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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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
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
where \( \textrm{Dom}[\nu ] \) is the domain of definition (and finiteness) of the function \( \nu \). One can add formally
Recall that if the function \( \ \nu (\cdot ) \ \) is convex and continuous then, by Fenchel-Moreau theorem,
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,
and, of course,
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
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
or
are generating functions.
If
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
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
and
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
where the the Grand Lorentz norm is defined by
We agree to write, as before, in the case when \( \ a = 1, \ b = \infty \),
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
The functional \( \ f \rightarrow ||f||^*_{T(H)} \ \) is non-negative and homogeneous:
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
Put
We introduce, for a measurable function \( \ f: \Omega \rightarrow \mathbb R \), a new norm defined by
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])
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 \(\mu (A) \in (0,1/2]\),
and
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
and
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
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
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 \(\ T[f](t) > 0 \),
From the last inequalities we deduce
which implies \(||f||^*_{T(H_G)}<\infty \).
Inversely, let the estimate (1.15) be given. As long as
we obtain for an arbitrary positive value \( \ v > 0 \ \)
Let us choose \( \ v = G^{-1}(\ln (2/\mu (A))); \ \) then we get
and, taking into account the restrictions on the function \( G(\cdot )\), we get
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
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.
Then
Assume also that \( \ f: \Omega \rightarrow \mathbb R \ \) is a (measurable) non-negative function satisfying the inequality
First of all we note that
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
Therefore \(g\ge 0\). For an arbitrary value \(z\ge 1\) we deduce
hence
Example 1.2
A tail estimate imposed on the measurable function \( \ f: \Omega \rightarrow \mathbb R\), where \( \mu (\Omega ) = 1\), of the form
is quite equivalent to the following inequality
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
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)
For instance,
or more generally
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,
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
holds, which implies
Inversely, assume that (2.3) holds. Define for \(\epsilon \in (0,1)\) the following quantities:
Suppose, in addition, that there exists a constant \(C_1>0\) such that
Then \(f \in G\psi [\nu ]\), i.e.
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),
By virtue of Tchebychev-Markov inequality
and, after the minimization over p, we have (2.3):
Therefore in the general case when \( \ 0 \ne f \in G\psi [\nu ] \ \)
The last inequality implies that
and also
since
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
We denote
It is easily seen that, under the condition (2.3), i.e.
we have
in fact
Put
which is equivalent to
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
Hence, since \(0<1-\varepsilon <1\),
Similarly, put
then
Hence
Therefore we have (2.6):
where
Finally, by the assumption (2.4):
and (2.6), we get \(f \in G\psi [\nu ]\), i.e.
\(\square \)
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
implies
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.
Consider the following conditions:
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
(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,
By (3.2) we have
which implies
Following,
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
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
Arguing as above we get
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
Therefore
Remark 3.2
Let \(\nu (\cdot )\) and \(\psi (\cdot )\) as in Theorem 3.1.
The condition (3.1) in (5):
is satisfied, for example, for the generating function of the form
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
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
but the inverse conclusion is not true.
Data Availability
Not applicable.
References
Ahmed, I., Fiorenza, A., Formica, M.R.: Interpolation of generalized gamma spaces in a critical case. J. Fourier Anal. Appl. 28, 54 (2022). https://doi.org/10.1007/s00041-022-09947-1
Ahmed, I., Fiorenza, A., Formica, M.R., Gogatishvili, A., Rakotoson, J.M.: Some new results related to Lorentz G-Gamma spaces and interpolation. J. Math. Anal. Appl. 483(2), 123623 (2020)
Ahmed, I., Fiorenza, A., Hafeez, A.: Some interpolation formulae for grand and small lorentz spaces. Mediterr. J. Math. 17(57), 1 (2020). https://doi.org/10.1007/s00009-020-1495-7
Anatriello, G., Fiorenza, A.: Fully measurable grand Lebesgue spaces. J. Math. Anal. Appl. 422(2), 783–797 (2015)
Anatriello, G., Formica, M.R.: Weighted fully measurable grand Lebesgue spaces and the maximal theorem. Ric. Mat. 65(1), 221–233 (2016)
Anatriello, G., Formica, M.R., Giova, R.: Fully measurable small Lebesgue spaces. J. Math. Anal. Appl. 447(1), 550–563 (2017)
Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, New York (1988)
Buldygin, V.V., Kozachenko, Yu. V.: Metric characterization of random variables and random processes. Transl. Math. Monographs 188, American Mathematical Soc. (2000)
Capone, C., Formica, M.R., Giova, R.: Grand lebesgue spaces with respect to measurable functions. Nonlinear Anal. 85, 125–131 (2013)
Carro, M. J., Raposo, J. A., Soria, J.: Recent Developments in the Theory of Lorentz Spaces and Weighted Inequalities. Mem. Am. Math. Soc., vol. 187, Providence (2007)
Cwikel, M.: The dual of weak \(L^{p}\). Ann. Inst. Fourier 25, 81–126 (1975)
Edmunds, D.E., Opic, B.: Equivalent quasi-norms on lorentz spaces. Proc. Amer. Math. Soc. 131(3), 745–754 (2003)
Ermakov, S.V., Ostrovsky, E.I.: Continuity Conditions, Exponential Estimates, and the Central Limit Theorem for Random Fields. VINITY, Moscow (1986). ((in Russian))
Fernández-Martínez, P., Signes, T.: Limit cases of reiteration theorems. Math. Nachr. 288(1), 25–47 (2015)
Fernández-Martínez, P., Signes, T.: Compactness results for a class of limiting interpolation methods. Mediterr. J. Math. 13(5), 2959–2979 (2016)
Fiorenza, A.: Duality and reflexivity in grand Lebesgue spaces. Collect. Math. 51(2), 131–148 (2000)
Fiorenza, A., Formica, M.R.: On the factor opposing the lebesgue norm in generalized grand lebesgue spaces. RM 76(2), 74 (2021). https://doi.org/10.1007/s00025-021-01375-9
Fiorenza, A., Formica, M.R., Gogatishvili, A.: On grand and small Lebesgue and Sobolev spaces and some applications to PDE’s. Differ. Equ. Appl. 10(1), 21–46 (2018)
Fiorenza, A., Formica, M.R., Gogatishvili, A., Kopaliani, T., Rakotoson, J.M.: Characterization of interpolation between grand, small or classical Lebesgue spaces. Nonlinear Anal. 177, 422–453 (2018)
Fiorenza, A., Formica, M.R., Rakotoson, J.M.: Pointwise estimates for \(G\Gamma \)-functions and applications. Diff. Int. Equ. 30(11–12), 809–824 (2017)
Fiorenza, A., Gupta, B., Jain, P.: The maximal theorem for weighted grand Lebesgue spaces. Studia Math. 188(2), 123–133 (2008)
Fiorenza, A., Karadzhov, G.E.: Grand and small Lebesgue spaces and their analogs. Z. Anal. Anwend. 23(4), 657–681 (2004)
Formica, M.R., Giova, R.: Boyd indices in generalized grand Lebesgue spaces and applications. Mediterr. J. Math. 12(3), 987–995 (2015)
Formica, M.R., Kozachenko, Y.V., Ostrovsky, E., Sirota, L.: Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables. Lith. Math. J. 60(3), 330–358 (2020)
Formica, M.R., Ostrovsky, E., Sirota, L.: Grand Lebesgue spaces are really Banach algebras relative to the convolution on unimodular locally compact groups equipped with Haar measure. Math. Nachr. 294(9), 1702–1714 (2021). https://doi.org/10.1002/mana.201900181
Grafakos, L.: Classical Fourier Analysis, Graduate Texts in Mathematics, 249, 2nd edn. Springer, New York (2008)
Greco, L., Iwaniec, T., Sbordone, C.: Inverting the \(p\)-harmonic operator. Manuscripta Math. 92(2), 249–258 (1997)
Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Rational Mech. Anal. 119(2), 129–143 (1992)
Iwaniec, T., Koskela, P., Onninen, J.: Mappings of finite distortion: monotonicity and continuity. Invent. Math. 144(3), 507–531 (2001)
Jain, P., Kumari, S.: On grand Lorentz spaces and the maximal operator. Georgian Math. J. 19(2), 235–246 (2012)
Jain, P., Molchanova, A., Singh, M., Vodopyanov, S.: On grand Sobolev spaces and pointwise description of Banach function spaces. Nonlinear Anal. 202, 112100, 17 (2021)
Kokilashvili, V., Meski, A.: Extrapolation in weighted classical and grand Lorentz spaces. Application to the boundedness of integral operators. Banach J. Math. Anal. 14(3), 1111–1142 (2020)
Kozachenko, Yu.V., Ostrovsky, E.I.: Banach Spaces of random variables of sub-Gaussian type. Teor. Veroyatn. Mat. Stat. Kiev 32(134), 42–53 (1985)
Kozachenko, Yu.V., Mlavets, YuYu., Yurchenko, N.V.: Weak convergence of stochastic processes from spaces \(F_\psi (\Omega )\). Stat. Optim. Inform. Comput. 6(2), 266–277 (2018)
Kozachenko, Yu. V., Ostrovsky, E., Sirota, L.: Relations between exponential tails, moments and moment generating functions for random variables and vectors. arXiv:1701.01901v1 [math.FA] 8 (2017)
Kozachenko, Yu. V., Ostrovsky, E., Sirota, L.: Equivalence between tails, Grand Lebesgue spaces and Orlicz norms for random variables without Cramer’s condition, arXiv: 1710.05260v1 [math.PR] 15 (2017)
Liflyand, E., Ostrovsky, E., Sirota, L.: Structural properties of bilateral grand Lebesque spaces. Turk. J. Math. 34(2), 207–219 (2010)
Lorentz, G.G.: Some new functional spaces. Ann. Math. 51(2), 37–55 (1950)
Lorentz, G.G.: On the theory of spaces \(\Lambda \). Pacific J. Math. 1(3), 411–429 (1951)
Ostrovsky, E.I.: Exponential Estimations for Random Fields, (Russian). Moscow - Obninsk, OINPE (1999)
Ostrovsky, E.: Exponential estimate in the Law of Iterated logarithm in Banach space. Math. Notes 56(5–6), 1165–1171 (1994)
Ostrovsky, E.I., Sirota, L.: Moment Banach spaces: theory and applications. HIAT J. Sci. Eng. C 4(1–2), 233–262 (2007)
Ostrovsky, E., Sirota, L.: Fundamental function for Grand Lebesgue Spaces, arXiv:1509.03644v1 [math.FA] 11 (2015)
Ostrovsky, E., Sirota, L.: Boundedness of operators in bilateral Grand Lebesgue Spaces, with exact and weakly exact constant calculation. arXiv:1104.2963v1 [math.FA] 15 (2011)
Rafeiro, H., Samko, S., Umarkhadzhiev, S.: Grand Lebesgue sequence spaces. Georgian Math. J. 25(2), 291–302 (2018)
Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker Inc, New York (1991)
Rockafellar, R.T.: Convex Analysis, Princeton Mathematical Series, No. 28 Princeton University Press, Princeton. (1970)
Samko, S.G., Umarkhadzhiev, S.M.: On Iwaniec-Sbordone spaces on sets which may have infinite measure. Azerb. J. Math. 1(1), 67–84 (2011)
Samko, S.G., Umarkhadzhiev, S.M.: On Iwaniec-Sbordone spaces on sets which may have infinite measure: addendum. Azerb. J. Math 1(2), 143–144 (2011)
Samko, S.G., Umarkhadzhiev, S.M.: On grand Lebesgue spaces on sets of infinite measure. Math. Nachr. 290(5–6), 913–919 (2017)
Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton mathematical series, 43. Monographs in harmonic analysis, III. Princeton University Press, Princeton (1993)
Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)
Unal, C., Aydin, I.: Inclusion theorems for Grand Lorentz Spaces. arXiv:1909.07743v1 [math.FA] 17 (2019)
Zeren, Y., Ismailov, M., Sirin, F.: On basicity of the system of eigenfunctions of one discontinuous spectral problem for second order differential equation for grand-Lebesgue space. Turk. J. Math. 44(5), 1595–1611 (2020)
Acknowledgements
M.R. Formica is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and member of the UMI group “Teoria dell’Approssimazione e Applicazioni (T.A.A.)”and is partially supported by the INdAM-GNAMPA project, Risultati di regolarità per PDEs in spazi di funzione non-standard, codice CUP_E53C22001930001 and partially supported by University of Naples “Parthenope”, Dept. of Economic and Legal Studies, project CoRNDiS, DM MUR 737/2021, CUP I55F21003620001.
Funding
Open access funding provided by Università Parthenope di Napoli within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Contributions
All authors contribute equally to the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The Authors declare that they have no conflict of interest.
Consent to participate
All authors agree with the content and give explicit consent to submit the present article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Formica, M.R., Ostrovsky, E. & Sirota, L. Connection Between Weighted Tail, Orlicz, Grand Lorentz And Grand Lebesgue Norms. Results Math 79, 103 (2024). https://doi.org/10.1007/s00025-024-02136-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-024-02136-0
Keywords
- Lebesgue-Riesz spaces
- lorentz spaces Lorentz
- orlicz spaces Orlicz
- grand Lorentz and Grand Lebesgue spaces
- tail function
- slowly varying at infinity function
- diffuse measure
- young-Fenchel transform Young
- fenchel-Moreau theorem Fenchel