Connection Between Weighted Tail, Orlicz, Grand Lorentz And Grand Lebesgue Norms

Abstract

We prove that the norm of functions in a suitable Grand Lorentz space built on a measure space, equipped with sigma finite diffuse measure, coincides with the norm in a suitable exponential Grand Lebesgue Space space as well as coincides with the so-called exponential tail norm, which may be quite described as norm in a suitable Banach rearrangement invariant space. We also exhibit comparisons with exponential Orlicz norms.

1 Introduction

Let $\ (\Omega = \{\omega \}, F,\mu ) \$ be a measure space equipped with sigma finite non-zero diffuse measure $\ \mu . \$ The diffuseness implies, as usually, that for an arbitrary measurable set $A \in F, \$ such that $\mu (A) > 0$, there exists a subset $\ A_1 \subset A \$ for which $\ \mu (A_1) = \mu (A)/2.\$

For any measurable numerical valued function $f: \Omega \rightarrow \mathbb R \$ we define its tail function

$$\begin{aligned} T[f](x) {\mathop {=}\limits ^{def}} \mu \{\omega : |f(\omega )| \ge x \} , \ \ \ x \ge 0. \end{aligned}$$

(1.1)

We introduce the so-called Grand Lorentz spaces, alike for the classical exponential Grand Lebesgue ones, built on the measure space $(\Omega , F,\mu )$ and we prove that they coincide, up equivalence of the norms, with suitable Grand Lebesgue Spaces (GLS) as well as coincides with the exponential Orlicz and tail spaces equipped with suitable norms.

Throughout the paper the letters $C,C_j(\cdot )$ will denote various positive constants which may differ from one formula to the next even within a single string of estimates and which do not depend on the variables. We make no attempt to obtain the best values for these constants.

For any function $\nu : \mathbb R \rightarrow \mathbb R$ we denote by $\nu ^*$ its Young-Fenchel, or Legendre transform

$$\begin{aligned} \nu ^*(\lambda ) {\mathop {=}\limits ^{def}} \sup _{y \in \textrm{Dom}[\nu ]}(\lambda y - \nu (y)), \end{aligned}$$

(1.2)

where $\textrm{Dom}[\nu ]$ is the domain of definition (and finiteness) of the function $\nu$. One can add formally

$$\begin{aligned} \nu (y) = + \infty , \ \ y \notin \textrm{Dom}[\nu ]. \end{aligned}$$

Recall that if the function $\ \nu (\cdot ) \$ is convex and continuous then, by Fenchel-Moreau theorem,

$$\begin{aligned} \nu ^{**} = [\nu ^*]^* = \nu , \end{aligned}$$

(1.3)

which we will use many times (see, e.g., [47, Section 31, pp. 327-341]).

1.1 Grand Lebesgue Spaces (GLS)

Let $p\ge 1$. The classical Lebesgue-Riesz space $L_p=L_p(\Omega ,F,\mu )=L_p(\Omega )$ consists of all measurable functions $\ f: \Omega \rightarrow \mathbb R \$ such that the norm $||f||_p = ||f||_{L_p(\mu )}$ is finite, where, as usually,

$$\begin{aligned} ||f||_p = ||f||_{L_p(\mu )} {\mathop {=}\limits ^{def}} \left( \int _{\Omega } \ |f(\omega )|^p \ \mu (d \omega ) \ \right) ^{1/p}, \ \ \ 1 \le p < \infty , \end{aligned}$$

and, of course,

$$\begin{aligned} ||f||_{\infty } = ||f||_{L_\infty (\mu )}={{\,\mathrm{ess\,sup}\,}}_{\omega \in \Omega }|f(\omega )|, \ \ \ p = \infty . \end{aligned}$$

We recall the definition of the Grand Lebesgue Spaces (GLS). Let $\psi = \psi (p)$ be a positive measurable numerical valued function, where $p \in (a,b)$, $1 \le a < b \le \infty$, not necessarily finite in every point, such that $\inf _{p \in (a,b)} \psi (p) > 0$.

The (Banach) Grand Lebesgue Space (GLS)   $G\psi =G\psi (a,b)$ consists of all the real (or complex) numerical valued measurable functions $f: \Omega \rightarrow \mathbb R \$ (or $f: \Omega \rightarrow \mathbb C$) defined on $\Omega$ and having finite norm

$$\begin{aligned} ||f||_{G\psi (a,b)}{\mathop {=}\limits ^{def}} \sup _{p \in (a,b)} \left[ \frac{||f||_p}{\psi (p)} \right] . \end{aligned}$$

(1.4)

We agree to write $G\psi$ in the case when $a = 1$ and $b =\infty$.

The function $\psi$ is named generating function for the space $G\psi$ and the set of all such generating functions $\psi$ will be denoted by $\{\psi (\cdot )\}$.

For instance

$$\begin{aligned} \psi (p) := p^{1/m}, \ \ \ p \in [1,\infty ), \ \ \ m > 0, \end{aligned}$$

(1.5)

or

$$\begin{aligned} \psi (p) := (p-a)^{-\alpha } \ (b-p)^{-\beta }, \ \ \ p \in (a,b), \ \ \ 1\le a<b<\infty , \ \ \ \alpha ,\beta \ge 0,\nonumber \\ \end{aligned}$$

(1.6)

are generating functions.

If

$$\begin{aligned} \psi (p) = 1, \ \ p = r; \ \ \ \ \psi (p) = +\infty , \ \ p \ne r, \ \ \ \ r\in [1,\infty ), \end{aligned}$$

where $C/\infty := 0, \ C \in \mathbb R$ (extremal case), then the corresponding $G\psi$ space coincides with the classical Lebesgue-Riesz space $L_r = L_r(\Omega )$.

Remark 1.1

Let $1<q<\infty$, $\theta \ge 0$ and define $\psi (p)=(q-p)^{-\theta /p}$, $p\in (1,q)$. Take $\Omega \subset \mathbb R^n, n\ge 1$, a measurable set with finite Lebesgue measure. Then, replacing p in (1.4) with $q-\varepsilon$, $\varepsilon \in (0,q-1)$, $b=q$, the space $G\psi (1,q)$ reduces to the classical Grand Lebesgue space $L^{q),\theta }(\Omega )$ defined by the norm

$$\begin{aligned} ||f||_{L^{q),\theta }(\Omega )}=||f||_{q),\theta }=\sup _{0<\varepsilon <q-1} \varepsilon ^{\frac{\theta }{q-\varepsilon }}||f||_{q-\varepsilon }, \end{aligned}$$

which, for $\theta =1$, is known as the space $L^{q)}(\Omega )$.

The Grand Lebesgue Spaces and several generalizations of them have been widely investigated, mainly in the case of GLS on sets of finite measure, (see, e.g., [9, 13, 16, 17, 23, 28, 29, 31, 37, 40], etc), while the case of Grand Lebesgue Spaces on sets of infinite measure was studied in [48,49,50]. They play an important role in the theory of Partial Differential Equations (PDEs) (see, e.g., [2, 18, 20, 27, 54], etc.), in interpolation theory (see, e.g., [1, 3, 14, 15, 19, 22]), in the theory of Probability ([24, 42, 44]), in Statistics [40, chapter 5], in theory of random fields [33, 43], in Functional Analysis [40, 41, 43] and so one.

These spaces are rearrangement invariant (r.i.) Banach functional (complete) spaces; their fundamental functions have been considered in [43]. They do not coincide, in the general case, with the classical Banach rearrangement functional spaces: Orlicz, Lorentz, Marcinkiewicz etc., see [37, 41]. The belonging of a function $f:\Omega \rightarrow \mathbb {R}$ to some $G\psi$ space is closely related with its tail function behavior as $\ t \rightarrow 0^+ \$ as well as when $\ t \rightarrow \infty , \$ see [33, 35].

1.2 Grand Lorentz Spaces

Recall now the classical definition of the Lorentz spaces (see [38, 39]). Let $\ p,q \in [1,\infty ]$. The Lorentz space $L_{p,q}=L_{p,q}(\Omega , F,\mu )=L_{p,q}(\Omega )$ consists of all measurable functions f on $\Omega$ for which $||f||_{p,q}$ is finite, where

$$\begin{aligned} ||f||_{p,q} {\mathop {=}\limits ^{def}} \left[ \ \int _0^{\infty } T^{q/p}[f](x) \ dx^q \ \right] ^{1/q}, \ \ \ \ 1\le q<\infty \end{aligned}$$

(1.7)

and

$$\begin{aligned} ||f||_{p,\infty } {\mathop {=}\limits ^{def}} \sup _{x > 0} \left[ \ x \ T^{1/p}[f](x) \ \right] , \ \ \ \ q=\infty . \end{aligned}$$

(1.8)

The functional $f \rightarrow ||f||_{p,q}$ is said Lorentz norm of the function $f: \Omega \rightarrow \mathbb R$. More precisely, this functional is not a norm, but there are norms to which it is equivalent (see., e.g., [26, p.53], [51, 52, pp.192-198]).

These spaces and their applications have been widely investigated (see, e.g., [7, 10, 12, 26], etc.)

Definition 1.1

We define here a (Grand) generalization of these spaces, alike one for the (Grand) generalization of classical Lebesgue-Riesz ones. Let again $\ \psi = \psi (p), \ p \in (a,b), \ 1 \le a < b \le \infty$, be some generating function from the set $\{\psi (\cdot )\}$. Let $q\in [1,\infty ]$. The Grand Lorentz Space is the rearrangement invariant (r.i.) space

$$\begin{aligned} WL[\psi ](q; a,b) {\mathop {=}\limits ^{def}} \left\{ \ f: \Omega \rightarrow \mathbb R \, \ ||f||_{W_q[\psi ](a,b)} < \infty \ \right\} , \end{aligned}$$

where the the Grand Lorentz norm is defined by

$$\begin{aligned} ||f||_{W_q[\psi ](a,b)} {\mathop {=}\limits ^{def}} \sup _{p \in (a,b)} \left\{ \ \frac{||f||_{p,q}}{\psi (p)} \ \right\} , \ \ \ 1 \le q \le \infty . \end{aligned}$$

(1.9)

We agree to write, as before, in the case when $\ a = 1, \ b = \infty$,

$$\begin{aligned} ||f||_{W_q[\psi ]} {\mathop {=}\limits ^{def}} ||f||_{W_q[\psi ](1,\infty )}. \end{aligned}$$

These spaces have been investigated in [3, 22, 30, 32, 53].

Note that this notion does not coincide with the definition of so-called strong Lorentz spaces (see, e.g., [30]).

1.3 Tail Norms and Tail Spaces

We introduce the tail norm for a (measurable) function $f: \Omega \rightarrow \mathbb R$. Let $\ H = H(x) \$ be a continuous non-negative and non-increasing function, defined on $\mathbb R^+$, such that $\ H(\infty ) = 0$. We define the tail quasi-norm $||f||^*_{T(H)}$ of the function f by

$$\begin{aligned} ||f||^*_{T(H)} {\mathop {=}\limits ^{def}} \inf \{ k> 0 \, : \, T[f](x) \le 2 H(x/k) \} , \ \ \forall x > 0. \end{aligned}$$

(1.10)

The functional $\ f \rightarrow ||f||^*_{T(H)} \$ is non-negative and homogeneous:

$$\begin{aligned} ||\alpha f||^*_{T(H)} = |\alpha | \,||f||^*_{T(H)}, \ \ \ \alpha \in \mathbb R, \end{aligned}$$

but it is not, in general, a norm. These functionals are used in the probability theory, of course in the case when $\ \mu (\Omega ) = 1$.

We intend to introduce a norm which is equivalent, under appropriate conditions, to the tail quasi-norm introduced above.

Definition 1.2

Let $G = G(x), \ x \ge 1$, be a strictly increasing continuous function such that

$$\begin{aligned} G(1) = 0, \ \ \ G(\infty ) = \infty . \end{aligned}$$

Put

$$\begin{aligned} H_G(x) = \exp (-G(x)), \ \ x \ge 1. \end{aligned}$$

We introduce, for a measurable function $\ f: \Omega \rightarrow \mathbb R$, a new norm defined by

$$\begin{aligned} ||f||_{T(H_G)} {\mathop {=}\limits ^{def}} \sup _{A \,:\, \mu (A) \in (0,1/2]} \left[ \ \frac{\displaystyle \int _A |f(\omega )| \ \mu (d\omega )}{\mu (A) \ G^{-1}(|\ln \mu (A)|)} \ \right] . \end{aligned}$$

(1.11)

The functional $\ f \rightarrow ||f||_{T(H_G)} \$ is really a norm, which is a slight generalization of the classical norm of Marcinkiewicz (see [11, p.82])

$$\begin{aligned} ||f||_{L^{p,\infty }}=\sup _{A\in F} \mu (A)^{\frac{1}{p}-1}\int _A |f(\omega )|\,\mu (d\omega ). \end{aligned}$$

Remark 1.2

Choosing $G(x)=p\ln x, \ x\ge 1, \ p>1$, the norm $||f||_{TH_G}$ is equivalent to $||f||_{L^{p,\infty }}$.

In fact in this case we have $G(1) = 0, \ \ \ G(\infty ) =\infty$, $H_G(x)=e^{-p\ln x}=x^{-p}$ and $G^{-1}(y)=e^{\frac{y}{p}}$,

which implies, recalling that $\mu (A) \in (0,1/2]$,

$$\begin{aligned} G^{-1}(|\ln \mu (A)|)=G^{-1}(-\ln \mu (A))=e^{-\frac{1}{p}\ln \mu (A)}=\mu (A)^{-\frac{1}{p}} \end{aligned}$$

and

$$\begin{aligned} ||f||_{T(H_G)}= & {} \sup _{A\,: \, \mu (A) \in (0,1/2]} \left[ \ \frac{\displaystyle \int _A |f(\omega )| \ \mu (d\omega )}{\mu (A) \cdot \mu (A)^{-\frac{1}{p}}} \ \right] \\ {}= & {} \sup _{A\,: \, \mu (A) \in (0,1/2]} \mu (A)^{\frac{1}{p}-1}\int _A |f(\omega )|\,\mu (d\omega ). \end{aligned}$$

Lemma 1.1

Let $(\Omega , F,\mu )$ be a measure space with probabilistic measure $\mu$, that is $\mu (\Omega ) = 1$. Let $G = G(x), \ x \ge 1$, be a strictly increasing continuous function such that $G(1) = 0, \ G(\infty ) = \infty$. Suppose also that

$$\begin{aligned} C_1 = C_1(G) := \sup _{z \ge \ln 2} G^{-1}(z + \ln 2)/G^{-1}(z) < \infty \end{aligned}$$

(1.12)

and

$$\begin{aligned} C_2 = C_2(G) := \sup _{z \ge \ln 2} \left\{ \frac{\displaystyle \int _{G^{-1}(z + \ln 2)}^{\infty } \ \exp (-G(s)) \, ds}{(\exp (-z) \ G^{-1}(z))} \ \right\} < \infty . \end{aligned}$$

(1.13)

Then, for any measurable function $f:\Omega \rightarrow \mathbb R$, the norm $||f||_{T(H_G)}$ and quasi-norm $||f||^*_{T(H_G)}$ are equivalent, that is

$$\begin{aligned} ||f||^*_{T(H_G)} \le ||f||_{T(H_G)} \le (C_1 + 2 C_2) \ ||f||^*_{T(H_G)}. \end{aligned}$$

(1.14)

Proof

Let us first consider $||f||_{T(H_G)} \in (0,\infty )$; one can assume, without loss of generality, $||f||_{T(H_G)} =1$. As long as the measure $\mu$ is continuous

$$\begin{aligned} \int _A |f| \ d \mu \le \mu (A) \ G^{-1} ( \ln (2/\mu (A))) \end{aligned}$$

for any Borelian set $A \in F$. (If $\mu (A) = 0$ then the right-hand side is also equal to zero).

Further we have, for an arbitrary positive value t such that $\ T[f](t) > 0$,

$$\begin{aligned} t \ T[f](t) \le \int _{\{\omega :\, |f(\omega )| > t\}} |f| \ d \mu \le \ T[f](t) \ G^{-1}(\ln (2/T([f](t)))). \end{aligned}$$

From the last inequalities we deduce

$$\begin{aligned} T[f](t) \le 2 \exp (-G(t))=2 H_G(t), \end{aligned}$$

(1.15)

which implies $||f||^*_{T(H_G)}<\infty$.

Inversely, let the estimate (1.15) be given. As long as

$$\begin{aligned} \int _A |f| \ d \mu = \int _0^{\infty } \mu \{\omega \in A :\, |f(\omega )| > t\} \ dt, \end{aligned}$$

we obtain for an arbitrary positive value $\ v > 0 \$

$$\begin{aligned} \int _A |f| \ d \mu \le v \ \mu (A) + 2 \int _v^{\infty } \exp (-G(s)) \ ds. \end{aligned}$$

Let us choose $\ v = G^{-1}(\ln (2/\mu (A))); \$ then we get

$$\begin{aligned} \int _A |f| \ d \mu \le \mu (A) \ G^{-1}(\ln (2/\mu (A))) + 2 \int _{G^{-1}(\ln (2/\mu (A)))}^{\infty } \exp (-G(s) ) \ ds \end{aligned}$$

and, taking into account the restrictions on the function $G(\cdot )$, we get

$$\begin{aligned} \int _A |f| \ d \mu \le (C_1 + 2 C_2) \cdot \mu (A) \ G^{-1}(|\ln \mu (A)|), \ \ \ A \in F, \end{aligned}$$

so that $||f||_{T(H_G)} \le C_1 + 2 C_2 < \infty$. $\square$

Example 1.1

The conditions of Lemma 1.1 are satisfied for the important class of functions $\ G(\cdot ) \$ having the form

$$\begin{aligned} G(x) = C_3 \,(x-1)^m \, \ln ^r(C_4 + x), \ \ \ \ x\ge 1, \ \ \ m> 0, \ \ r \in \mathbb R, \ \ C_3>0, \end{aligned}$$

and $C_4= C_4(m,r) \$ is a positive constant sufficiently large.

Remark 1.3

Let G and $H_G$ be functions as in Definition 1.2. Suppose in addition that the function $h=\ln H_G$ is sub-additive, i.e.

$$\begin{aligned} h(t+s)\le h(t)+h(s), \ \ \ t,s\ge 1. \end{aligned}$$

Then

$$\begin{aligned} H_G(t+s) \le H_G(t) H_G(s), \ \ \ t,s \ge 1. \end{aligned}$$

(1.16)

Assume also that $\ f: \Omega \rightarrow \mathbb R \$ is a (measurable) non-negative function satisfying the inequality

$$\begin{aligned} T[f](t) \le C \, H_G(t), \ \ \ t \ge 1, \ \ C = \textrm{const}> 1. \end{aligned}$$

First of all we note that

$$\begin{aligned} T[f](t) \le \min \{C\, H_G(t),1\}, \ \ \ t \ge 1. \end{aligned}$$

Let $t_0>1$ be the value for which $C\,H_G(t_0)=1$; this value exists and is unique. Consider the measurable function $g=f-t_0$. Then

$$\begin{aligned} \textbf{P} (g \le 0) = 0. \end{aligned}$$

Therefore $g\ge 0$. For an arbitrary value $z\ge 1$ we deduce

$$\begin{aligned} T[g](z)= & {} \textbf{P}(f > z + t_0) \le C H_G(z + t_0) = \frac{H_G(t_0 + z)}{H_G(t_0)} \le \frac{H_G(t_0)H_G(z)}{H_G(t_0)}\\ {}= & {} H_G(z), \end{aligned}$$

hence

$$\begin{aligned} T[g](t)\le H_G(t), \ \ \ t\ge 1. \end{aligned}$$

Example 1.2

A tail estimate imposed on the measurable function $\ f: \Omega \rightarrow \mathbb R$, where $\mu (\Omega ) = 1$, of the form

$$\begin{aligned} T[f](x) \le \exp \left( - (x/K)^m \right) , \ \ \forall x \ge 0, \ \ \ m,K = \textrm{const}\in (0,\infty ), \end{aligned}$$

is quite equivalent to the following inequality

$$\begin{aligned} \sup _{t \in (0,1/2)} \left\{ \ t^{-1} \ |\ln t|^{-1/m} \ \left[ \ \sup _{A: \mu (A) \le t} \int _A |f| \ d \mu \ \right] \ \right\} \le c(m) K < \infty , \end{aligned}$$

for some non-zero finite constant $\ c(m) \in (0,\infty )$.

2 The equivalence Between Tail Behavior and Grand Lebesgue Norm

We intend to obtain some results in this topic similarly to the ones in [8, 13, 35].

It is convenient to represent the generating function $\ \psi (\cdot ) \$ in the following exponential form

$$\begin{aligned} \psi (p) = \psi [\nu ](p) = \exp \left( \ \frac{\nu (p)}{p} \ \right) \end{aligned}$$

(2.1)

for some continuous convex function $\ \nu = \nu (p)$, $1 \le p < \infty$, or at least for $p \in (p_0, \infty )$, where $p_0 = \textrm{const}> 1$.

Of course, it follows from (2.1)

$$\begin{aligned} \nu (p):= \nu [\psi ](p) = p \ \ln \psi (p). \end{aligned}$$

For instance,

$$\begin{aligned} \psi (p)= \psi _{m,r}(p) = p^{1/m} \, \ln ^r(p + 1), \ \ m = {\textrm{const}} > 0, \ \ r = {\textrm{const}} \in \mathbb R, \end{aligned}$$

or more generally

$$\begin{aligned} \psi (p) = \psi _{m,r,L}(p) = p^{1/m} \ \ln ^r(p+1) \ L(\ln (p + 1)), \end{aligned}$$

(2.2)

where $\ L = L(z), \ z \ge 1$, is a positive twice continuous differentiable slowly varying at infinity function.

Remark 2.1

Since the case when $\displaystyle \sup _{p \ge p_0} \nu (p) < \infty \$ is trivial, one can assume, without loss of generality,

$$\begin{aligned} \lim \limits _{p \rightarrow \infty } \nu (p) = \infty . \end{aligned}$$

Definition 2.1

The Grand Lebesgue Space $G\psi [\nu ]\$ with generating function $\ \psi (\cdot ) \$ of the form (2.1) is named exponential GLS, as well as other spaces considered henceforth.

Theorem 2.1

Let $\nu (\cdot )$ be a non-negative even function, continuous and convex and let $\psi [\nu ](\cdot )$ be defined as in (2.1).

If $f \in G\psi [\nu ]$, $f\ne 0$, then the following inequality

$$\begin{aligned} T[f](x) \le \exp \left( \, - \nu ^*(\ln x) \, \right) , \ \ \ x > 1 \end{aligned}$$

(2.3)

holds, which implies

$$\begin{aligned} ||f||^*_{T(H)}<\infty , \ \ \hbox {where} \ \ H(x)=\exp (-\nu ^*(\ln x))/2. \end{aligned}$$

Inversely, assume that (2.3) holds. Define for $\epsilon \in (0,1)$ the following quantities:

$$\begin{aligned}{} & {} K(\varepsilon ):= \int _\mathbb R \exp \left( \ -\varepsilon \nu ^*(y/(1 - \varepsilon ) \ ) \right) \ dy,\\{} & {} Z(\varepsilon ):= \int _\mathbb R \exp \left[ \ \nu ^*((1 - \varepsilon ) y ) - \nu ^*(y) \ \right] \ dy,\\{} & {} \Theta (p) = \Theta [\nu ](p) {\mathop {=}\limits ^{def}} \inf _{\varepsilon \in (0,1)} \left\{ \ \min \left( \frac{K(\varepsilon )}{1-\varepsilon }, Z(\varepsilon )\right) \cdot \exp (\nu (p/(1 - \varepsilon ))) \ \right\} . \end{aligned}$$

Suppose, in addition, that there exists a constant $C_1>0$ such that

$$\begin{aligned} {[}\Theta (p)]^{1/p} \le C_1 \ \psi (p), \ \ \ p \ge 1. \end{aligned}$$

(2.4)

Then $f \in G\psi [\nu ]$, i.e.

$$\begin{aligned} ||f||_{G\psi [\nu ]} = \sup _{p \ge 1} \left\{ \frac{||f||_p}{\psi [\nu ](p)} \ \right\} < \infty . \end{aligned}$$

(2.5)

Proof

Let $0 \ne f \in G\psi [\nu ]$, that is $||f||_{G\psi [\nu ]}<\infty$; one can assume without loss of generality $\ ||f||_{G\psi [\nu ]} = 1$. We have, by the definition of the norm (1.4),

$$\begin{aligned} \int _{\Omega } |f|^p \, d\mu \le \exp (\nu (p)). \end{aligned}$$

By virtue of Tchebychev-Markov inequality

$$\begin{aligned} T[f](x) \le \frac{\exp (\nu (p))}{x^p } = \exp [ -(p \ln x - \nu (p)) ], \ \ \ x > 1, \end{aligned}$$

and, after the minimization over p, we have (2.3):

$$\begin{aligned} T[f](x) \le \exp \left( \, - \nu ^*(\ln x) \, \right) , \ \ \ x > 1. \end{aligned}$$

Therefore in the general case when $\ 0 \ne f \in G\psi [\nu ] \$

$$\begin{aligned} T[f](x) \le \exp \left( \, - \nu ^*(\ln (x/||f||_{G\psi [\nu ]})) \, \right) , \ \ \ x > ||f||_{G\psi [\nu ]}. \end{aligned}$$

The last inequality implies that

$$\begin{aligned} ||f||^*_{T(H)}<\infty , \ \ \hbox {where} \ \ H(x)=\exp (-\nu ^*(\ln x))/2, \end{aligned}$$

and also

$$\begin{aligned} ||f||^*_{T(H_G)}<\infty , \ \ \textrm{where} \ \ H_G(x)=\exp (-G(x)), \ \ \ G(x)= \nu ^*(\ln x), \end{aligned}$$

since

$$\begin{aligned} T[f](x){} & {} \le \exp \left( \, - \nu ^*(\ln (x/||f||_{G\psi [\nu ]})) \, \right) <2 \exp \left( \, - \nu ^*(\ln (x/||f||_{G\psi [\nu ]})) \, \right) , \ \ \\\ {}{} & {} \quad x > ||f||_{G\psi [\nu ]} \end{aligned}$$

and $G(x)= \nu ^*(\ln x)$ is strictly increasing being $\nu ^*$ strictly increasing and $G(1)=\nu ^*(0)=0$ (see [46, p.6]).

Inversely, let the inequality (2.3) be given for a function f.

Similarly as done in [35, 36], we prove that

$$\begin{aligned} ||f||_p \le C \ [\Theta (p)]^{1/p}. \end{aligned}$$

(2.6)

We denote

$$\begin{aligned} I(p)=\int _{-\infty }^{+\infty } \exp (py-\nu ^*(y)\,)\,dy. \end{aligned}$$

It is easily seen that, under the condition (2.3), i.e.

$$\begin{aligned} T[f](x) \le \exp \left( \, - \nu ^*(\ln x) \, \right) , \ \ \ x > 1, \end{aligned}$$

we have

$$\begin{aligned} ||f||_p\le p^{1/p} [I(p)]^{1/p}, \end{aligned}$$

in fact

$$\begin{aligned} \begin{aligned} ||f||_p&= p^{1/p} \left( \int _0^\infty x^{p-1}\,T[f](x)\,dx\right) ^{\frac{1}{p}}\\&\le p^{1/p} \left( \int _0^\infty x^{p-1}\,\exp \left( \, - \nu ^*(\ln x) \, \right) \,dx\right) ^{\frac{1}{p}}\\&= p^{1/p} \left( \int _{-\infty }^{\infty } \exp \left( py - \nu ^*(y) \, \right) \,dy\right) ^{\frac{1}{p}}. \end{aligned} \end{aligned}$$

Put

$$\begin{aligned} \sigma _\varepsilon (dy)=\frac{\exp (-\varepsilon \nu ^*(y))(1-\varepsilon )}{K(\varepsilon )}\,dy \end{aligned}$$

which is equivalent to

$$\begin{aligned} \sigma _\varepsilon (A)=\frac{\displaystyle \int _A \exp (-\varepsilon \nu ^*(y))(1-\varepsilon )\, dy}{K(\varepsilon )} \end{aligned}$$

for any $A\in F$, $\varepsilon \ge \varepsilon _0$, $\varepsilon _0\in (0,1)$. Then $\sigma _\varepsilon (dy)$ is probabilistic, i.e. $\int _\mathbb R \sigma _\varepsilon (dy) =1, \ \ 0<\varepsilon <1$.

We get

$$\begin{aligned} \begin{aligned} \frac{I(p)}{K(\varepsilon )}&=\frac{1}{1-\varepsilon }\int _{-\infty }^{+\infty } \exp (py-\nu ^*(y)) \, \exp (\varepsilon \nu ^*(y) )\, \sigma _\varepsilon (dy)\\&=\frac{1}{1-\varepsilon }\int _{-\infty }^{+\infty } \exp (py-(1-\varepsilon )\nu ^*(y)) \,\sigma _\varepsilon (dy)\\&\le \frac{1}{1-\varepsilon } \,\exp \left\{ \sup _{y\in \mathbb R}(py-(1-\varepsilon )\nu ^*(y))\right\} \int _{-\infty }^{+\infty }\sigma _\varepsilon (dy)\\&= \frac{1}{1-\varepsilon } \,\exp \left\{ \sup _{y\in \mathbb R}\left( (1-\varepsilon )\left( \frac{p}{1-\varepsilon }y-\nu ^*(y)\right) \right) \right\} \\&= \frac{1}{1-\varepsilon } \exp \left\{ (1-\varepsilon )\nu ^{**}\left( \frac{p}{1-\varepsilon }\right) \right\} \\&= \frac{1}{1-\varepsilon } \exp \left\{ (1-\varepsilon )\nu \left( \frac{p}{1-\varepsilon }\right) \right\} . \end{aligned} \end{aligned}$$

Hence, since $0<1-\varepsilon <1$,

$$\begin{aligned} ||f||_p\le p^{1/p} [I(p)]^{1/p}\le p^{1/p} \left( \frac{1}{1-\varepsilon }\right) ^{1/p}(K(\varepsilon ))^{1/p} \left[ \exp \left\{ \nu \left( \frac{p}{1-\varepsilon }\right) \right\} \right] ^{1/p} \end{aligned}$$

Similarly, put

$$\begin{aligned} \overline{\sigma }_\varepsilon (dy)=\frac{\exp (\nu ^*((1-\varepsilon ) y)-\nu ^*(y))}{Z(\varepsilon )}\,dy, \end{aligned}$$

then

$$\begin{aligned} \begin{aligned} \frac{I(p)}{Z(\varepsilon )}&=\int _{-\infty }^{+\infty } \exp (py-\nu ^*((1-\varepsilon )y)\,\overline{\sigma }_\varepsilon (dy)\\&\le \exp \left\{ \sup _{y\in \mathbb R}(py-\nu ^*((1-\varepsilon )y))\right\} \\&= \exp \left\{ \sup _{y\in \mathbb R}\left( p\frac{y}{1-\varepsilon }-\nu ^*(y)\right) \right\} \\&= \exp \left\{ \nu ^{**}\left( \frac{p}{1-\varepsilon }\right) \right\} = \exp \left\{ \nu \left( \frac{p}{1-\varepsilon }\right) \right\} \end{aligned} \end{aligned}$$

Hence

$$\begin{aligned} ||f||_p\le p^{1/p} [I(p)]^{1/p}\le p^{1/p} (Z(\varepsilon ))^{1/p} \left[ \exp \left\{ \nu \left( \frac{p}{1-\varepsilon }\right) \right\} \right] ^{1/p}. \end{aligned}$$

Therefore we have (2.6):

$$\begin{aligned} ||f||_p \le C \ [\Theta (p)]^{1/p} \end{aligned}$$

where

$$\begin{aligned} \Theta (p) = \Theta [\nu ](p) = \inf _{\varepsilon \in (0,1)} \left\{ \ \min \left( \frac{K(\varepsilon )}{1-\varepsilon }, Z(\varepsilon )\right) \cdot \exp (\nu (p/(1 - \varepsilon ))) \ \right\} . \end{aligned}$$

Finally, by the assumption (2.4):

$$\begin{aligned} {[}\Theta (p)]^{1/p} \le C_1 \ \psi (p), \ \ \ p \ge 1, \end{aligned}$$

and (2.6), we get $f \in G\psi [\nu ]$, i.e.

$$\begin{aligned} ||f||_{G\psi [\nu ]} = \sup _{p \ge 1} \left\{ \frac{||f||_p}{\psi [\nu ](p)} \ \right\} < \infty . \end{aligned}$$

$\square$

From Theorem 2.1 and the equivalence given in Lemma 1.1 we have the following:

Corollary 2.1

If all the conditions of Theorem 2.1 and Lemma 1.1 are satisfied, then both the norms $||f||_{G\psi [\nu ]}$ and $||f||_{T(H_G)}$ are equivalent, where $||f||_{T(H_G)}$ is defined in (1.11).

Remark 2.2

The conditions of Theorem 2.1 are satisfied for example for the $\psi$-functions of the form $\ \psi (p) = \psi _{m,r,L}(p)$, defined in (2.2).

3 Main Results. The Equivalence Between Grand Lorentz, Tail and Grand Lebesgue Norms

We suppose henceforth that the function $\ \psi (p) = \psi [\nu ](p) \$ is defined as in (2.1). We assume that $\mu (\Omega ) \in (0,\infty ]$.

Recall that for a measurable non-zero function $f:\Omega \rightarrow \mathbb R$ the condition

$$\begin{aligned} T[f](x) \le \exp \{-\nu ^*(\ln x)\}, \ \ x \ge 1 \end{aligned}$$

implies

$$\begin{aligned} ||f||_{T(H_G)} < \infty , \ \ \hbox {where} \ \ H_G(x)=\exp (-G(x)), \ \ \ G(x)= \nu ^*(\ln x), \ \ x>1. \end{aligned}$$

Theorem 3.1

Let $f:\Omega \rightarrow \mathbb R$ a measurable non-zero function. Let $\nu (\cdot )$ be a non-negative even function, continuous and convex and let $\psi [\nu ](\cdot )$ be defined as in (2.1), i.e.

$$\begin{aligned} \psi [\nu ](p)=\exp (\nu (p)/p), \ \ \ 1\le p<\infty . \end{aligned}$$

Consider the following conditions:

$$\begin{aligned} \begin{aligned}&{{\textbf {A}}.} \ \ \ \ \ T[f](x) \le \exp \{-\nu ^*(\ln x)\}, \ \ x \ge 1\\ \\&{{\textbf {B}}.} \ \ \ \ \ \exists \, q \in [1,\infty ] \ : \ ||f||_{W_q[\psi [\nu ]]}< \infty ;\\ \\&{{\textbf {C}}.} \ \ \ \ \ ||f||_{W_q[\psi [\nu ]]}< \infty \ \ \ \forall q \in [1,\infty ] \ ;\\ \\&{{\textbf {D}}.} \ \ \ \ \ ||f||_{G\psi [\nu ]} < \infty . \end{aligned} \end{aligned}$$

Then

• (1) $\textbf{D} \rightarrow \textbf{A}$

• (2) $\textbf{A} \rightarrow \textbf{D}$ if the condition (2.4) holds.

• (3) $\textbf{C} \rightarrow \textbf{B}$

• (4) $\textbf{B} \rightarrow \textbf{A}$

• (5) If the condition

  $$\begin{aligned} \displaystyle \int _{-\infty }^{\infty } \exp \left[ \ p^{-1} \, \left( p y - \nu ^*(y) \ \right) \ \right] \ dy \le C \ \exp ( \nu ^{**}(p) /p ) \end{aligned}$$

  (3.1)

  holds, then

  $$\begin{aligned} \textbf{A} \rightarrow \textbf{C} \ \ \ \hbox {and} \ \ \ \textbf{D} \rightarrow \textbf{C}. \end{aligned}$$

Proof

The implications (1) and (2) have already been proved in Sect. 2.

The implication in (3) $\ \textbf{C} \rightarrow \textbf{B} \$ is obvious.

We will use the following classical embedding inclusions for the Lorentz spaces

$$\begin{aligned} ||f||_{p,q_2} \le ||f||_{p,q_1}, \ \ \ 1 \le p < \infty , \ \ \ 1 \le q_1 \le q_2 \le \infty , \end{aligned}$$

(3.2)

(see e.g. [51, p. 265], [52, p. 192], [7, p. 217]).

Let us now show the implication (4): $\ \textbf{B} \rightarrow \textbf{A}$.

Suppose that there exists $q\in [1,\infty ]$ such that $||f||_{W_q[\psi [\nu ]]}<\infty$. One can assume, without loss of generality,

$$\begin{aligned} ||f||_{W_q[\psi [\nu ]]}=\sup _{p \in [1,\infty ) } \left[ \ \frac{||f||_{p,q}}{\psi [\nu ](p)} \ \right] = 1. \end{aligned}$$

By (3.2) we have

$$\begin{aligned} ||f||_{p,\infty } =\sup _{x > 0} \, x \ \left[ \ T^{1/p}[f](x) \ \right] \le ||f||_{p,q}\le \psi [\nu ](p), \end{aligned}$$

which implies

$$\begin{aligned} T[f](x) \le \exp (-p \ln x + \nu (p)), \ \ \ x \ge 1. \end{aligned}$$

Following,

$$\begin{aligned} T[f](x) \le \inf _p \exp (-p \ln x + \nu (p)) = \exp \{ -\nu ^*(\ln x) \}, \ \ \ x \ge 1. \end{aligned}$$

Now we prove (5): $\ \textbf{A} \rightarrow \textbf{C}$, under condition (3.1).

Assume that A holds, i.e. $T[f](x) \le \exp \{ \ -\nu ^*(\ln x) \}, \ x \ge 1$ holds. It is enough, by virtue of embedding inclusions (3.2), to consider only the value $\ q = 1. \$ By the assumption and using Fenchel-Moreau theorem (see e.g. [35]) we deduce

$$\begin{aligned} \begin{aligned} ||f||_{p,1}&= \int _0^{\infty } T^{1/p}[f](x) \ dx \le \int _0^{\infty }\exp \left[ - \nu ^*(\ln x)/p \right] \ dx\\ \\&= \int _{-\infty }^{\infty } \exp \left( \ y - \frac{\nu ^*(y)}{p} \ \right) \ dy = \int _{-\infty }^{\infty } \exp \left[ \ p^{-1} \, \left( p y - \nu ^*(y) \ \right) \ \right] \ dy \\ \\&\le C \ \exp ( \nu ^{**}(p) /p ) = C \ \exp ( \nu (p) /p ) = C \psi [\nu ](p). \end{aligned} \end{aligned}$$

So we have $||f||_{W_1[\psi [\nu ]]}<\infty$. Therefore $||f||_{W_q[\psi [\nu ]]}\le ||f||_{W_1[\psi [\nu ]]}<\infty$ for any $q\in [1,\infty ]$.

We prove (5): $\ \textbf{D} \rightarrow \textbf{C} \$, under condition (3.1).

As seen in Theorem 2.1 we have

$$\begin{aligned} T[f](x) \le \exp \left( - \nu ^*(\ln x) \right) , \ \ x \ge 1. \end{aligned}$$

Arguing as above we get

$$\begin{aligned} ||f||_{p,1}\le C \psi [\nu ](p). \end{aligned}$$

Therefore $||f||_{p,q} \le ||f||_{p,1}\le C \psi [\nu ](p)$ for any $q\in [1,\infty ]$, and finally $||f||_{W_q[\psi [\nu ]]}<\infty$ for any $q\in [1,\infty ]$.

This completes the proof. $\square$

Remark 3.1

Let $\nu (\cdot )$ and $\psi (\cdot )$ as in Theorem 3.1.

If $f\in G\psi [\nu ]$ then $f\in W_\infty [\psi [\nu ]]$.

In fact, if $f\in G\psi [\nu ]$ then $T[f](x) \le \exp \left( - \nu ^*(\ln x) \right) , \ x \ge 1$ holds. Hence

$$\begin{aligned} \begin{aligned} ||f||_{p,\infty }&=\sup _{x> 0} \, x \ \left[ \ T^{1/p}[f](x) \ \right] \le \sup _{x \ge 1} \left[ \ x \left\{ \ \exp ( \ - \nu ^*(\ln x)/p ) \ \right\} \ \right] \\&= \sup _{x \ge 1} \left\{ \ \exp \left\{ \ln x - p^{-1}\nu ^*(\ln x) \right\} \ \right\} \\ {}&= \sup _{y \ge 0} \exp \left( \ y - p^{-1} \nu ^*(y) \ \right) = \left[ \ \exp \left( \sup _{y > 0}(py - \nu ^{*}(p))\right) \ \right] ^{1/p}\\&= \left[ \ \exp (\nu ^{**}(p)) \ \right] ^{1/p} = \exp (\nu ^{**}(p) / p)=\exp (\nu (p) / p) = \psi [\nu ](p). \end{aligned} \end{aligned}$$

Therefore

$$\begin{aligned} ||f||_{W_\infty [\psi [\nu ]]}=\sup _{p \in [1,\infty ) } \left[ \ \frac{||f||_{p,\infty }}{\psi [\nu ](p)} \ \right] <\infty . \end{aligned}$$

Remark 3.2

Let $\nu (\cdot )$ and $\psi (\cdot )$ as in Theorem 3.1.

The condition (3.1) in (5):

$$\begin{aligned} \int _{-\infty }^{\infty } \exp \left[ \ p^{-1} \, \left( p y - \nu ^*(y) \ \right) \ \right] \ dy \le C \ \exp ( \nu ^{**}(p) /p ) \end{aligned}$$

is satisfied, for example, for the generating function of the form

$$\begin{aligned} \psi (p) = C p^{1/m} \ \ln ^{\gamma }(p) \ L(\ln (p + 1)), \ \ p \in [1,\infty ),\ \ m = \textrm{const}>0, \ \gamma \ \textrm{const}, \end{aligned}$$

where $\ L = L(x) \$ is a positive continuous slowly varying function.

4 Remark about Coincidence with Exponential Orlicz Spaces

We retain all the notations and restrictions of the previous sections. Define the following Young-Orlicz function

$$\begin{aligned}{} & {} N(u) = N[\nu ](u) = \exp \left( \ \nu ^*(\ln |u|) \ \right) , \ \ |u| \ge e;\\{} & {} N(u) = N[\nu ](u) = \exp \left( \ \nu ^*(1) \ \right) \, u^2, \ \ |u| < e. \end{aligned}$$

It is proved in [36] that, under some natural conditions, the exponential Orlicz space built over $\ (\Omega , F, \mu ) \$ equipped with the corresponding Young - Orlicz function $\ N = N[\nu ](u) \$ coincides, up to norm equivalence, with the Grand Lebesgue space $\ G\psi [\nu ]. \$

Therefore, it coincides also with the Grand Lorentz space introduced in the previous sections, as well as with suitable tail space, see Theorem 3.1.

5 Concluding Remark

It is easy to make sure that the exponential condition is in the general case very important. It is sufficient to consider for instance the classical Lebesgue - Riesz - Orlicz space $\ L_p = L_p(\Omega ,\mu ), \ p \ge 1 \$, still for the probabilistic measure $\ \mu . \$ Indeed, if the non - zero function $\ f: \Omega \rightarrow \mathbb R \$ belongs to the space $\ L_p$ such that $\ ||f||_p = 1, \$ then

$$\begin{aligned} T[f](x) \le x^{-p}, \ \ \ x \ge 1, \end{aligned}$$

but the inverse conclusion is not true.