} ^ n} \sup_ {r> 0} r ^ {- \lambda} \| f \chi (B (x, r)) | L ^ p \|.\end{aligned}$$

We note that if λ = 0, then $M_{\lambda , L_p} = L_p,$ if $\lambda = {{n} \over {p}}$, then $M_{\lambda , L_p} = L_ \infty ,$ if λ < 0 or $\lambda > {{n} \over {p}}$, then $M_{\lambda , L_p}$ consists only of functions equivalent to zero.

If we now replace the Lebesgue space L ^p in the definition of the classical Morrey space by an ideal space X, we obtain the Morrey space M _λ,X constructed from the ideal space X in which the norm is defined by the equality

$$\displaystyle \begin{aligned}\| f | M_{\lambda, X} \| = \sup_ {x \in \mathbb{R} ^ n} \sup_ {r> 0} r ^ {- \lambda} \| f \chi (B (x, r)) | X \|.\end{aligned}$$

Next step in the extension of Morrey spaces consists of the replacement of the outer sup-norm by the norm in the ideal space L and the replacement of the balls B(0, r) by homothetic sets $U(0, r) \subset \mathbb {R}^n.$ Below, we always assume that 0 ∈ U(0, 1) and μ(U(0, 1))) ∈ (0, ∞). Moreover, we assume that U(0, 1) is star-shaped with respect to the point 0, that is, if t ∈ U(0, 1), then γt ∈ U(0, 1) for all γ ∈ (0, 1). In general, the star-shapedness assumption is not necessary, but sometimes is useful. Next, make the parameter r discrete.

We denote by Υ the set of non-negative numerical sequences τ = {τ _i} each of which satisfies the conditions

$$\displaystyle \begin{aligned}\forall i: \ \ \ \ \tau_i <\tau_ {i + 1}, \ \ \ \ \bigcup_ {i} (\tau_i, \tau_ {i + 1}] = \mathbb{R}_ +.\end{aligned}$$

When τ _i+1 = ∞, we assume that (τ _i, ∞] = (τ _i, ∞). For every sequence τ = {τ _i} we construct a family of sets U(0, τ _i) and a family of disjoint annuli D _i = U(0, τ _i)∖U(0, τ _i−1).

Definition 2.3 ([4])

Let an ideal space X on $\mathbb {R}^n$, an ideal space l of two-sided sequences with the standard basis {e ⁱ} and a sequence τ ∈ Υ be given.

By the local Morrey space $M ^ {\tau }_{l, X}$ we mean the set of all functions $f \in L ^ {1, loc} (\mathbb {R} ^ n)$ for each of which the following norm is finite:

$$\displaystyle \begin{aligned}\| f | M ^ {\tau}_{l, X} \| = \| \sum_{i=-\infty}^{\infty} e ^ i \| f \chi (U (0, \tau_i)) | X \| | l \|. \end{aligned}$$

By the approximation local Morrey space $\overline {M ^ {\tau }_{l, X}}$ we mean the set of all functions $f \in L ^ {1, loc} (\mathbb {R} ^ n)$ for each of which the following norm is finite:

$$\displaystyle \begin{aligned}\| f | \overline{M ^ {\tau}_{l, X}} \| = \| \sum_{i=-\infty}^{\infty} e ^ i \| f \chi (D_i)) | X \| | l \|. \end{aligned}$$

By the global Morrey space $GM ^{\tau } _{l, X}$ we mean the set of all functions $f \in L ^ {1, loc} (\mathbb {R} ^ n)$ for each of which the following norm is finite:

$$\displaystyle \begin{aligned}\| x | GM ^{\tau} _{l, X} \| = \sup_{t \in \mathbb{R} ^ n} \| \sum_i e ^ i \| x (t +.) \chi (U (0, r_i)) | X \| | l \|. \end{aligned}$$

Discussion of interconnections of spaces ${M ^ {\tau }_{l, X}}$, $\overline {M ^ {\tau }_{l, X}}$ and their examples are given in [4].

We only note that the embedding $M ^ {\tau }_{l, X} \subseteq \overline {M ^ {\tau }_{l, X}}$ is obvious and the reverse embedding, which plays a key role in the theory of discrete Morrey spaces, is given in the following theorem.

Theorem B ([4])

Let an ideal space X on $\mathbb {R}^n$ , an ideal space L on $\mathbb {R}_+$ and a set $U(0, 1) \subset \mathbb {R}^n$ for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞) and a sequence τ ∈ Υ be given. Let spaces $M_ {l, X} ^ \tau$ and $\overline {M_ {l, X} ^ \tau }$ be constructed from spaces X and l, the set U(0, 1) and the sequence τ ∈ Υ. We introduce the operator T : l → l by the equality

$$\displaystyle \begin{aligned} T (\sum_{i=- \infty}^{ \infty} e ^ i x_i) = \sum_{k=- \infty}^{ \infty} e ^ k y_k, \mathit{\text{where}} \ \ \ y_k = \sum_{i= - \infty} ^ k x_i. \end{aligned} $$

(2.1)

When ∥T|l → l∥ = c ₀ < ∞, spaces $M_ {l, X} ^ \tau$ and $\overline {M_ {l, X} ^ \tau }$ have the same set of elements and the following inequalities hold:

$$\displaystyle \begin{aligned}\| f | \overline{ M_ {l, X} ^ \tau} \| \le \| f | M_ {l, X} ^ \tau \| \le c_0 \| f | \overline{ M_ {l, X} ^ \tau} \|.\end{aligned}$$

Note that coincidence conditions do not contain restrictions on the space X and the sequence τ ∈ Υ. There is only a restriction on the sequence space l.

Everywhere below c, possibly with indices, we will denote constants whose exact value are not important.

Discrete spaces are more convenient to consider at least for following reasons. Firstly, all classical Morrey spaces can be realized as discrete Morrey spaces (see the example below), and secondly, one does not need to think about the measurability of the function ∥x(t + .)χ(B(0, r))|X∥.

Note that all discrete Morrey spaces are ideal.

If $l = l ^ p_ \nu$, then it is useful to have in mind that for the norm of the operator T in space $l ^ p _{\nu }$ for p ∈ (1, ∞) the relation is true [11]:

$$\displaystyle \begin{aligned} \| T | l ^ p _{\nu} \to l ^ p _{\nu} \| \approx \left \{ \begin{array}{ll} \sup_k (\sum _{- \infty} ^ k ({{1} \over{\nu_j}}) ^{p ^{\prime}}) ^{1 / p ^{\prime}} \cdot (\sum ^{\infty} _k{\nu_j} ^{p}) ^{1 /{p}}, & \text{for} \ \ p \in (1, \infty), \ {{1} \over{p ^{\prime}}} +{{1} \over{p}} = 1; \\ \sup_k{{1} \over{\nu_k}} \cdot (\sum ^{\infty} _k{\nu_j}), & \text{for} \ \ p = 1; \\ \sup_k \nu_k (\sum _{- \infty} ^ k{{1} \over{\nu_j}}), & \text{for} \ \ p = \infty. \end{array} \right. \end{aligned} $$

(2.2)

The following example shows that most recently investigated Morrey spaces can be implemented as discrete Morrey spaces.

Example 2.1 ([4].)

Let U(0, 1) be a star-shaped set of a positive measure, λ > 0, p ∈ [1, ∞], and the ideal space X the space M _λ,p;X, the norm in which is given by the equality

$$\displaystyle \begin{aligned} \| x | M _{\lambda, p; X} \| = \left \{ \begin{array}{ll} \int_0 ^{\infty} (r ^{- \lambda} \| x \chi (U (0, r)) | X \|) ^ p {{dr} \over{r}} \big{)} ^{1 / p}, & \ \text{for} \ \ p \in [1, \infty); \\ \sup_r \{r ^{- \lambda} \| x \chi (U (0, r)) | X \| \}, & \ \ \text{for} \ \ p = \infty \end{array} \right. \end{aligned}$$

be given.

If p ∈ [1, ∞), then for each function x ∈ M _λ,p;X inequalities hold:

$$\displaystyle \begin{aligned} 2 ^{- \lambda} (ln2) ^{1 / p} (\sum_{i} (2 ^{- i \lambda} \| x \chi (U (0,2 ^{i})) | X \|) ^ p) ^{1 / p} \le \| x | M _{\lambda, p; X} \| \le \end{aligned}$$

$$\displaystyle \begin{aligned} 2 ^{\lambda} \cdot (ln2) ^{1 / p} (\sum_{i} (2 ^{- i \lambda} \| x \chi (U (0,2 ^{i })) | X \|) ^ p) ^{1 / p}. \end{aligned}$$

Thus, for p ∈ [1, ∞) using the equality

$$\displaystyle \begin{aligned}\| x | M _{\lambda, p; X} \|{}_b = (\sum_{i} (2 ^{- \lambda i} \| x \chi (U (0,2 ^{i})) | X \|) ^ p) ^{1 / p}\end{aligned}$$

on the space M _λ,p;X we can introduce an equivalent norm.

If p = ∞, then for each x ∈ M _λ,∞;X inequalities hold:

$$\displaystyle \begin{aligned}2 ^{- \lambda} \sup_{i} 2 ^{- i \lambda} \| x \chi (U (0,2 ^{i})) | X \| \le \| x | M _{\lambda, \infty; X} \| \le 2 ^{\lambda} \sup_{i} 2 ^{- i \lambda} \| x \chi (U (0,2 ^{i})) | X \|.\end{aligned}$$

So using equality

$$\displaystyle \begin{aligned}\| x | M _{\lambda, \infty; X} \|{}_b = \sup_{i} 2 ^{- i \lambda} \| x \chi (U (0,2 ^{i})) | X \|\end{aligned}$$

on the space M _λ,∞;X an equivalent norm can be introduced.

Put τ _i = 2ⁱ, $(i \in \mathbb {Z})$, according to a sequence of points $\{\tau _i \} _{- \infty } ^{\infty }$ organize the partition τ for R ₊ and the equality ω _λ(i) = 2^−λi, $(i \in \mathbb {Z})$ we define a weight sequence. Then we get that for all p ∈ [1, ∞] up to equivalent norms are valid equality:

$$\displaystyle \begin{aligned}M ^{\tau} _{l ^ p _{\omega _{\lambda}}, X} = M _{\lambda, p; X}. \end{aligned}$$

From the explicit form of the weight sequence ω _λ and (2.2), it follows that the operator T defined in Theorem B1 equality (2.1), is bounded from $l ^ p _{\omega _{\lambda }}$ in $l ^ p _{\omega _{\lambda }}$. It follows that up to equivalent norms, the equality holds:

$$\displaystyle \begin{aligned}M ^{\tau} _{l ^ p _{\omega _{\lambda}}, X} = \overline{M ^{\tau} _{l ^ p _{\omega _{\lambda}}, X}}.\end{aligned} $$

3 Main Results

3.1 Geometric Properties of Local Morrey Spaces

We begin by characterizing local approximation Morrey spaces with the A—property.

Theorem 3.1

Let an ideal space X on $\mathbb {R}^n$ , an ideal space of sequences l, a set $U(0, 1) \subset \mathbb {R}^n$ , for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞), and a sequence τ ∈ Υ be given.

The space $\overline {M_ {l, X} ^ \tau }$ has an absolutely continuous norm, if and only if the ideal space of functions X and the ideal space of sequences l have absolutely continuous norm.

Proof

Let $x \in \overline {M_ {l, X} ^ \tau }$ be given, for which

$$\displaystyle \begin{aligned}\| x | \overline{M_ {l, X} ^ \tau} \| = \| \varSigma_{- \infty} ^ {\infty} \| x \chi (D_i) | X \| e ^ i | l \| = 1. \end{aligned}$$

Let $V \subset \mathbb {R}^n$. Fix $k \in \mathbb {N}$ and define

$$\displaystyle \begin{aligned}s_1 = \| \varSigma_{- \infty} ^ {- k} \| x \chi (D_i) \chi (V) | X \| e ^ i | l \|, \ \ \ s_2 = \| \varSigma_{- k} ^ {k} \| x \chi (D_i) \chi (V) | X \| e ^ i | l \|,\end{aligned}$$

$$\displaystyle \begin{aligned}\ \ \ s_3 = \| \varSigma_ {k} ^ {\infty} \| x \chi (D_i) \chi (V) | X \| e ^ i | l \|. \end{aligned}$$

Then following inequalities are true

$$\displaystyle \begin{aligned} \ max \{s_1, s_2, s_3 \} \le \| x \chi (V) | \overline{M_ {l, X} ^ \tau} \| \le s_1 + s_2 + s_3. \end{aligned} $$

(3.1)

Let ideal spaces X and l have absolutely continuous norm. Since the space l has absolutely continuous norm, then lim_k→∞ s ₁ + s ₃ = 0. Since the space X has absolutely continuous norm, for each fixed $k \in \mathbb {N}$ lim_{μ(V )→0} s ₂ = 0. This implies the sufficiency of the conditions of the lemma.

Let the ideal space $\overline {M_ {l, X} ^ \tau }$ has absolutely continuous norm. The proof that both spaces X and l have absolutely continuous norm is analogous to the proof of sufficiency, one only needs to use the left inequality in (3.1). □

From Theorem B and Theorem 3.1 we obtain the following theorem.

Theorem 3.2

Let an ideal space X on $\mathbb {R}^n$ , an ideal space of sequences l, a set $U(0, 1) \subset \mathbb {R}^n$ , for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞), and a sequence τ ∈ Υ be given. Let the operator T : l → l, defined by equality (2.1), is bounded.

The space ${M_ {l, X} ^ \tau }$ has an absolutely continuous norm, if and only if the ideal space of functions X and the ideal space of sequences l have absolutely continuous norm.

To characterize local approximation Morrey spaces to have the Fatou property, we need one definition.

Definition 3.1

Let an ideal space X on $\mathbb {R}^n$ and a sequence τ ∈ Υ be given.

An ideal space X ⊂ S(μ,  Ω) has the τ—Fatou property if for every $i \in \mathbb {Z}$ from 0 ≤ x _n ↑ x; x _n ∈ X, supp x _n ∈ D _i and sup_n∥x _n|X∥ < ∞ it follows that x ∈ X and ∥x|X∥ =sup_n∥x _n|X∥.

For each $D \subseteq \mathbb {R }^{n}$ we denote by χ(D)X the ideal space consisting of restrictions of functions x ∈ X to the set D with norm ∥χ(D)x|χ(D)X∥ = ∥χ(D)x|X∥. Then Definition 3.1 actually means that for any $i \in \mathbb {Z}$ every ideal space χ(D _i)X from of the set of ideal spaces $\{\chi (D_i) X \} _{- \infty } ^{\infty }$ have the Fatou property. Since from 0 ≤ x _n ↑ x, x _n ∈ X, it follows that χ(D _i)x _n ↑ χ(D _i)x for any $i \in \mathbb {Z}$, then every ideal space X with the Fatou property also has the τ—Fatou property.

The following example shows that the opposite is not true.

Example 3.1

We denote by $L ^{\infty } _0 [0,1]$ the subspace of L ^∞[0, 1], consisting of functions for each of which the condition is satisfied

$$\displaystyle \begin{aligned} \lim _{\tau \to 0} ess \sup_{t \in [0, \tau]} | x (t) | = 0. \end{aligned}$$

Let the sequence be given 0… < τ _i−1 < τ _i < …, by which the partition τ is constructed. Then the ideal space $L ^{\infty } _0 [0,1]$ has the τ—Fatou property. On the other hand, an example of the sequence x _n = χ(n ⁻¹, 1) shows that $L ^{\infty } _0 [0,1]$ have not the Fatou property.

We will need another definition.

Definition 3.2 ([12])

Let an ideal space X in S(μ) and a set $D \subseteq \mathbb {R}^n$ be given. A function e _D(.) ∈ X, which is positive almost everywhere on D and outside it is equal to zero, is called the unit on D in X.

It is well known [12] that for any D units exist.

The following lemma will be necessary for us to verify the Fatou property.

Lemma 3.1

Fix a measurable set D. Let the sequence 0 ≤ a ₁ < a ₂ < …a _n < a _n+1 < …, lim _n→∞ a _n = a ₀ < ∞ be given.

Then there exists a sequence of {x _n} elements of X such that: 0 ≤ x ₁ ≤ x ₂ ≤… ≤ x _n ≤ x _n+1 ≤…, for each n supp x _n ⊆ D and the equality follows ∥x _n|X∥ = a _n.

Proof

Fix the unit e _D(.) on D. First, we choose a non-negative element x ₁ ∈ X with support in D for which ∥x ₁|X∥ = a ₁ and define the function $\varphi : \mathbb {R } _ + \to \mathbb {R } _ +$ by the equality φ(x ₁, α) = ∥x ₁ + αe _D(.)|X∥. This function does not decrease with α, φ(x ₁, 0) = ∥x ₁|X∥ and $\lim _{\alpha \to \infty } \varphi (x_1, \alpha ) = \infty$. From the triangle inequality, we obtain the inequality φ(α+δ)−φ(α) = ∥x ₁ +(α+δ)e _D(.)|X∥−∥x ₁ +αe _D(.)|X∥≤ δ∥e _D(.)|X∥, which implies Lipschitz property of the function φ(x ₁, α) with respect to α. We determine the number α ₁ from the equality φ(x ₁, α ₁) = a ₂ and put x ₂ = x ₁ + α ₁ e _D(.). Next, we define the function φ ₁(x ₂, α) = ∥x ₂ + αe _D(.)|X∥. This function has the same properties as the function φ(x ₁, α) = ∥x ₁ + αe _D(.)|X∥. We determine the number α ₂ from the equality φ(x ₂, α ₂) = a ₃ and put x ₃ = x ₂ + α ₂ e _D(.), etc.

The desired sequence is built. □

Theorem 3.3

Let an ideal space X on $\mathbb {R}^n$ , an ideal space of sequences l, a set $U(0, 1) \subset \mathbb {R}^n$ , for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞), and a sequence τ ∈ Υ be given.

The space $\overline {M_ {l, X} ^ \tau }$ has the Fatou property, if and only if

$$\displaystyle \begin{aligned} \mathit{\text{the ideal sequence space}} \ \ l \ \ \mathit{\text{have the Fatou property}}, \end{aligned} $$

(3.2)

$$\displaystyle \begin{aligned} \mathit{\text{the ideal space}} \ \ X \ \ \mathit{\text{have the}} \ \tau - \mathit{\text{Fatou property.}} \end{aligned} $$

(3.3)

Proof

Sufficiency. Let $x_n \in \overline {M_{l, X} ^ \tau }$, $\| x_n | \overline {M_{l, X} ^ \tau } \| \le 1$ and x _n ↑ x. Then for any $i \in \mathbb {Z}$ the relation x _n χ(D _i) ↑ xχ(D _i) holds. Since X has the τ—Fatou property, then ∥x _n χ(D _i)|X∥↑∥xχ(D _i)|X∥ are fulfilled.

Therefore, for a numerical sequence, the condition holds

$$\displaystyle \begin{aligned} \varSigma _{- \infty} ^{\infty} \| x_n \chi (D_i) | X \| e ^ i \uparrow \varSigma _{- \infty} ^{\infty} \| x \chi (D_i) | X \| e ^ i. \end{aligned}$$

Since l has the Fatou property, then

$$\displaystyle \begin{aligned} \lim_{n \to \infty} \| \varSigma _{- \infty} ^{\infty} \| x_n \chi (D_i) | X \| e ^ i | l \| = \| \varSigma _{- \infty} ^{\infty} \| x \chi (D_i) | X \| e ^ i | l \|, \end{aligned}$$

This implies that conditions (3.2)–(3.3) are sufficient.

Necessity.

We show that X has the τ—Fatou property. Let be x _n χ(D _i) ↑ xχ(D _i), a _n = ∥x _n χ(D _i)|X∥, a = ∥xχ(D _i)|X∥. Then $x_n \chi (D_i) \in \overline {M_{l, X} ^ \tau }$. Since the space $\overline {M_{l, X} ^ \tau }$ has the Fatou property, relations $\| x_n \chi (D_i) | \overline {M_{l, X} ^ \tau } \| = \| x_n \chi (D_i) | X \| \| | e ^{i} | l \| \uparrow \| x \chi (D_i) | X \| \| | e ^{i} | l \|$ are fulfilled. This means that the space X has τ—Fatou property.

We show that l has the Fatou property. Let be $b_n = \sum _{- \infty } ^{\infty } b_{i, n} e ^{i} \uparrow \overline {b} = \sum _{- \infty } ^{\infty } \overline { b_{i}} e ^{i}$. Then for each $i \in \mathbb {Z}$ for a numerical sequence conditions $b_{i, n} \uparrow \overline {b_{i}}$ are satisfied. Using Lemma 3.1, for each $i \in \mathbb {Z}$ construct a sequence of nonnegative elements $\{x_{n, i} \} _{- \infty } ^{\infty }$ such that: supp x _n,i ⊆ D _i, x _n,i ↑, ∥x _n,i|X∥ = b _i,n. We defined elements y _n by the equality

$$\displaystyle \begin{aligned} y_n = \sum _{- \infty} ^{\infty} x_{n, i}. \end{aligned} $$

Then relations are fulfilled:

$$\displaystyle \begin{aligned} \| y_n | \overline{M_{l, X} ^ \tau} \| = \| \sum _{- \infty} ^{\infty} \| x_{n, i} | X \| e ^{i} | l \| = \| \sum _{- \infty} ^{\infty} b_{i, n} e ^{i} | l \|; \ \ \ y_n \uparrow \overline{y}. \end{aligned} $$

(3.4)

Since the space $\overline {M_{l, X} ^ \tau }$ has the Fatou property, relations $\overline {y} \in \overline {M_{l, X} ^ \tau }$ and $\lim _{n \to \infty } \| y_n | \overline {M_{l, X} ^ \tau } \| = \| \overline {y} | \overline {M_{l, X} ^ \tau } \|$ are fulfilled.

Put $\overline {b_{i}} = \lim _{n \to \infty } \| y_n \chi (D_i) | X \|$ (these limits exist) and define the vector $\overline {b}$ by the equality

$$\displaystyle \begin{aligned} \overline{b} = \sum _{- \infty} ^{\infty} \overline{b_{i}}. \end{aligned} $$

Then, by construction, the relation $b_n = \sum _{- \infty } ^{\infty } b_{i, n} e ^{i} \uparrow \overline {b}$ holds. From (3.4 ) we get that

$$\displaystyle \begin{aligned} \| b_n | l \| = \| \sum _{- \infty} ^{\infty} b_{i, n} e ^{i} | l \| = \| y_n | \overline{M_{l, X} ^ \tau} \| \uparrow \| \overline{y} | \overline{M_{l, X} ^ \tau} \| = \| \sum _{- \infty} ^{\infty} \overline{b_{i}} e ^{i} | l \| = \| \overline{b} | l \|. \end{aligned}$$

□

Remark 3.1

Often, along with spaces have the Fatou property, they consider a class of ideal spaces in which an unit ball is closed with respect to convergence in measure. The closure property of an unit ball with respect to convergence in measure is often called the BC—property.

The analogue of Definition 3.1 for the BC—property looks like this.

An ideal space X ∈ S(μ,  Ω) has the τBC—property if for every $i \in \mathbb {Z}$ from 0 ≤ x _n ↑ x; x _n ∈ X, supp x _n ∈ D _i and sup_n∥x _n|X∥ < ∞ it follows that x ∈ X and ∥x|X∥ =sup_n∥x _n|X∥.

By the scheme of the proof of Theorems 3.3, we can prove the following theorem.

Theorem 3.4

Let an ideal space X on $\mathbb {R}^n$ , an ideal space of sequences l, a set $U(0, 1) \subset \mathbb {R}^n$ , for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞), and a sequence τ ∈ Υ be given.

The space $\overline {M_ {l, X} ^ \tau }$ has the BC—property, if and only if

$$\displaystyle \begin{aligned} \mathit{\text{the ideal sequence space}} \ \ l \ \ \mathit{\text{have the }}\,\,BC \mathit{\text{-property}}, \end{aligned}$$

$$\displaystyle \begin{aligned} \mathit{\text{the ideal space}} \ \ X \ \ \mathit{\text{have the}} \ \tau\,\,BC \mathit{\text{-property.}} \end{aligned}$$

Now we are ready to characterize reflexive local approximation Morrey spaces.

Theorem 3.5

Let an ideal space X on $\mathbb {R}^n$ , an ideal space of sequences l, a set $U(0, 1) \subset \mathbb {R}^n$ , for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞), and a sequence τ ∈ Υ be given.

The space $\overline {M_{l, X} ^ \tau }$ will be reflexive if and only if

$$\displaystyle \begin{aligned} \mathit{\text{the ideal sequence space}} \ \ l\,\,\mathit{\text{ is reflexive}}, \end{aligned} $$

(3.5)

$$\displaystyle \begin{aligned} \mathit{\text{the ideal space}} \ \ X\,\,\mathit{\text{has the }}A\mathit{\text{ - property and the }\ }\tau\mathit{\text{ - Fatou property}}, \end{aligned} $$

(3.6)

$$\displaystyle \begin{aligned} \mathit{\text{the dual space}} \ \ X^{\prime}\,\,\mathit{\text{has the }}A\mathit{\text{ - property}}. \end{aligned} $$

(3.7)

Proof

The following criterion of reflexivity for the ideal space X is well known (see, for example, [12]): X is reflexive if and only if both spaces X and $X ^{' }$ have the absolutely continuous norm and X has the Fatou property. We will use this criterion.

According to Theorem 3.1, the Morrey space $\overline {M_{l, X} ^ \tau }$ has the absolutely continuous norm if and only if both spaces l and X have the absolutely continuous norm.

By Theorem 3.3, the Morrey space $\overline {M_{l, X} ^ \tau }$ has the Fatou property if and only if spaces l has the Fatou property and X has the τ—Fatou property.

According to Theorem 2 [4] the dual space to $\overline {M_{l, X} ^ \tau }$ coincides with the space $\overline {M_{l ^{\prime }, X ^{\prime }} ^ \tau }$.

According to Theorem 3.1 the Morrey space $\overline {M_{l ^{\prime }, X ^{\prime }} ^ \tau }$ has the absolutely continuous norm if and only if both spaces l ^′ and X ^′ have the absolutely continuous norm.

Using the criterion of reflexivity of an ideal space again, we’ll find that the reflexivity of the space $\overline {M_{l, X} ^ \tau }$ is equivalent to the fulfillment of conditions (3.5)–(3.7) □

Corollary 3.1

Let an ideal space X on $\mathbb {R}^n$ , an ideal space of sequences l, a set $U(0, 1) \subset \mathbb {R}^n$ , for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞), and a sequence τ ∈ Υ be given.

If spaces X and l are reflexive, then the space $\overline {M_{l, X} ^ \tau }$ is reflexive too.

Proof

If the space X is reflexive, then, as indicated above, this is equivalent to following three conditions being satisfied: both ideal spaces X and X ^′ have the absolutely continuous norm, the ideal space X has the Fatou property. This and the reflexivity of l imply that conditions (3.5)–(3.7) are satisfied and, therefore, the space $\overline {M_{l, X} ^ \tau }$ is reflexive too. □

From Theorem B and Theorem 3.5 we get the following corollary.

Corollary 3.2

Let an ideal space X on $\mathbb {R}^n$ , an ideal space of sequences l, a set $U(0, 1) \subset \mathbb {R}^n$ , for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞), and a sequence τ ∈ Υ be given. Let the operator T : l → l, defined by equality (2.1), be bounded.

The local Morrey space ${M_{l, X} ^ \tau }$ is reflexive if and only if conditions (3.5)–(3.7) are satisfied.

In particular, if spaces X and l are reflexive, then the space ${M_{l, X} ^ \tau }$ is reflexive too.

3.2 Geometric Properties of Global Morrey Spaces

Now we turn to a discussion of geometric properties of global Morrey spaces ${GM_{l, X} ^ \tau }$.

Practically verbatim repeating the proof of sufficiency in Theorem 3.3, we can prove the following theorem.

Theorem 3.6

Let an ideal space X on $\mathbb {R}^n$ , an ideal space of sequences l, a set $U(0, 1) \subset \mathbb {R}^n$ , for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞), and a sequence τ ∈ Υ be given.

If both ideal spaces l and X have the Fatou property, then the space $G{M_{l, X} ^ \tau }$ has the Fatou property too.

Theorem 3.6 has a complete analogue for the BC—property.

Theorem 3.7

Let an ideal space X on $\mathbb {R}^n$ , an ideal space of sequences l, a set $U(0, 1) \subset \mathbb {R}^n$ , for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞), and a sequence τ ∈ Υ be given.

If both ideal spaces l and X have the BC—property, then the space $G{M_{l, X} ^ \tau }$ has the BC—property too.

We turn to the study of the A—property for global Morrey spaces.

We will start with the construction of the example. This example will be important when investigating the A—property for global Morrey spaces ${GM_{l, X} ^ \tau }$.

Let a weight sequence $\{\omega (.) \} _{- \infty } ^{\infty }$ be given. Denote by $c ^{0} _{\omega }$ sequence space $\{a_i \} _{- \infty } ^{\infty }$, each of which satisfies conditions:

$$\displaystyle \begin{aligned} \lim_{i \to - \infty} a_i \omega (i) = 0, \ \ \lim_{i \to \infty} a_i \omega (i) = 0, \end{aligned}$$

and a norm of $\{a_i \} _{- \infty } ^{\infty }$ is given by the equality:

$$\displaystyle \begin{aligned}\| \sum _{- \infty} ^{\infty} a_i e ^{i} | c ^{0} _{\omega} \| = \max_{i} | a_i | \omega (i).\end{aligned}$$

It is easy to see that the space $c ^{0} _{\omega }$ have the absolutely continuous norm.

Now we show that for any symmetric space X the Morrey space ${GM ^{\tau } _{c ^{0} _{\omega }, X}}$ under minimal conditions on the weight sequence have not the absolutely continuous norm.

The main restriction on the weight sequence will follow from following definitions.

Definition 3.3

Let an ideal space X on $\mathbb {R}^n$, an ideal space of sequences l, a set $U(0, 1) \subset \mathbb {R}^n$, for which 0 ∈ U(0, 1) and μ(U(0, 1)) ∈ (0, ∞), and a sequence τ ∈ Υ be given.

We say that the weight {ω(.)} belongs to the class B(l, X), ({ω(.)}∈ B(l, X)) if for each $i \in \mathbb {Z}$ the equality holds

$$\displaystyle \begin{aligned} \| \chi (U (0, \tau _i)) |G{M ^{\tau} _{l _{\omega}, X}} \| = \omega (i) \| \chi (U (0, \tau _i)) | X \|. \end{aligned} $$

(3.8)

If l = c ⁰, then the condition (3.8) means that relations are fulfilled

$$\displaystyle \begin{aligned}\omega (i-1) \| \chi (U (0, \tau _{i-1}) | X \| \le \omega (i) \| \chi (U (0, \tau _i)) | X \|, \ \ \ (i \in \mathbb{Z});\end{aligned}$$

$$\displaystyle \begin{aligned}\lim_{i \to - \infty} \omega (i) \| \chi (U (0, \tau _i)) | X \| = 0.\end{aligned}$$

Definition 3.4

We say that the weight {ω(.)} satisfies the δ ₂—condition with respect to the system {U(0, τ _i)} if there is m ₀ ∈ N such that for any i < j the inequality $\omega (i) \le \frac {1}{2} \omega (j)$ follows from the inequality μ(U(0, τ _i)) ≥ m ₀ μ(U(0, τ _j)).

Let the sequence {ω(.)} is defined by the equality ω(i) = ϖ(μ(U(0, τ _i))). Then the δ ₂—condition with respect to the system {U(0, τ _i)} is closely related to the inequality $\sup _{t \in R_ +} \frac {\varpi (m_0 t)}{\varpi (t)} \le \frac {1}{ 2}$.

Now everything is ready for us to prove the fact that the space ${GM ^{\tau } _{c ^{0} _{\omega }, X}}$ have not absolutely continuous norm. We restrict ourselves to the one-dimensional case; moreover, we assume that all our functions outside the segment [−1, 1] are determined by zero. Denote by U(0, τ) a segment centered at zero of length 2τ.

Theorem 3.8

Let a symmetric space X on [−1, 1] and numerical sequences τ ∈ Υ, {ω(.)}, for which

$$\displaystyle \begin{aligned} 0.5> \tau _{- 1}> \tau _{- 2}> \ldots> \tau _{- i}> \ldots> 0, \ \ \ \lim_{i \to - \infty} \tau _i = 0; \ \ \ \lim_{i \to - \infty} \omega (.) = \infty, \end{aligned} $$

(3.9)

be given. Let a sequence {ω(.)} satisfies the δ ₂ —condition for the system {U(0, τ _i)}, and {ω(.)}∈ B(c ⁰, X). Let the space $G{M_{c ^{0} _{\omega }, X} ^ \tau }$ be constructed from spaces X, $c ^{0} _{\omega }$ and the sequence τ ∈ Υ.

Then the space $G{M_{c ^{0} _{\omega }, X} ^ \tau }$ have not the absolutely continuous norm.

Remark 3.2

Note, that Theorem 3.8 holds for symmetric spaces with the A—property, for example, we can put X = L ^p for p ∈ [1, ∞). It follows that even if both ideal spaces l and X in the definition of a global Morrey space $G {M_{l, X}^\tau }$ have the A—property, then the space $G{M_{l, X}^\tau }$ may have not this property. This is the fundamental difference between Theorems 3.8 and 3.1. Classical global Morrey spaces $M_{\lambda , L ^ p}$, $0< \lambda < {{n} \over {p}}$ ($G M _{\lambda , \infty ; L^p}$ in the notation of this article, see Example 2.1) have not the A –property. This fact can be obtained from the result [6]. It follows from Example 2.1 that the space $M_{\lambda , L ^ p}$ coincides with the space $G {M_{l^{\infty }_{\omega _\lambda }, L^{p}}^\tau }$, $\omega _\lambda (i) = 2^{-i \lambda },\ \ \lambda >0, \ \ i \in \mathbb {Z}$. Since $G {M_{c^{0}_{\omega _\lambda }, L^{p}}^\tau } \subset G {M_{l^{\infty }_{\omega _\lambda }, L^{p}}^\tau }$, our theorem is stronger than the result of Y. Savano [6].

Proof

Let’s choose a subsequence of negative numbers {n _i} so that conditions were met:

$$\displaystyle \begin{aligned} \omega (n_j) \sum_{k = j + 1} ^{\infty} \frac{1}{\omega (n_k)} \le (\frac{1}{2}) ^{j}, \ \ \ j = 1,2, \ldots..; \end{aligned} $$

(3.10)

$$\displaystyle \begin{aligned} \sum_{i = 1} ^{\infty} (m_0 ^{i} +2) \mu (U (0, r_{n_i})) <1. \end{aligned} $$

(3.11)

The possibility of choosing such a subsequence of numbers {n _i} follows from (3.9).

We define the function $f: [-1,1] \to \mathbb {R} _ +$ as follows. Put $s_1 = \frac {1}{2} \mu (U (0, r_{n_1}))$, $t_1 = \mu (U (0, r_{n_1}))$. We define the function f on J ₁ = [t ₁ − s ₁, t ₁ + s ₁] by the equality:

$$\displaystyle \begin{aligned}f (t) = \frac{\chi (U (t_1, r_{n_1}))}{\| \chi (U (0, r_{n_1})) | X \| \omega (n_1)} .\end{aligned}$$

Put $s_2 = \frac {1}{2} \mu (U (0, r_{n_2}))$, t ₂ = t ₁ + (2m ₀ + 1)s ₁ + s ₂ and define the function f on J ₂ = [t ₂ − s ₂, t ₂ + s ₂] by the equality:

$$\displaystyle \begin{aligned}f (t) = \frac{\chi (U (t_2, r_{n_2}))}{\| \chi (U (0, r_{n_2})) | X \| \omega (n_2)} .\end{aligned}$$

Put $s_3 = \frac {1}{2} \mu (U (0, r_{n_3}))$, $t_3 = t_2 + (2 m_0 ^{2} +1) s_2 + s_3$ and define the function f on J ₃ = [t ₃ − s ₃, t ₃ + s ₃] by the equality:

$$\displaystyle \begin{aligned}f (t) = \frac{\chi (U (t_3, r_{n_2}))}{\| \chi (U (0, r_{n_2})) | X \| \omega (n_2)}.\end{aligned}$$

For arbitrary k ≥ 2 we put $s_k = \frac {1}{2} \mu (U (0, r_{n_k})$, $t_k = t_{k-1} + (2 m_0 ^{ k-1} +1) s_{k-1} + s_{k}$ and define the function f on J _k = [t _k − s _k, t _k + s _k] by the equality:

$$\displaystyle \begin{aligned}f (t) = \frac{\chi (U (t_k, r_{n_k}))}{\| \chi (U (0, r_{n_k})) | X \| \omega (n_k)}.\end{aligned}$$

Etc.

Outside of ⋃_k J _k, we define the function f to zero.

Note that the distance between J _k−1 and J _k is exactly equal to $d (J_{k-1}, J_k) = 2 m_0 ^{k-1} s_{k-1} = m_0 ^{k-1} \mu (U (0, r_{n_{k-1}}).$ Therefore From this relation, equalities

$$\displaystyle \begin{aligned}t_k = t_{k-1} + (2 m_0 ^{k-1} +1) s_{k-1} + s_{k} = t_1 + \sum_{i = 1} ^{k- 1} (2 m_0 ^{i} +1) s_{i} + \sum_{i = 2} ^{k} s_i =\end{aligned}$$

$$\displaystyle \begin{aligned}\mu (U (0, r_{n_1})) + \sum_{i = 1} ^{k-1} (m_0 ^{i} + \frac{1}{2}) \mu (U (0, r_{n_i})) + \frac{1}{2} \sum_{i = 2} ^{k} \mu (U (0, r_{n_i})) =\end{aligned}$$

$$\displaystyle \begin{aligned}\frac{3}{2} \mu (U (0, r_{n_1})) + \sum_{i = 1} ^{k-1} m_0 ^{i} \mu (U (0, r_{n_i })) + \sum_{i = 2} ^{k-1} \mu (U (0, r_{n_i})) + \frac{1}{2} \mu (U (0, r_{n_k} ))\end{aligned}$$

and inequalities (3.11), it follows that $f: [-1,1] \to \mathbb {R} _ +$ is defined correctly.

Let us show that $f \in G{M_{c ^{0} _{\omega }, X} ^ \tau }$ and

$$\displaystyle \begin{aligned} \| f | G{M_{c ^{0} _{\omega}, X} ^ \tau} \| \le 1. \end{aligned} $$

(3.12)

For the proof (3.12), we estimate the value:

$$\displaystyle \begin{aligned}S = \omega (i) \| f \chi (U (t, r_{i})) | X \| = \omega (i) \| \chi (U (t, r_{i})) \sum_{k \ge 1} \frac{\chi (U (t_k, r_{n_k}))}{\| \chi (U (0, r_{ n_k})) | X \| \omega (n_k)} | X \| =\end{aligned}$$

$$\displaystyle \begin{aligned}\omega (i) \| \sum_{k \ge 1} \frac{\chi (U (t, r_{i})) \chi (U (t_k, r_{n_k}))}{\| \chi (U (0, r_{ n_k})) | X \| \omega (n_k)} | X \|. \end{aligned}$$

When l + 1 summands in this sum are no equality zero, then they go in a row due to the construction (l can take the value ∞, arguments are is valid in this case as well). Let’s denote the number of the first non-zero summand by n _j, and the last one by n _j+l. Let’s first l ≥ 1. Then

$$\displaystyle \begin{aligned} S \le \omega (i) \sum_{k = j} ^{j + l} \frac{\| \chi (U (t, r_{i})) \bigcap (U (t_2, r_{n_k}) ) | X \|}{\| \chi (U (0, r_{n_k})) | X \| \omega (n_k)} \le \omega (i) \sum_{k = j} ^{j + l} \frac{1}{\omega (n_k)}. \end{aligned} $$

(3.13)

Since U(t, r _i) intersects $U (t_j, r_{n_j})$ and $U (t_{j + 1}, r_{n_{j + 1}})$, then the inequality $\mu (U (t, r_{i})) \ge d (J_{j}, J_{j + 1}) = m_0 ^{j} \mu (U (0, r_{n_{j}} )$ holds. Therefore, the inequality $\omega (i) \le (\frac {1}{2}) ^{j} \omega (n_j)$ follows from the δ ₂—condition. From the last ratio and (3.10) we get:

$$\displaystyle \begin{aligned} S \le (\frac{1}{2}) ^{j} \omega (n_j) \sum_{k = j} ^{j + l} \frac{1}{\omega (n_k)} \le 1 . \end{aligned} $$

(3.14)

Let’s now analyze the case of l = 0. Then the sum will consist of a single term. Let’s evaluate this term:

$$\displaystyle \begin{aligned}S = \frac{\omega (i)}{\omega (n_j)} \frac{\| \chi (U (t, r_{i})) \bigcap (U (t_j, r_{n_j})) | X \|}{\| \chi (U (0, r_{n_j})) | X \|}. \end{aligned}$$

If i ≥ n _j, then the monotonicity of the weight sequence implies the inequality

$$\displaystyle \begin{aligned} S \le \frac{\omega (i)}{\omega (n_j)} \frac{\| (U (t_j, r_{n_j})) | X \|}{\| \chi (U (0, r_{n_j})) | X \|} = \frac{\omega (i)}{\omega (n_j)} \le 1. \end{aligned} $$

(3.15)

If i < n _j, then from ω(.) ∈ B(c ⁰, X) we get:

$$\displaystyle \begin{aligned} S \le \frac{\omega (i)}{\omega (n_j)} \frac{\| \chi (U (t, r_{i})) | X \|}{\| \chi (U (0, r_{n_j})) | X \|} \le 1. \end{aligned} $$

(3.16)

From (3.14), (3.15), and (3.16) it follows the inequality (3.12).

Checking that the condition is met:

$$\displaystyle \begin{aligned} \lim_{i \to - \infty} \omega (i) \| f \chi (U (t, r_i)) | X \| = 0. \end{aligned} $$

(3.17)

Put $M = \overline {\bigcup _{k \ge 1} J_k}$. The set $t \in \mathbb {R} \backslash M$ is open. So if $t \in \mathbb {R} \backslash M$, the condition (3.17) will be met. Let t ∈ J _k. Then for i →−∞ the analogue of the inequality (3.16) holds. Using (3.16) and ω(.) ∈ B(c ⁰, X), we obtain the equality

$$\displaystyle \begin{aligned}\lim_{i \to - \infty} \omega (i) \| f \chi (U (t, r_i)) | X \| = \lim_{i \to - \infty} \frac{\omega (i)}{\omega (n_k)} \frac{\| \chi (U (t, r_{i})) | X \|}{\| \chi (U (0, r_{n_k}))} = 0. \end{aligned}$$

It remains to consider the point t _∗ =lim_k→−∞ t _k. Fix U(t _∗, r _i).

Denote by n _j(i) the number of the first set for which $U(t_{*}, r_i)\bigcap U(t_j,r_{n_j(i)}) \neq \emptyset$. Then, by analogy with the inequality (3.14), we get

$$\displaystyle \begin{aligned}\omega (i) \| f \chi (U (t, r_i)) | X \| \le (\frac{1}{2}) ^{j (i)} \omega (n_j (i)) \sum_{k = j (i)} ^{\infty} \frac{1}{\omega (n_k)} \to 0 \ \ \text{for} \ \ i \to- \infty. \end{aligned}$$

Therefore (3.17) also holds at the point t _∗.

Thus, $f \in G{M_{c ^{0} _{\omega }, X} ^ \tau }$.

We show that for every n ∈ N the inequality $\| \chi (t _{*} - n ^{- 1}, t _{*} + n ^{- 1}) f | G{M_{c ^{0} _{\omega }, X} ^ \tau } \| \ge 1$ holds. From this it follows that $G {M_{c^{0}_{\omega }, X}^\tau }$ have not the absolutely continuous norm.

Indeed, the interval (t _∗− n ⁻¹, t _∗ + n ⁻¹) contains the set $U (t_k, r_{n_k})$ for a sufficiently large k. Therefore

$$\displaystyle \begin{aligned}\| \chi (t _{*} - n ^{- 1}, t _{*} + n ^{- 1}) f | G{M_{c ^{0} _{\omega}, X} ^ \tau} \| \ge \| \frac{\chi (U (t_k, r_{n_k}))}{\| \chi (U (0, r_{n_k})) | X \| \omega (n_k)} | G{M_{c ^{0} _{\omega}, X} ^ \tau} \| = 1.\end{aligned}$$

□

Corollary 3.3

Suppose that conditions of Theorem 3.8 are satisfied.

Then the global Morrey space $G{M_{c ^{0} _{\omega }, X} ^ \tau }$ contains an isometric copy of the space l ^∞.

Proof

It is enough to define the operator $S \sum a_k e ^{k}: l ^{\infty } \to G{M_{c ^{0} _{\omega }, X} ^ \tau }$, which performs the isometry by the equality:

$$\displaystyle \begin{aligned}S \sum a_k e ^{k} = \sum a_k \frac{\chi (U (t_k, r_{n_k}))}{\| \chi (U (0, r_{n_k})) | X \| \omega (n_k)}. \end{aligned}$$

□

Remark 3.3

In this article we considered Morrey spaces of functions defined on $\mathbb {R}^n$. If we consider Morrey spaces of functions on a subset $\Omega \subset \mathbb {R}^n$, (0 ∈ Ω), then in Definition 3.1 is necessary to replace U(0, τ) by U(0, τ) ∩ Ω. All results will remain true.