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On the relationship between embeddings and coverings of cones of functions E. G. Bakhtigareevaa, M. L. Goldmanb a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia b Peoples' Friendship University of Russia, Moscow, Russia
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    § 1. IntroductionThe paper is devoted to pointwise and integral coverings for cones of nonnegative measurable functions equipped with nondegenerate positively homogeneous functionals. The connection of these coverings with estimates for pointwise and integral majorants on these cones and with embeddings of the cones in ideal function spaces is investigated. Note that the general properties of cones of nonnegative measurable functions and, first of all, cones of functions with monotonicity properties and cones of decreasing rearrangements in connection with the theory of Lorentz spaces, were considered by a number of authors (see, for example, [1]–[8]). Applications of such cones to the theory of generalized Bessel–Riesz potentials were studied in [1], [2] and [9]–[12]. The basic properties of the classical Bessel and Riesz potentials are presented in the books [13]–[15] by Maz’ya, Nikol’skii and Stein. In § 2 of this paper we study the concepts of pointwise and integral coverings of cones and their connection with estimates for pointwise and integral majorants on the cones. In § 3 we use the results presented to study the relationship between cones of decreasing rearrangements of generalized Bessel–Riesz potentials. These results play an important role in the investigation of the integral properties of potentials. Section 4 is devoted to embeddings of cones in ideal spaces. Some general properties of ideal spaces are briefly discussed, in particular, the relationship between such characteristics as the Fatou and Riesz–Fischer properties and the equivalence of the Riesz–Fischer property to the completeness of an ideal space. The main attention is paid to the question of when an embedding of a covering cone in an ideal space implies a similar embedding for the covered cone. This question is considered both for the pointwise and integral versions of a covering. In the case of an integral covering the additional condition of the compatibility of the quasinorm on the ideal space with the integral estimate is imposed. Section 5 is devoted to the construction of ideal spaces with these compatibility properties. It substantiates an algorithm for constructing a new ideal space the quasinorm in which is compatible with the integral estimate, from a general ideal space, in which such a compatibility need not take place. We show that if the original ideal space has the Fatou or Riesz–Fischer property, then the new space we construct also has these properties. § 2. Pointwise and integral coverings and majorants for cones2.1. Definitions of coverings and majorants for conesLet $q \in (0, \infty)$ and $T \in (0, \infty]$. Throughout this paper we assume that $\mu$ is a countably additive measure on $(0, T)$ such that
$$
\begin{equation}
0<M_q(t):=\biggl(\int_{(0, t]}d\mu\biggr)^{1/q}< \infty \quad \forall\, t \in (0, T)
\end{equation}
\tag{2.1}
$$
and
$$
\begin{equation}
\biggl(\int_{(0, T)}d\mu\biggr)^{1/q}< \infty \quad \text{if } T< \infty.
\end{equation}
\tag{2.2}
$$
We denote by $L_{0, \mu}(0, T)$ the set of all (in general, complex-valued) functions on $(0, T)$ that are measurable with respect to the measure $\mu$. Let
$$
\begin{equation}
L_{0, \mu}^{+}(0, T)=\bigl\{ f \in L_{0, \mu}(0, T)\colon f(t)\geqslant 0,\, t \in (0, T)\bigr\};
\end{equation}
\tag{2.3}
$$
let $K$ be a cone in $L_{0, \mu}^{+}(0, T)$ equipped with a functional $\rho_K$:
$$
\begin{equation}
\rho_K\colon K\,{\to}\, [0, \infty), \qquad h \in K, \quad\alpha \in [0, \infty) \quad\Longrightarrow\quad \alpha h \in K, \qquad\rho_K(\alpha h)=\alpha \rho_K(h);
\end{equation}
\tag{2.4}
$$
$$
\begin{equation}
\rho_K(h)=0 \quad\Longrightarrow\quad h=0 \quad \mu\text{-a.e. on } (0, T).
\end{equation}
\tag{2.5}
$$
For the cone $K$ we introduce the pointwise and $q$-integral majorants:
$$
\begin{equation}
\lambda_K(t)=\sup \bigl\{ h(t)\colon h \in K;\,\rho_K(h) \leqslant 1\bigr\}, \qquad t\in (0, T),
\end{equation}
\tag{2.6}
$$
and
$$
\begin{equation}
\widetilde{\lambda}_{K,q}(t)=\sup \biggl\{ M_q(t)^{-1}\biggl(\int_{(0, t]}h^q\,d\mu\biggr)^{1/q}\colon h \in K;\, \rho_K(h) \leqslant 1\biggr\}, \qquad t\in (0, T).
\end{equation}
\tag{2.7}
$$
If the functions in $K$ are decreasing, then
$$
\begin{equation}
\lambda_K(t)\leqslant \widetilde{\lambda}_{K,q}(t), \qquad t\in (0, T),
\end{equation}
\tag{2.8}
$$
since for $0\leqslant h\!\downarrow$ we have the bound
$$
\begin{equation*}
\begin{gathered} \, M_q(t)^{-1}\biggl(\int_{(0, t]}h(\tau)^q\,d\mu(\tau)\biggr)^{1/q} \geqslant M_q(t)^{-1}\biggl(\int_{(0, t]}h(t)^q\,d\mu(\tau)\biggr)^{1/q}=h(t), \\ t\in (0, T). \end{gathered}
\end{equation*}
\notag
$$
Definition 1. A cone $M$ equipped with a functional $\rho_M$ covers pointwise a cone $K$ equipped with a functional $\rho_K$, with covering constants $c_0\in (0, \infty)$ and $c_1\in [0, \infty)$, if for each function $h_1 \in K$ there exists a function $h_2 \in M$ such that $\rho_M(h_2)\leqslant c_0\rho_K(h_1)$ and for all $\tau \in (0, T)$ Remark 1. In Definition 1 we assume that Definition 2. Let $q, r \in (0, \infty)$. A cone $M \subset L^{+}_{0,\mu}(0, T)$ equipped with a functional $\rho_M$ covers a cone $K$ equipped with a functional $\rho_K$ $(q,r)$-integrally, with covering constants $c_0 \in (0, \infty)$ and $c_1 \in [0, \infty)$, if for each function $h_1 \in K$ there exists a function $h_2 \in M$ such that $\rho_M(h_2)\leqslant c_0\rho_K(h_1)$ and for all $t \in (0, T)$ Remark 2. In (2.11) we assume that Remark 3. If $c_1=0$ in (2.11), then this inequality takes the form We introduce the notation $K\leqslant M(c_0, c_1)$ for a pointwise covering and ${K\underset{(q,r)}{\prec} M(c_0, c_1)}$ for a $(q,r)$-integral covering. The equivalence (pointwise or $(q,q)$-integral) of cones $K\cong M$ or $K\underset{(q,q)}{\approx}M$, respectively, means the mutual covering of these cones, that is,
$$
\begin{equation}
K\cong M \quad\Longleftrightarrow\quad K\leqslant M(c_0, c_1), \qquad M\leqslant K(\widetilde{c}_0, \widetilde{c}_1),
\end{equation}
\tag{2.14}
$$
and
$$
\begin{equation}
K\underset{(q,q)}{\approx}M \quad\Longleftrightarrow\quad K\underset{(q,q)}{\prec} M(c_0, c_1), \qquad M\underset{(q,q)}{\prec}K(\widetilde{c}_0, \widetilde{c}_1).
\end{equation}
\tag{2.15}
$$
Theorem 1. In the notation (2.1)–(2.15) the following implications hold. 1. For $0<r\leqslant q<\infty$ a pointwise covering of cones implies a $(q,r)$-integral covering for these cones:
$$
\begin{equation}
K\leqslant M(c_0, c_1) \quad\Longrightarrow\quad K\underset{(q,r)}{\prec}M(c_0, c_1).
\end{equation}
\tag{2.16}
$$
2. For $0<s\leqslant r\leqslant q<\infty$ the following relations hold for integral coverings:
$$
\begin{equation}
K\underset{(q,q)}{\prec} M(c_0, c_1) \quad\Longrightarrow\quad K\underset{(q,r)}{\prec} M(c_0, c_1) \quad\Longrightarrow\quad K\underset{(q,s)}{\prec} M(c_0, c_1).
\end{equation}
\tag{2.17}
$$
3. For $0<q\leqslant r\leqslant \sigma<\infty$ the following relations hold for integral coverings: Proof. 1. For $0<r\leqslant q<\infty$ the following bound holds by Hölder’s inequality:
$$
\begin{equation}
\biggl(\int_{(0, t]}h_1^r\,d\mu\biggr)^{1/r}M_r(t)^{-1} \leqslant \biggl(\int_{(0, t]} h_1^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}, \qquad t \in (0, \infty).
\end{equation}
\tag{2.19}
$$
A pointwise covering $K\leqslant M(c_0, c_1)$ implies (2.9). Then in $L_q$, for $t \in (0, \infty)$ we have
$$
\begin{equation*}
(2.9)\ \ \Longrightarrow\ \ \biggl(\int_{(0, t]}h_1^q\,d\mu\biggr)^{1/q}M_q(t)^{-1} \,{\leqslant}\, \biggl(\int_{(0, t]} \biggl(h_2\,{+}\,c_1\rho_K(h_1)\biggr)^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}.
\end{equation*}
\notag
$$
This, in combination with (2.19), gives the right-hand covering in (2.16).
2. Similarly to (2.19), the following estimates hold for $0<s\leqslant r\leqslant q<\infty$ and $t \in (0, \infty)$:
$$
\begin{equation}
\biggl(\int_{(0, t]}h_1^s\,d\mu\biggr)^{1/s}M_s(t)^{-1} \leqslant \biggl(\int_{(0, t]}h_1^r\,d\mu\biggr)^{1/r}M_r(t)^{-1} \leqslant \biggl(\int_{(0, t]} h_1^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}.
\end{equation}
\tag{2.20}
$$
By Definition 2, for $r=q$ the covering $K\underset{(q,q)}{\prec} M(c_0, c_1)$ means that for each $h_1\in K$ there exists $h_2 \in M$ such that $\rho_M(h_2)\leqslant c_0\rho_K(h_1)$, and for all $t\in (0, T)$
$$
\begin{equation}
\biggl(\int_{(0, t]}h_1^q\,d\mu\biggr)^{1/q}M_q(t)^{-1} \leqslant \biggl(\int_{(0, t]} \biggl(h_2+c_1\rho_K(h_1)\biggr)^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}.
\end{equation}
\tag{2.21}
$$
However, then from the right-hand bound in (2.20) and (2.21) we obtain ${K\underset{(q,r)}{\prec} M(c_0, c_1)}$.
Similarly, for $0<s\leqslant r\leqslant q<\infty$
$$
\begin{equation*}
K\underset{(q,r)}{\prec} M(c_0, c_1) \quad\Longrightarrow\quad K\underset{(q,s)}{\prec} M(c_0, c_1).
\end{equation*}
\notag
$$
3. The following bounds hold for $0<q\leqslant r\leqslant \sigma<\infty$:
$$
\begin{equation*}
\biggl(\int_{(0, t]}h_1^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\leqslant \biggl(\int_{(0, t]}h_1^r\,d\mu\biggr)^{1/r}M_r(t)^{-1}\leqslant \biggl(\int_{(0, t]} h_1^{\sigma}\,d\mu\biggr)^{1/\sigma}M_{\sigma}(t)^{-1}.
\end{equation*}
\notag
$$
They imply the coverings (2.18) similarly to how (2.20) implies (2.17). 2.2. Estimates of pointwise and integral majorantsTheorem 2. Let $T \in (0, \infty]$, let $K$ and $ M$ be cones in $L_0^{+}(0,T)$ equipped with functionals $\rho_K$ and $\rho_M$, respectively, and let $c_0 \in (0, \infty)$ and $c_1 \in [0, \infty)$. 1. If there is a pointwise covering $K\leqslant M(c_0, c_1)$, then the following bound holds for the pointwise majorants (2.6):
$$
\begin{equation}
\lambda_K(t)\leqslant c_0\lambda_M(t)+c_1, \qquad t \in (0, T).
\end{equation}
\tag{2.22}
$$
2. If there is a $(q,r)$-integral covering $K\underset{(q,r)}{\prec}M(c_0, c_1)$ for $0<r,q<\infty$, then the following bound holds for the pointwise majorants (2.7): or Proof. We carry out the proof for integral majorants in the case when $0<q<1$, that is, we prove (2.23) (in other cases the reasoning is similar).
It follows from an integral covering $K\underset{(q,r)}{\prec}M(c_0, c_1)$ that for any function $h_1 \in K$ there exists a function $h_2 \in M$ with the properties $\rho_M(h_2)\leqslant c_0\rho_K(h_1)$ and
$$
\begin{equation*}
\biggl(\int_{(0, t]}h_1^r\,d\mu\biggr)^{1/r} M_r(t)^{-1} \leqslant \biggl\{ \int_{(0, t]}(h_2+c_1\rho_K(h_1))^q\,d\mu\biggr\}^{1/q}M_q(t)^{-1}, \qquad t \in (0, T).
\end{equation*}
\notag
$$
We assume here that the condition $\rho_K(h_1)\leqslant 1$ holds. Then
$$
\begin{equation}
\biggl(\int_{(0, t]}h_1^r\,d\mu\biggr)^{1/r} M_r(t)^{-1}\leqslant \biggl\{ \int_{(0, t]} (h_2+c_1)^q\,d\mu\biggr\}^{1/q}M_q(t)^{-1}, \qquad t \in (0, T).
\end{equation}
\tag{2.25}
$$
Set $\widetilde{h}_2=c_0^{-1}h_2 \in M$. Then $\rho_M(\widetilde{h}_2)=c_0^{-1}\rho_M(h_2)\leqslant\rho_K(h_1)\leqslant 1$. In addition, for $0<q<1$ and $t \in (0, T)$ inequality (2.25) implies the following bound:
$$
\begin{equation}
\biggl(\int_{(0, t]}h_1^r\,d\mu\biggr)^{1/r} M_r(t)^{-1} \leqslant \biggl\{ c_0^q\biggl[ \biggl( \int_{(0, t]}\widetilde{h}_2^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr]^q +c_1^q\biggr\}^{1/q}.
\end{equation}
\tag{2.26}
$$
However, by the definition (2.7) of the integral majorant $\widetilde{\lambda}_{M,q}(t)$ we have
$$
\begin{equation*}
\widetilde{h}_2 \in M, \qquad \rho_M(\widetilde{h}_2)\leqslant 1 \quad\Longrightarrow\quad M_q(t)^{-1}\biggl(\int_{(0, t]}\widetilde{h}_2^q\,d\mu\biggr)^{1/q}\leqslant \widetilde{\lambda}_{M,q}(t).
\end{equation*}
\notag
$$
Hence (2.26) implies that
$$
\begin{equation}
\biggl(\int_{(0, t]}h_1^r\,d\mu\biggr)^{1/r}M_r(t)^{-1} \leqslant \bigl\{ c_0^q\widetilde{\lambda}_{M,q}(t)^q+c_1^q\bigr\}^{1/q}.
\end{equation}
\tag{2.27}
$$
Thus, inequality (2.27) is valid for any function $h_1\in K$ satisfying $\rho_K(h_1)\leqslant 1$. Therefore,
$$
\begin{equation*}
\begin{aligned} \, \widetilde{\lambda}_{K,r}(t) &=\sup\biggl\{ \biggl(\int_{(0, t]}h_1^r\,d\mu\biggr)^{1/r}M_r(t)^{-1} \colon h_1\in K, \rho_K(h_1)\leqslant 1\biggr\} \\ &\leqslant\bigl\{ c_0^q\widetilde{\lambda}_{M,q}(t)^q+c_1^q\bigr\}^{1/q}. \end{aligned}
\end{equation*}
\notag
$$
This completes the proof of Theorem 2. Corollary 1. Under the assumptions of Theorem 2 let $r=q$ and $K\underset{(q,q)}\cong M$, that is, let there exist $c_0, \widetilde{c}_0 \in (0, \infty)$, $c_1, \widetilde{c}_1 \in [0, \infty)$ such that $K\underset{(q,q)}\prec M(c_0, c_1)$ and $M\underset{(q,q)}\prec K(\widetilde{c}_0, \widetilde{c}_1)$. Then for $t\in (0, T)$ Let $c_1=\widetilde{c}_1=0$ in these bounds. Then
$$
\begin{equation}
\widetilde{\lambda}_{K,q}(t)\leqslant c_0\widetilde{\lambda}_{M,q}(t)\quad\text{and} \quad \widetilde{\lambda}_{M,q}(t)\leqslant \widetilde{c}_0\widetilde{\lambda}_{K,q}(t), \quad t\in (0, \infty).
\end{equation}
\tag{2.32}
$$
Remark 4. For cones the concept of covering is much more general than the concept of embedding. Recall that a cone $K$ is embedded in a cone $M$ with embedding constant $c_0 \in (0, \infty)$ if $K\subset M$ and the functionals $\rho_K$ and $\rho_M$ satisfy the bound We denote this embedding by $K\stackrel{\to}{\subset} M(c_0)$. An embedding of cones implies their pointwise covering:
$$
\begin{equation*}
K\stackrel{\to}{\subset} M(c_0) \quad\Longrightarrow\quad K\leqslant M(c_0, 0).
\end{equation*}
\notag
$$
There are many cases in which a covering of cones does not reduce to an embedding. One of such examples, which is important for the theory of generalized Bessel–Riesz potentials, is considered in the next section. § 3. Example. Cones of decreasing rearrangements for potentialsConsider the space of potentials $H_E^G=H_E^G(\mathbb{R}^n)$ over a rearrangement-invariant base space $E=E(\mathbb{R}^n)$. The definitions and general properties of decreasing rearrangements and rearrangement-invariant spaces are considered in [1]–[3], [10]–[12], [16] and [17]. We have
$$
\begin{equation}
H_E^G(\mathbb{R}^n)=\bigl\{ u=G\ast f\colon f \in E(\mathbb{R}^n)\bigr\}
\end{equation}
\tag{3.1}
$$
and
$$
\begin{equation}
\|u\|_{H_E^G}=\inf \bigl\{ \|f\|_E\colon f \in E(\mathbb{R}^n);\, G\ast f=u\bigr\}.
\end{equation}
\tag{3.2}
$$
The kernel $G$ of an integral representation is said to be admissible if where $E'(\mathbb{R}^n)$ is the associate space of $E(\mathbb{R}^n)$ (see [16]). The convolution $G\ast f$ is defined byThe general properties of the potentials introduced were considered in [1], [2] and [9]–[11]. In the general case, given $u \in H_E^G(\mathbb{R}^n)$, the function $f \in E(\mathbb{R}^n)$ providing the representation $G\ast f=u$ is not necessarily unique. Therefore, the lower bound over all functions $f \in E(\mathbb{R}^n)$ providing this representation (the quotient norm) is taken in (3.2). We present here one of the general results of the theory. Theorem 3 ([11]). Let $G$ be an admissible kernel. Then the integral (3.4) converges for almost all $x \in \mathbb{R}^n$. The space $H_E^G(\mathbb{R}^n)$ is (quasi-)Banach, and Remark 5. In the case of an admissible kernel, for a potential $u \in H_E^G(\mathbb{R}^n)$ we can define its decreasing rearrangement $u^{*}$ (see [2] and [11]). Namely, consider the Lebesgue distribution function of the potential and set These functions are not identically equal to infinity, and $0\leqslant u^{*}\!\downarrow$. Here we consider the weighted $r$-integral means of decreasing rearrangements for $r \in (0,\infty)$. To this end we introduce a Lebesgue-measurable positive weight function $w$ on $(0,\infty)$ such that for $t \in (0,\infty)$
$$
\begin{equation*}
0<W_r(t):=\biggl(\int_{0}^{t}w(\tau)\,d\tau\biggr)^{1/r}<\infty,
\end{equation*}
\notag
$$
and we set
$$
\begin{equation}
u_{r,w}^{**}(t)=W_r(t)^{-1}\biggl(\int_{0}^{t}u^{*}(\tau)^rw(\tau)\,d\tau\biggr)^{1/r}, \qquad t \in (0,\infty).
\end{equation}
\tag{3.7}
$$
A similar construction of weighted integral means for decreasing rearrangements was introduced in [7] and [8] for $r=1$. In particular, for $r=1$ and $w(\tau)=1$ we obtain the known integral means of decreasing rearrangements
$$
\begin{equation*}
u_{r,w}^{**}(t)=u^{**}(t)=t^{-1}\int_{0}^{t}u^{*}(\tau)\,d\tau, \qquad t \in (0,\infty).
\end{equation*}
\notag
$$
We note that
$$
\begin{equation}
0\leqslant u^{*}\!\downarrow \quad\Longrightarrow\quad u_{r,w}^{**}(t)\geqslant W_r(t)^{-1}\biggl( \int_{0}^{t}u^{*}(t)^rw(\tau)\,d\tau\biggr)^{1/r} =u^{*}(t), \qquad t \in (0,\infty).
\end{equation}
\tag{3.8}
$$
We introduce cones of decreasing rearrangements for potentials $u\in H_E^G(\mathbb{R}^n)$: the cone
$$
\begin{equation}
K=\bigl\{ h(t)=u^{*}(t)\colon u \in H_E^G(\mathbb{R}^n),\, t \in (0,\infty)\bigr\},
\end{equation}
\tag{3.9}
$$
which we equip with the positively homogeneous nondegenerate functional
$$
\begin{equation}
\rho_{K}(h)=\inf \bigl\{ \|u\|_{H_E^G}(\mathbb{R}^n);\, u^{*}(t)=h(t),\, t \in (0, \infty)\bigr\},
\end{equation}
\tag{3.10}
$$
and the cone of integral means of rearrangements
$$
\begin{equation}
M=M_{r,w}=\bigl\{ h(t)=u_{r,w}^{**}(t)\colon u \in H_E^G(\mathbb{R}^n),\, t \in (0,\infty)\bigr\},
\end{equation}
\tag{3.11}
$$
equipping it with the positively homogeneous nondegenerate functional
$$
\begin{equation}
\rho_{M}(h)=\inf \bigl\{ \|u\|_{H_E^G}\colon u \in H_E^G(\mathbb{R}^n);\, u_{r,w}^{**}(t)=h(t),\, t \in (0,\infty)\bigr\}.
\end{equation}
\tag{3.12}
$$
These cones play an important role in the study of the integral properties of potentials. For example, criteria for embedding potentials in rearrangement-invariant spaces are expressed in their terms (see, for example, [1] and [9]). The following statements describe the properties of pointwise and integral coverings of these cones. Theorem 4. In the notation (3.1)–(3.12) the following pointwise covering holds for any $\varepsilon>0$: Proof. According to (3.10), for $\varepsilon>0$ and any function $h_1 \in K$ there exists a potential $u_{\varepsilon} \in H_E^G$ such that
$$
\begin{equation}
h_1(t)= u_{\varepsilon}^{*}(t), \quad t \in (0,\infty)\quad\text{and} \quad \|u_{\varepsilon}\|_{H_E^G}\leqslant(1+\varepsilon)\rho_K(h_1).
\end{equation}
\tag{3.14}
$$
We set $h_2(t)=(u_{\varepsilon})_{r,w}^{**}(t)$. Then
$$
\begin{equation}
h_2 \in M_{r,w}\quad\text{and} \quad \rho_M(h_2)\leqslant \|u_{\varepsilon}\|_{H_E^G}\leqslant(1+\varepsilon)\rho_K(h_1).
\end{equation}
\tag{3.15}
$$
It follows from (3.8) that
$$
\begin{equation}
h_1(t)= u_{\varepsilon}^{*}(t)\leqslant (u_{\varepsilon})_{r,w}^{**}(t)=h_2(t), \qquad t \in (0,\infty).
\end{equation}
\tag{3.16}
$$
Inequalities (3.15) and (3.16) prove (3.13), which completes the proof. Corollary 2. Under the assumptions of Theorem 4, for $0<r\leqslant q<\infty$ there is a $(q,r)$-integral covering Indeed, (3.17) follows from (3.13) and Theorem 1 (part 1). Consider the case where $\mu$ is a weighted Lebesgue measure on $(0,\infty)$: where the weight satisfies
$$
\begin{equation}
0<v \in L_1(0, t) \quad \forall\, t \in (0, \infty).
\end{equation}
\tag{3.19}
$$
Moreover, let the weight functions $v$ and $w$ be related as follows: there exists a constant $c \in (0, \infty)$ such that for any $t\in (0, \infty)$
$$
\begin{equation}
\int_{\xi}^t\biggl(\int_{0}^{\tau}w(\rho)\,d\rho\biggr)^{-1}v(\tau)\, d\tau\leqslant cw(\xi)^{-1}v(\xi), \qquad \xi \in (0, t).
\end{equation}
\tag{3.20}
$$
For example, let $w(\tau)=1$, let (3.19) hold, and let
$$
\begin{equation}
0<v(\tau)\tau^{\alpha}\! \downarrow, \qquad \alpha\in (0, 1).
\end{equation}
\tag{3.21}
$$
Then condition (3.20) is satisfied for $c=\alpha^{-1}$. Indeed, for $\xi \in (0, t)$ the left-hand side of (3.20) is equal to
$$
\begin{equation*}
\begin{aligned} \, \int_{\xi}^t \tau^{-1} v(\tau)\,d\tau &=\int_{\xi}^t \tau^{-1-\alpha} (\tau^{\alpha}v(\tau))\,d\tau\leqslant \int_{\xi}^t \tau^{-1-\alpha} (\xi^{\alpha}v(\xi))\,d\tau \\ &=(\xi^{\alpha}v(\xi))\alpha^{-1}(\xi^{-\alpha}-t^{-\alpha})<\alpha^{-1}v(\xi). \end{aligned}
\end{equation*}
\notag
$$
Note that for $\mu$ as above and $q \in (0, \infty)$ the equality
$$
\begin{equation}
M_q(t)=\biggl(\int_{(0,t]}v(\tau)\,d\tau\biggr)^{1/q}, \qquad t\in (0, \infty),
\end{equation}
\tag{3.22}
$$
holds, and thus conditions (3.18) and (3.19) are equivalent to (2.1) and (2.2), respectively. Theorem 5. Let $q \in (0, \infty)$ and $r \in (0, q]$. In the notation and conditions (3.1)–(3.12), provided that (3.18)–(3.20) are satisfied, a $(q,r)$-integral covering of cones Proof. According to (3.12), for any $\varepsilon>0$ and $h_1 \in M_{r,w}$ there exists $u_{\varepsilon} \in H_E^G$ with the following properties:
We set $h_2(\tau)=u_{\varepsilon}^{*}(\tau)$. Then
$$
\begin{equation*}
\begin{gathered} \, h_2 \in K, \qquad \rho_{K}(h_2)\leqslant \|u_{\varepsilon}\|_{H^G_E}\leqslant (1+\varepsilon)\rho_{M}(h_1), \\ \begin{split} h_1(\tau)&=(u_{\varepsilon})^{**}_{r,w}(\tau) =W_r(\tau)^{-1}\biggl(\int_{0}^{\tau}u_{\varepsilon}^{*}(\xi)^rw(\xi)\,d\xi\biggr)^{1/r} \\ &=W_r(\tau)^{-1}\biggl( \int_{0}^{\tau}h_2(\xi)^rw(\xi)\,d\xi\biggr)^{1/r} \end{split} \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\int_{0}^{t} h_1(\tau)^r\,d\mu(\tau) =\int_{0}^{t} h_1(\tau)^rv(\tau)\,d\tau =\int_{0}^{t}W_r(\tau)^{-r}\biggl( \int_{0}^{\tau}h_2(\xi)^rw(\xi)\,d\xi\biggr)v(\tau)\,d\tau.
\end{equation*}
\notag
$$
Changing the order of integration in the multiple integral, we obtain
$$
\begin{equation*}
\int_{0}^{t} h_1(\tau)^r\,d\mu(\tau)=\int_{0}^{t}h_2(\xi)^rw(\xi) \biggl( \int_{\xi}^{t}W_r(\tau)^{-r}v(\tau)\,d\tau\biggr)\, d\xi.
\end{equation*}
\notag
$$
If condition (3.20) is satisfied, then the following bounds hold:
$$
\begin{equation*}
\int_{\xi}^{t}W_r(\tau)^{-r}v(\tau)\,d\tau=\int_{\xi}^{t}\biggl( \int_0^{\tau}w(\rho)\,d\rho\biggr)^{-1}v(\tau)\,d\tau\leqslant cw(\xi)^{-1}v(\xi)
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\int_{0}^{t} h_1(\tau)^r\,d\mu(\tau)\leqslant c \biggl( \int_{0}^{t}h_2(\xi)^rv(\xi)\,d\xi\biggr)=c\int_{0}^{t}h_2(\xi)^r\,d\mu(\xi).
\end{equation}
\tag{3.25}
$$
We introduce the function $\widetilde{h}_2=c^{1/r}h_2 \in K$. According to (3.24),
$$
\begin{equation}
\rho_K(\widetilde{h}_2)=c^{1/r}\|u_{\varepsilon}\|_{H_E^G}\leqslant c^{1/r}(1+\varepsilon)\rho_M(h_1).
\end{equation}
\tag{3.26}
$$
Further, it follows from (3.25) for $q \in (0, \infty)$ and $r \in (0,q]$ that
$$
\begin{equation}
\biggl(\int_{0}^{t} h_1^r\,d\mu\biggr)^{1/r}M_r(t)^{-1}\leqslant \biggl( \int_{(0,t]}\widetilde{h}_2^r\,d\mu\biggr)^{1/r}M_r(t)^{-1}\leqslant \biggl( \int_{(0,t]}\widetilde{h}_2^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}.
\end{equation}
\tag{3.27}
$$
Thus, for any function $h_1 \in M_{r,w}$ we have found a function $\widetilde{h}_2 \in K$ such that (3.26) and (3.27) hold. This proves the covering (3.23) and completes the proof of Theorem 5. Corollary 3. Under the assumptions of Theorem 5 let $r=q$. Then $M_{q,w}\approx K$, that is, These coverings follow from (3.17) and (3.23) for $r=q$. The coverings (3.28) imply the mutual bounds (2.28)–(2.31) for integral majorants, where $T=\infty$, ${c_1=\widetilde{c}_1=0}$ and $t \in (0, \infty)$, that is, for all $\varepsilon>0$ § 4. The relationship between coverings of cones and their embeddings in ideal spaces4.1. The concept of an ideal spaceLet $T \in (0, \infty]$, and let $\mu$ be a countably additive measure on $(0,T)$ satisfying conditions (2.1) and (2.2). Definition 3. Let $F=F(0,T)\subset L_{0,\mu}(0,T)$ be the space of functions satisfying the condition Then $F$ is called an ideal space. The convergence $f_n\to f$ in $F$ means, as usual, that $\|f-f_n\|_F\to 0$ as $n\to \infty$. Definition 4. Let $F$ be an ideal space in which the following condition is also satisfied: if $h_n \in F$, $0\leqslant h_n\leqslant h_{n+1}$, $n \in \mathbb{N}$, and $\lim_{n\to \infty}h_n=f$ $\mu$-almost everywhere on $(0, T)$, then Then we call $F$ an ideal space with the Fatou property. Remark 6. In formula (4.3) the monotonicity of the sequence $\{h_n\}$ implies its convergence to a measurable limit function $f$. Moreover, by property (5) the sequence of nonnegative quantities $\|h_n\|_F$ is also increasing and, consequently, the finite or infinite limit $\lim_{n\to \infty}\|h_n\|_F$ exists. The Fatou property means that this limit coincides with $\|f\|_F$. Its finiteness is equivalent to the inclusion $f \in F$. Remark 7. For $F=L_p(0,T)$, $0<p\leqslant \infty$, all these properties hold. In particular, for $0<p<1$, property (4.2) holds for $c_0=1$ and $q_0=p$. Indeed, (4.2) for $q_0=p$ and $c_0=1$ follows from the well-known inequality For $p\geqslant 1$, (4.2) holds for $q_0=1$ and $c_0=1$: The Fatou property of $F=L_p(E)$ is a consequence of Levi’s theorem for $0<p<\infty$. It also holds for $p=\infty$, but this requires a separate justification.Remark 8. In the theory of ideal spaces the triangle inequality is often written in the form where $C_F\geqslant 1$ ($\|f\|_F$ is a norm for $C_F=1$ and a quasinorm for $C_F>1$). The following result of Aoki and Rolewicz is known, which allows us to go from a relation of the form (4.6) to a relation of the form (4.2). Theorem 6 (Aoki–Rolewicz theorem). In the definition of an ideal space let (4.2) be replaced by inequality (4.6). Let $q_0 \in (0,1]$ be defined by $(2C_F)^{q_0}= 2$. Then Property (4.2) is thus established with the universal constant $c_0= 2^{1/q_0}$ when the triangle inequality holds in the form (4.6). Note that $C_F=1$ $\Longrightarrow$ $q_0=1$ and $C_F>1$ $\Longrightarrow$ $q_0 \in (0,1)$. The question of the sharp constant in (4.7) is apparently open in the general case. For a number of concrete spaces constants can be specified. The concept of an ideal space extends the axiomatics of Banach function spaces developed by Bennett and Sharpley (see [16]; normed spaces with the Fatou property are considered there). It is also related to concepts in the theory of normed ideal structures considered by Krein, Petunin and Semenov in [17]. We consider significantly more general two-parameter $(q_0, q_1)$-spaces of the type of ideal spaces in a separate paper. Let us formulate some basic properties of ideal spaces. Definition 5. Let $F$ be an ideal space in Definition 3. We say that $F$ has the Riesz–Fischer property if for all $f_s \!\in\! F$, $s\!=\!0,1,\dots$, such that ${(\sum_{s=0}^{\infty}\|f_s\|_F^{q_0})^{1/q_0}\!<\!\infty}$ the series $\sum_{s=0}^{\infty}f_s$ converges almost everywhere and in $F$ to a function $f \in F$ and Theorem 7. Let $F$ be an ideal space with the Fatou property. Then the Riesz–Fischer property with constants $c_0$ and $q_0$ in (4.2) is valid for $F$. Theorem 8. For an ideal space $F$ to be complete it is necessary and sufficient that it have the Riesz–Fischer property. These results are proved by generalizing to ideal spaces the arguments of Bennett and Sharpley [16] in the case of Banach function spaces. 4.2. The relationship between embeddings in ideal spaces and pointwise coverings of conesDefinition 6. Let $K$ be a cone in $L_{0,\mu}^{+}(0,T)$ equipped with a functional $\rho_K$, and let $F\subset L_{0,\mu}(0,T)$ be an ideal space. We say that $K$ is an embedding in $F$ with embedding constants $c \in (0, \infty)$ and $d \in [0,\infty)$ if $K\subset F$ and the following bound holds: Remark 9. In the case of $\|\chi_{(0,\infty)}\|_F=\infty$ we assume in (4.9) that $d=0$ for $T=\infty$, so that (4.9) takes the form We denote the embedding of $K$ in the ideal space $F$ with embedding constants $c$ and $d$ by Theorem 9. Let $T \,{\in}\, (0,\infty]$, let $\mu$ be a measure on $(0,T)$ satisfying conditions (2.1) and (2.2), let $K$ and $M$ be cones in $L_{0,\mu}^+(0,T)$ equipped with functionals $\rho_K$ and $\rho_M$, respectively; let $c \in (0,\infty)$, $d \in [0, \infty)$, and let $K\leqslant M(c, d)$ be a pointwise covering, where $d=0$ for $T=\infty$. Let $F\subset L_{0,\mu}(0,T)$ be an ideal space, and let $M\stackrel{\to}{\subset}F(c_M,d_M)$, where $d_M=0$ if $T=\infty$. Then Proof. It follows from the pointwise covering $K\leqslant M(c,d)$ that for each function $h_1 \in K$ there exists $h_2 \in M$ with the properties
It follows from the embedding $ M\stackrel{\to}{\subset}F(c_M,d_M)$ that $h_2 \in F$ and
$$
\begin{equation}
\|h_2\|_F\leqslant \bigl(c_M+d_M \|\chi_{(0,T)}\|_F\bigr)\rho_M(h_2)
\end{equation}
\tag{4.15}
$$
(for $T=\infty$ the second term in parentheses is absent). By the monotonicity of the quasinorm and the triangle inequality in $F$ it follows from the second bound in (4.14) that
$$
\begin{equation*}
h_1\in F\quad\text{and} \quad \|h_1\|_F\leqslant C_F\bigl(\|h_2\|_F+d\rho_K(h_1) \|\chi_{(0,T)}\|_F\bigr)
\end{equation*}
\notag
$$
(for $T=\infty$ there is no second term in parentheses). We substitute here (4.15):
$$
\begin{equation*}
\|h_1\|_F\leqslant C_F\bigl[(c_M+d_M \|\chi_{(0,T)}\|_F)\rho_M(h_2)+d\rho_K(h_1)\|\chi_{(0,T)}\|_F\bigr].
\end{equation*}
\notag
$$
Hence from the first bound in (4.14) we obtain
$$
\begin{equation*}
\|h_1\|_F\leqslant C_F \bigl[ c(c_M+d_M \|\chi_{(0,T)}\|_F)+d\|\chi_{(0,T)}\|_F\bigr]\rho_K(h_1).
\end{equation*}
\notag
$$
These are the required relations (4.12) and (4.13), which completes the proof of Theorem 9. 4.3. The relationship between embeddings in ideal spaces and integral coverings of conesIn the study of the relationship between integral coverings of cones and their embeddings in ideal space, we have to impose more restrictive conditions on the ideal space $F$: we replace condition (4.1) of the monotonicity of the quasinorm with respect to pointwise inequalities with the stronger condition of the compatibility of the quasinorm with an integral bound. Definition 7. Let $q \in (0,\infty)$ and $T \in (0,\infty]$, let $\mu$ be a countably additive measure on $(0,T)$ satisfying (2.1) and (2.2), and let $F\subset L_{0,\mu}(0,T)$ be an ideal space. A quasinorm in $F$ is said to be compatible with the $q$-integral bound if the following implication holds for $g \in L_{0,\mu}(0,T)$: Theorem 10. Let $q \in (0,\infty)$, $r \in [q,\infty)$ and $T \in (0,\infty]$, let $K$ and $M$ be cones in $L^{+}_{0,\mu}(0,T)$ equipped with functionals $\rho_{K}$ and $\rho_{M}$, respectively, let $c \in (0,\infty)$, $d \in [0,\infty)$, and let $K\underset{(q,r)}{\prec}M(c,d)$ be a $(q,r)$-integral covering. Let $F=F(0,T)\subset L_{0,\mu}(0,T)$ be an ideal space with quasinorm compatible with the $q$-integral bound, and let $M\stackrel{\to}{\subset}F(c_M, d_M)$. Then where Proof. By (2.18) the covering $K\underset{(q,r)}{\prec}M(c,d)$ for $r\geqslant q$ implies that $K\underset{(q,q)}{\prec}M(c,d)$. Then for any function $h_1 \in K$ there exists $h_2 \in M$ such that $\rho_{M}(h_2)\leqslant c\rho_{K}(h_1)$ and
$$
\begin{equation}
\biggl(\int_{(0,t]}h_1^{q}\,d\mu\biggr)^{1/q}M_q(t)^{-1} \leqslant \biggl( \int_{(0,t]}(h_2+d\rho_{K}(h_1))^{q}\,d\mu\biggr)^{1/q}M_q(t)^{-1} \quad \forall\, t \in (0, T).
\end{equation}
\tag{4.19}
$$
Set
$$
\begin{equation}
h_3(\tau)=h_2(\tau)+d\rho_{K}(h_1), \qquad \tau \in (0, T).
\end{equation}
\tag{4.20}
$$
Then inequality (4.19) takes the form
$$
\begin{equation*}
\biggl(\int_{(0,t]}h_1^{q}\,d\mu\biggr)^{1/q}M_q(t)^{-1} \leqslant \biggl( \int_{(0,t]}h_3^{q}\,d\mu\biggr)^{1/q}M_q(t)^{-1}, \qquad t \in (0, T).
\end{equation*}
\notag
$$
It follows from the compatibility property (4.16) that $\|h_1\|_F\leqslant \|h_3\|_F$. From the triangle inequality (4.6) in the ideal space $F=F(0,T)$ we obtain
$$
\begin{equation*}
\|h_3\|_F\leqslant C_F\bigl(\|h_2\|_F+d\rho_{K}(h_1)\|\chi_{(0,T)}\|_F\bigr).
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation}
\|h_1\|_F\leqslant\|h_3\|_F\leqslant C_F\bigl(\|h_2\|_F+d\rho_{K}(h_1)\|\chi_{(0,T)}\|_F\bigr).
\end{equation}
\tag{4.21}
$$
It follows from the embedding $M\stackrel{\to}{\subset}F(c_M, d_M)$ that $h_2 \in F$ and
$$
\begin{equation}
\|h_2\|_F\leqslant (c_M+d_M\|\chi_{(0,T)}\|_F)\rho_{M}(h_2).
\end{equation}
\tag{4.22}
$$
We substitute this into (4.21):
$$
\begin{equation*}
\|h_1\|_F\leqslant C_F\bigl[ (c_M+d_M\|\chi_{(0,T)}\|_F)\rho_{M}(h_2)+d\rho_{K}(h_1)\|\chi_{(0,T)}\|_F\bigr].
\end{equation*}
\notag
$$
Hence from the inequality $\rho_{M}(h_2)\leqslant c\rho_{K}(h_1)$ we obtain $K\stackrel{\to}{\subset}F(c_K, d_K)$ and
$$
\begin{equation*}
\|h_1\|_F\leqslant C_F\bigl[c (c_M+d_M\|\chi_{(0,T)}\|_F)+d\|\chi_{(0,T)}\|_F\bigr]\rho_{K}(h_1).
\end{equation*}
\notag
$$
These are the required relations (4.17) and (4.18), which completes the proof of Theorem 10. § 5. Constructing an ideal space with a quasinorm compatible with an integral boundThis section describes a natural way to construct, on the basis of an ideal space $F_0$, a new ideal space $F$ in which the quasinorm is compatible with a $q$-integral bound. Let $q \in (0,\infty)$ and $T \in (0, \infty]$. Throughout this section we assume that $\mu$ and $v$ are countably additive measures on $(0,T)$ such that
$$
\begin{equation}
0< \biggl(\int_{(0, t]}d\mu\biggr)<\infty \quad \forall\, t \in (0,T),\quad T \in (0, \infty) \quad\Longrightarrow\quad \biggl(\int_{(0, T)}d\mu\biggr)<\infty,
\end{equation}
\tag{5.1}
$$
and
$$
\begin{equation}
0< \biggl(\int_{(0, t]}dv\biggr)<\infty \quad \forall\, t \in (0,T), \quad T \in (0, \infty) \quad\Longrightarrow\quad \biggl(\int_{(0, T)}dv\biggr)<\infty.
\end{equation}
\tag{5.2}
$$
Here $L_{0,\mu}(0,T)$ and $L_{0,v}(0,T)$ are the spaces of measurable functions with respect to the measures $\mu$ and $v$, respectively. Let $F_0=F_0(0,T)\subset L_{0,v}(0,T)$ be an ideal space of functions with finite quasinorm $\|f\|_{F_0}$ satisfying the triangle inequality
$$
\begin{equation}
\|f+g\|_{F_0}\leqslant C_{F_0}\bigl(\|f\|_{F_0}+\|g\|_{F_0}\bigr),
\end{equation}
\tag{5.3}
$$
where $C_{F_0}\geqslant 1$; $\|f\|_{F_0}$ is a norm for $C_{F_0}=1$ and a quasinorm for $C_{F_0}>1$. The following bound holds in the case of a quasinorm:
$$
\begin{equation}
\biggl\|\sum_{k=1}^{n}f_k\biggr\|_{F_0}\leqslant 2^{1/q_0}\biggl(\sum_{k=1}^{n}\|f_k\|_{F_0}^{q_0}\biggr)^{1/q_0},
\end{equation}
\tag{5.4}
$$
where $q_0$ is defined by $(2C_{F_0})^{q_0}=2$ (see the Aoki–Rolewicz theorem). In addition, if $T=\infty$, then assume that
$$
\begin{equation}
\bigl\|\chi_{(r,\infty)}(\cdot)M_q(\cdot)^{-1}\bigr\|_{F_0}\leqslant \infty \quad \forall\, r \in (0, \infty).
\end{equation}
\tag{5.5}
$$
Note that the ideal space $F_0$ has the Riesz–Fischer property with constants $c_0$ and $q_0$ if for $a_s \in F_0$, $s=0,1, \dots$, it follows from the condition $(\sum_{s=0}^{\infty}\|a_s\|_{F_0}^{q_0})^{1/q_0}<\infty$ that the series $\sum_{s=0}^{\infty}a_s$ converges $v$-almost everywhere and in $F_0$ to some function $f \in F_0$ and
$$
\begin{equation}
\|f\|_{F_0}=\biggl\|\sum_{s=0}^{\infty}a_s\biggr\|_{F_0} \leqslant c_0\biggl(\sum_{s=0}^{\infty}\|a_s\|_{F_0}^{q_0}\biggr)^{1/q_0}.
\end{equation}
\tag{5.6}
$$
Theorem 11. 1. In the notation of this section assume that $q \in (0, \infty)$ and conditions (2.1), (2.2) and (5.1)–(5.5) are satisfied. Consider the space $F=F(0,T)\subset L_{0, \mu}(0,T)$ of functions with finite quasinorm 2. Under the assumptions of part 1 let the ideal space $F_0$ have the Fatou property. Then $F$ also has the Fatou property. 3. Under the assumptions of part 1 let $q\geqslant 1$ and, in addition, let the ideal space $F_0$ have the Riesz–Fischer property with constants $c_0$ and $q_0$. Then $F$ also has the Riesz–Fischer property with constants $c_0$ and $q_0$. Remark 10. We note that for $f \in L_{0, \mu}(0,T)$ the function Proof of Theorem 11. 1. We claim that under the assumptions of part 1 of the theorem $F=F(0,T)$ is an ideal space. We need to prove that properties (1)–(6) in Definition 3 hold.
(1) It is clear that $\|f\|_F\geqslant 0$ for all $f\in F$ and $\|0\|_F=0$. Further, by the definition of $F(0,T)$ we have
$$
\begin{equation}
\begin{aligned} \, \|f\|_{F}=0 &\quad\Longrightarrow\quad \biggl\| \biggl( \int_{(0,t]}|f|^{q}\,d\mu\biggr)^{1/q}M_q(t)^{-1} \biggr\| _{F_0}=0 \nonumber \\ &\quad\Longrightarrow\quad\biggl(\int_{(0,t]}|f|^{q}\,d\mu\biggr)^{1/q}=0 \quad v\text{-a.e. on } (0,T). \end{aligned}
\end{equation}
\tag{5.10}
$$
Since this function is increasing on $(0,T)$, (5.10) is valid for each $t \in (0,T)$. But then from the properties of the integral we see that $f(\tau)=0$ for $\mu$-almost every $\tau \in (0,t]$, for each $t \in (0,T)$. This means that $f=0$ $\mu$-almost everywhere on $(0,T)$.
Thus, $\|f\|_F=0$ $\Longrightarrow$ $f=0$ $\mu$-almost everywhere on $(0,T)$. (2) Next, according to (5.9),
$$
\begin{equation}
f \in F \quad\Longrightarrow\quad \biggl(\int_{(0,t]}|f|^{q}\,d\mu\biggr)^{1/q}<\infty \quad v\text{-a.e. on } (0,T).
\end{equation}
\tag{5.11}
$$
Since this function increases on $(0,T)$, (5.11) is valid for each $t \in (0,T)$. Then by the properties of the integral $|f(\tau)|<\infty$ for $\mu$-almost every $\tau \in (0,t]$, for each $t \in (0, T)$. Hence $|f(\tau)|<\infty$ $\mu$-almost everywhere on $(0,T)$.
Thus, $f \in F$ $\Longrightarrow$ $|f(\tau)|<\infty$ $\mu$-almost everywhere on $(0,T)$. Moreover, it follows from (5.7) that $\|\alpha f\|_F=|\alpha|\|f\|_F$ for any $\alpha \in \mathbb{C}$ and $f \in F$. (3) We claim that
$$
\begin{equation}
\|\chi_{(0,T)}\|_F<\infty\quad\text{for } T \in (0,\infty)
\end{equation}
\tag{5.12}
$$
and
$$
\begin{equation}
\|\chi_{(0,r]}\|_F<\infty \quad\text{for } T=\infty \quad \forall\, r \in (0,\infty).
\end{equation}
\tag{5.13}
$$
For $T \in (0,\infty)$ we have
$$
\begin{equation*}
\|\chi_{(0,T)}\|_F=\biggl\|\biggl(\int_{(0,t]}1^{q}\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr\|_{F_0} =\|\chi_{(0,T)}\|_{F_0(0,T)}<\infty.
\end{equation*}
\notag
$$
Here we have taken into account property (4) in Definition 3 as applied to the ideal space $F_0$. For $T=\infty$, for any $r \in (0,\infty)$ we have
$$
\begin{equation*}
\begin{aligned} \, \|\chi_{(0,r]}\|_F &=\biggl\| \biggl(\int_{(0,t]}\chi_{(0,r]}^{q}\, d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr\|_{F_0(0,T)} \biggl(\int_{(0,t]} \chi_{(0,r]}^{q}\,d\mu\biggr)^{1/q}M_q(t)^{-1} \\ &=\begin{cases} 1, &t \in (0, r], \\ M_q(r)M_q(t)^{-1}, &t \in (r, \infty). \end{cases} \end{aligned}
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\biggl( \int_{(0,t]}\chi_{(0,r]}^{q}\,d\mu\biggr)^{1/q}M_q(t)^{-1} =\chi_{(0,r]}(t)+M_q(r)M_q(t)^{-1}\chi_{(r,\infty)}(t).
\end{equation*}
\notag
$$
As a result,
$$
\begin{equation*}
\|\chi_{(0,r]}\|_F=\bigl\|\chi_{(0,r]}(t)+M_q(r)M_q(t)^{-1}\chi_{(r,\infty)}(t)\bigr\|_{F_0}
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\|\chi_{(0,r]}\|_F\leqslant C_{F_0}\bigl( \|\chi_{(0,r]}(t)\|_{F_0}+M_q(r)\|M_q(t)^{-1}\chi_{(r,\infty)}(t)\|_{F_0}\bigr).
\end{equation}
\tag{5.14}
$$
By the general properties of the ideal space $F_0$ and (5.5), the right-hand side of (5.14) is finite for all $r \in (0, \infty)$.
(4) To derive the triangle inequality in $F$, note that the following bound holds in the space $L_q(0,t]$, $t \in (0,T)$:
$$
\begin{equation*}
\begin{gathered} \, \biggl(\int_{(0, t]}|f+g|^q\,d\mu\biggr)^{1/q}\leqslant A_q \biggl[ \biggl(\int_{(0, t]}|f|^q\,d\mu\biggr)^{1/q} +\biggl(\int_{(0, t]}|g|^q\,d\mu\biggr)^{1/q}\biggr], \\ A_q=2^{1/q-1}, \quad 0<q<1, \qquad A_q=1, \quad 1\leqslant q<\infty. \end{gathered}
\end{equation*}
\notag
$$
From this bound, by the monotonicity properties of the quasinorm in $F_0$ we obtain
$$
\begin{equation*}
\begin{aligned} \, & \biggl\|\biggl(\int_{(0, t]}|f+g|^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr\|_{F_0} \\ &\qquad \leqslant A_q\biggl\|\biggl[\biggl(\int_{(0, t]}|f|^q\,d\mu\biggr)^{1/q} +\biggl(\int_{(0, t]}|g|^q\,d\mu\biggr)^{1/q}\biggr]M_q(t)^{-1}\biggr\|_{F_0}. \end{aligned}
\end{equation*}
\notag
$$
By the triangle inequality in $F_0$ (see (4.6))
$$
\begin{equation*}
\begin{aligned} \, &\biggl\|\biggl(\int_{(0, t]}|f+g|^q\,d\mu\biggr)^{1/q} M_q(t)^{-1}\biggr\|_{F_0} \\ &\quad\leqslant A_qC_{F_0} \biggl(\biggl\| \biggl(\int_{(0, t]}|f|^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr\|_{F_0}+\biggl\| \biggl(\int_{(0, t]}|g|^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr\|_{F_0}\biggr). \end{aligned}
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
f,g \in F \quad\Longrightarrow\quad \|f+g\|_F\leqslant A_qC_{F_0}\bigl(\|f\|_F+\|g\|_F\bigr).
\end{equation*}
\notag
$$
It follows from the above that $F$ is an ideal space with constant $C_F=A_qC_{F_0}$ in the triangle inequality (4.6). Remark 11. For $1\leqslant q<\infty$ we see that $C_F=A_qC_{F_0}=C_{F_0}$. In particular, if $F_0$ is a normed ideal space, that is, $C_{F_0}=1$, then $F$ is also a normed ideal space. (5) Finally, it follows from the inequality
$$
\begin{equation*}
\biggl(\int_{(0, t]}|g|^q\,d\mu\biggr)^{1/q}\leqslant \biggl(\int_{(0, t]}|f|^q\,d\mu\biggr)^{1/q} \quad \forall\, t \in (0,T)
\end{equation*}
\notag
$$
and the monotonicity of the quasinorm $\|\cdot\|_{F_0}$ that
$$
\begin{equation*}
\|g\|_F=\biggl\| \biggl(\int_{(0, t]}|g|^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr\|_{F_0} \leqslant \biggl\|\biggl(\int_{(0, t]}|f|^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr\|_{F_0}=\|f\|_F.
\end{equation*}
\notag
$$
Thus, condition (4.16) of the compatibility of the quasinorm $\|\cdot\|_{F}$ with the integral bound is satisfied, and part 1 of Theorem 11 is proved. 2. Let us show that if $F_0$ has the Fatou property, then the ideal space $F$ also has it. Let $f_n \in F$, $0\leqslant f_n\leqslant f_{n+1}$, $n \in \mathbb{N}$, and let $\lim_{n\to \infty} f_n=f$ $\mu$-almost everywhere on $(0, T)$. Then $0\leqslant f \in L_{0, \mu}(0, T)$, $0\leqslant f^{q}_{n}\leqslant f^{q}_{n+1}$, $n \in \mathbb{N}$, and $\lim_{n\to \infty} f_n^{q}=f^{q}$. We set
$$
\begin{equation*}
\psi_n(t)= \biggl(\int_{(0, t]}f_n^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}, \qquad t \in (0, T).
\end{equation*}
\notag
$$
Then $0\leqslant \psi_n\leqslant \psi_{n+1}$, and by Beppo Levi’s theorem, for the limit function
$$
\begin{equation*}
\psi(t)=\lim_{n\to \infty} \psi_n=\lim_{n\to \infty} \biggl[\biggl(\int_{(0, t]}f_n^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr]
\end{equation*}
\notag
$$
we have the equality
$$
\begin{equation*}
\psi(t)= \biggl(\int_{(0, t]}\lim_{n\to \infty}(f_n^q)\,d\mu\biggr)^{1/q}M_q(t)^{-1} =\biggl(\int_{(0, t]}f^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}, \qquad t \in (0, T).
\end{equation*}
\notag
$$
From the Fatou property of the ideal space $F_0$ we obtain $\|\psi\|_{F_0} =\lim_{n\to \infty}\|\psi_n\|_{F_0}$, and thus
$$
\begin{equation*}
\begin{aligned} \, \|f\|_F &=\biggl\|\biggl(\int_{(0, t]}|f|^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr\|_{F_0} =\lim_{n\to \infty}\biggl\|\biggl( \int_{(0, t]}f_n^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr\|_{F_0} \\ &=\lim_{n\to \infty}\|f_n\|_F. \end{aligned}
\end{equation*}
\notag
$$
This means that the ideal space $F$ has the Fatou property. 3. Let $q\geqslant 1$ under the assumptions of part 1 of the theorem and, additionally, let the ideal space $F_0$ have the Riesz–Fischer property with constants $c_0$ and $q_0$. Then the following implication holds: if $h_s \in F_0$, $s=0,1,\dots$, and ${(\sum_{s=0}^{\infty}\|h_s\|^{q_0}_{F_0})^{1/q_0}<\infty}$, then the series $\sum_{s=0}^{\infty}h_s$ converges $v$-almost everywhere and in $F_0$ and
$$
\begin{equation}
\biggl\| \sum_{s=0}^{\infty}h_s\biggr\|_{F_0}\leqslant c_0\biggl( \sum_{s=0}^{\infty}\|h_s\|^{q_0}_{F_0}\biggr)^{1/q_0}<\infty.
\end{equation}
\tag{5.15}
$$
Now let
$$
\begin{equation}
f_s\in F, \quad s=0,1,\dots, \quad\text{and}\quad1 \biggl(\sum_{s=0}^{\infty}\|f_s\|^{q_0}_{F}\biggr)^{1/q_0}<\infty.
\end{equation}
\tag{5.16}
$$
We set
$$
\begin{equation}
h_s(t)=\biggl(\int_{(0, t]}|f_s|^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}, \qquad s=0,1,\dots, \quad t \in (0,T).
\end{equation}
\tag{5.17}
$$
Then
$$
\begin{equation}
0\leqslant h_s\in F_0, \ \ \ \|h_s\|_{F_0}=\|f_s\|_{F}\ \ \ \text{and}\ \ \ \biggl(\sum_{s=0}^{\infty}\|h_s\|^{q_0}_{F_0}\biggr)^{1/q_0}=\biggl( \sum_{s=0}^{\infty}\|f_s\|^{q_0}_{F}\biggr)^{1/q_0}<\infty.
\end{equation}
\tag{5.18}
$$
It follows from the bound (5.15) and relations (5.16)–(5.18) that
$$
\begin{equation}
\biggl\|\sum_{s=0}^{\infty}h_s\biggr\|_{F_0} =\biggl\|\sum_{s=0}^{\infty}\biggl(\int_{(0, t]}|f_s|^q\,d\mu\biggr)^{1/q} M_q(t)^{-1}\biggr\|_{F_0}\leqslant c_0 \biggl( \sum_{s=0}^{\infty}\|f_s\|^{q_0}_{F}\biggr)^{1/q_0}<\infty.
\end{equation}
\tag{5.19}
$$
For $1\leqslant q< \infty$ the following inequality holds:
$$
\begin{equation}
\biggl(\int_{(0, t]} \biggl( \sum_{s=0}^{\infty}|f_s|\biggr)^q\,d\mu\biggr)^{1/q}M_q(t)^{-1} \leqslant \sum_{s=0}^{\infty}\biggl(\int_{(0, t]}|f_s|^{q}\,d\mu\biggr)^{1/q}M_q(t)^{-1}.
\end{equation}
\tag{5.20}
$$
By the monotonicity of the quasinorm in the ideal space $F_0$ it follows from this estimate that
$$
\begin{equation*}
\begin{aligned} \, \biggl\|\sum_{s=0}^{\infty}|f_s|\biggr\|_{F} &=\biggl\|\biggl(\int_{(0, t]}\biggl( \sum_{s=0}^{\infty}|f_s|\biggr)^q\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr\|_{F_0} \\ &\leqslant \biggl\|\biggl[\sum_{s=0}^{\infty}\biggl(\int_{(0, t]}|f_s|^{q}\,d\mu\biggr)^{1/q}M_q(t)^{-1}\biggr]\biggr\|_{F_0} . \end{aligned}
\end{equation*}
\notag
$$
Now taking (5.17) into account we obtain
$$
\begin{equation}
\biggl\|\sum_{s=0}^{\infty}|f_s|\biggr\|_{F}\leqslant \biggl\|\sum_{s=0}^{\infty}|h_s|\biggr\|_{F_0}.
\end{equation}
\tag{5.21}
$$
Hence it follows from (5.19) that
$$
\begin{equation}
\biggl\|\sum_{s=0}^{\infty}|f_s|\biggr\|_{F}\leqslant c_0\biggl( \sum_{s=0}^{\infty}\|h_s\|^{q_0}_{F_0}\biggr)^{1/q_0}=c_0\biggl( \sum_{s=0}^{\infty}\|f_s\|^{q_0}_{F}\biggr)^{1/q_0}<\infty.
\end{equation}
\tag{5.22}
$$
Thus,
$$
\begin{equation}
\varphi:=\sum_{s=0}^{\infty}|f_s| \in L_{0,\mu}^{+}(0,T) \quad\text{and}\quad \|\varphi\|_F=\biggl\|\sum_{s=0}^{\infty}|f_s|\biggr\|_F<\infty \quad\Longrightarrow\quad \varphi \in F.
\end{equation}
\tag{5.23}
$$
As proved in part 1 of Theorem 11, $F$ is an ideal space. By the properties of ideal spaces, for $0\leqslant\varphi \in F$ we have $0\leqslant \varphi(t)< \infty$ for $\mu$-almost all $t \in (0,T)$. Hence, the series $\sum_{s=0}^{\infty}f_s$ converges absolutely $\mu$-almost everywhere on $(0, T)$ and $\sum_{s=0}^{\infty}f_s \in L_{0,\mu}(0,T)$. Moreover, from the inequality
$$
\begin{equation*}
\biggl|\sum_{s=0}^{\infty}f_s\biggr| \leqslant \sum_{s=0}^{\infty}|f_s| \in F,
\end{equation*}
\notag
$$
by the monotonicity property of the quasinorm in the ideal space $F$ we obtain
$$
\begin{equation*}
\sum_{s=0}^{\infty}f_s \in F, \qquad \biggl\|\sum_{s=0}^{\infty}f_s\biggr\|_F\leqslant \biggl\|\sum_{s=0}^{\infty}|f_s|\biggr\|_{F}<\infty.
\end{equation*}
\notag
$$
It follows from this bound and (5.22) that
$$
\begin{equation}
\biggl\|\sum_{s=0}^{\infty}f_s\biggr\|_{F}\leqslant c_0\biggl( \sum_{s=0}^{\infty}\|f_s\|^{q_0}_{F}\biggr)^{1/q_0}<\infty.
\end{equation}
\tag{5.24}
$$
We claim that the series $\sum_{s=0}^{\infty}f_s$ converges in $F$. Setting ${f_0=f_1=\dots=f_n=0}$ in (5.24), from the convergence of the series on the right-hand side of (5.24) we obtain
$$
\begin{equation*}
\biggl\|\sum_{s=n+1}^{\infty}f_s\biggr\|_{F}\leqslant c_0\biggl( \sum_{s=n+1}^{\infty}\|f_s\|^{q_0}_{F}\biggr)^{1/q_0}\to 0 \qquad (n\to\infty).
\end{equation*}
\notag
$$
It follows from what we proved above that the series $\sum_{s=0}^{\infty}f_s$ converges almost everywhere on $(0,T)$. The above estimate shows that for $f=\sum_{s=0}^{\infty}f_s$ we have
$$
\begin{equation*}
\biggl\|f-\sum_{s=0}^{n}f_s\biggr\|_{F} =\biggl\|\sum_{s=n+1}^{\infty}f_s\biggr\|_{F} \leqslant c_0\biggl(\sum_{s=n+1}^{\infty}\|f_s\|^{q_0}_{F}\biggr)^{1/q_0}\to 0 \qquad (n\to\infty),
\end{equation*}
\notag
$$
and thus the series also converges in $F$. These arguments and (5.24) prove that the ideal space $F$ has the Riesz–Fischer property with constants $c_0$ and $q_0$. This completes the proof of Theorem 11. Remark 12. Under the assumptions of part 3 of Theorem 11, if the space $F_0$ is complete, then $F$ is too, since for an ideal space completeness is equivalent to the Riesz–Fischer property (see Theorem 8).
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\Bibitem{BakGol25} |
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