Maximal and other operators in exponential Orlicz and Grand Lebesgue Spaces

TL;DR

This paper provides precise non-asymptotic estimates for the norms of maximal and similar operators within exponential Orlicz and Grand Lebesgue Spaces, advancing the understanding of these operators in probabilistic functional analysis.

Contribution

It introduces exact non-asymptotic bounds for nonlinear operators in exponential Orlicz and Grand Lebesgue Spaces, utilizing the theory of GLS.

Findings

Exact estimates for maximal operator norms in exponential Orlicz spaces

Non-asymptotic bounds for operators in Grand Lebesgue Spaces

Application of GLS theory to operator norm estimation

Abstract

We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic rearrangement invariant norm: exponential Orlicz and Grand Lebesgue Spaces. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of measurable functions.

Equations

$$|f{|}_p=|f|L(p)=|f|L_p(X,\mu ):={\left[{\int }_X|f(x){|}^p\mu (dx)\right]}^{1/p},p\in [1,\infty ).$$

$$g(x)=Q(\vec{f})=Q(f_1,f_2,\dots ,f_N),N=1,2,\dots ,\infty ,$$

$$\psi (p):=\underset{i}{\sup }|f_i{|}_p,$$

$$g=\underset{i=1}{\overset{N}{\max }}{f}_i.$$

$$s_M(x)=\sum _{k=0}^M(g,{\varphi }_k){\varphi }_k(x)$$

$$g(x):=\underset{M\ge 1}{\sup }|s_M(x)|;$$

$$f^o(t):=\inf \{y,y>0:\mu \{s:|f(s)|>y\}\le t\},t\in (0,1]$$

$$f^{oo}(t):=\frac{1}{t}{\int }_0^t{f}^o(x)dx.$$

$$H(E):=\{f,f\in L_1:f^{oo}\in E\}\subset E$$

$$A=A_{\theta }=A_{\theta }[f](t)=f(\theta (t)),$$

$$H(L{\ln }^{n-1}L)=L{\ln }^{n}L,n\ge 1,$$

$$\exists b=\mathrm{const}\in (1,\infty ],\forall p\in [1,\mathrm{b})\Rightarrow {\psi }_{\mathrm{S}}(\mathrm{p}):=\underset{\mathrm{z}\in \mathrm{S}}{\sup }|\eta (\mathrm{z}){|}_{\mathrm{p}}<\infty .$$

$$\underset{z\in S}{\sup }||\eta (z)||G{\Psi }_S=1.$$

$${\psi }_{\Delta }(p):=|\Delta {|}_p,$$

$$T_f(y)\stackrel{def}{=}\max (\mu (f\ge y),\mu (f\le -y)),y\ge 0.$$

$$|f{|}_p^p={\int }_X|f{|}^p(x)\mu (dx)=p{\int }_0^{\infty }{y}^{p-1}{T}_f(y)dy$$

$$v(p)=v_{\psi }(p):=p\ln \psi (p).$$

$$f^{\ast }(u)\stackrel{def}{=}\underset{x\in \mathrm{Dom}(\mathrm{f})}{\sup }(xu-f(x)).$$

Full text

**Maximal and other operators in exponential

**

**Orlicz and Grand Lebesgue Spaces.

**

**Ostrovsky E., Sirota L.

**

Bar-Ilan University, department of Mathematic and Statistics, ISRAEL, 59200.

E-mails: eugostrovsky@list.ru,       sirota3@bezeqint.net

Abstract

We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic rearrangement invariant norm: exponential Orlicz and Grand Lebesgue Spaces.

We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of measurable functions.

Key words and phrases: Measure and probability, measurable functions, random variable (r.v.), operators, maximal functional operator, tail of distribution, contraction, Lebesgue-Riesz, Orlicz and Grand Lebesgue Spaces (GLS), martingales, Doob’s inequality and theorem, Dunford-Schwartz operator, generating function, Lyapunov’s inequality, Young-Orlicz function, conditional expectation, Young-Fenchel transform, rearangement invariant (r.i.) space.

AMS 2000 subject classification: Primary: 60E15, 60G42, 60G44; secondary: 60G40.

1. 1.  Definitions. Notations. Previous results. Statement of problem.

Let \ (X,{B},{\bf\mu})\ be a probability space: \ \mu(X)=1.\ We will denote by \ |f|_{p}=|f|L(p)\ the ordinary Lebesgue - Riesz \ L(p)\ norm of arbitrary measurable numerical valued function \ f:X\to R:\

[TABLE]

**Definition 1.1. ** The operator \ Q:L(p)\to L(p),\ p\in(1,\infty),\ not necessary to be linear, acting from any \ L(p)\ to one, is said to be of a type \ \lambda,\nu;\lambda,\nu=\rm const,\ \lambda\geq\nu\geq 0,\ write

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iff for some finite constant \ Z=Z[Q]\ and for certain interval \ p\in[1,b),\ b=\rm const\in(1,\infty)\

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or equvalently

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Note that the function \ f(\cdot)\ in the left-hand side of inequality (1.1) may be vector - function; moreover, one can consider the relation of the form

[TABLE]

such that

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for certain positive continuous function \ \psi=\psi(p),\ p\in[1,b).\ The concrete form of these function will be clarified below. For instance, one can choose the function \ \psi(\cdot)\ as a natural function for the family of the r.v. \ \{f_{i}\}:\

[TABLE]

if it is finite at last for one value \ p=b,\ b\in(1,\infty].\

This approach may be used for instance in the martingale theory, see [8], where \ \lambda=\nu=1,\ and

[TABLE]

There are many examples for such operators satisfying the estimate (1.2) (or (1.1)): Doob’s inequality for martingales [8], [13]; singular integral operators of Hardy-Littlewood type [30], [31], [24], Fourier integral operators [27],[30], pseudodifferential operators [33], theory of Sobolev spaces [1] etc.

Note that in the last two examples, as well as in [31], [24],   \ \lambda=\nu=1.\

Especially many examples are delivered to us the theory of the so-called maximal operators, see [1], [8], [27] etc. For instance, let \ \{\phi_{k}\},\ k=0,1,2,\ldots\ be ordinary complete orthonormal trigonometric system on the set \ [-\pi,\pi]\ equipped with normalized Lebesgue measure \ d\mu=dx/(2\pi)\ and let \ f\in L(p),\ p\in(1,\infty).\  Denote by

[TABLE]

the partial Fourier sum for \ f(\cdot)\ and

[TABLE]

then the inequality (1.2) holds true and herewith $\ \lambda=4;\ \nu=3;$ see [27].

Note that there are several examples in which \ \lambda>0,\ but \ \nu=0;\ see e.g. [32], [28].

Let us turn now our attention on the ergodic theory, see e.g. [3], [20], [21] etc. To be more concrete, suppose \ T=[0,1]\ with the classical Lebesgue measure \ \mu.\ Let \ f:T\to R\ be certain measurable function. Denote as ordinary

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and

[TABLE]

Introduce for arbitrary rearrangement invariant (r.i.) space \ E\ builded over \ (T,\mu)\ by \ H(E)\ another (complete) r.i. space as follows

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equipped with the norm

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Further, let an operator \ A\ be an \ L_{1}\ -\ L_{\infty}\ contraction, for instance,

[TABLE]

where \ \theta(\cdot)\ is an invertible ergodic measure preserving transformation of the set \ [0,1].\   Define the following maximal Dunford-Schwartz operator, not necessary to be linear

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M.Braverman in [3] proved that

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In particular, if \ E=L_{p}(T),\ 1<p<\infty,\ the estimate (1.5) takes the form

[TABLE]

So, the inequality (1.1) is satisfied for the operator \ Q=B_{A}\ again with the parameters \ \lambda=\nu=1.\

The lower bounds for the inequalities of the form (1.1), (1.2), i.e. the lower bounds for the operator \ Q\ with at the same parameters \ \lambda,\nu\ may be found, for instance, in [11], [15].

Notice [3], [9] that there are rearrangement invariant spaces \ E\ for which the norms \ ||\cdot||E\ and \ ||\cdot||H(E)\ are not equivalent. For example,

[TABLE]

see [3], proposition 1.2.

**We intend in this preprint to extend the inequality (1.1) (or (1.2)) into the wide class of another rearrangement invariant Banach functional spaces: exponential Orlicz spaces and into Grand Lebesgue Spaces. **

In detail, let \ Y_{1},\ Y_{2}\ be two rearrangement invariant (r.i.) Banach functional spaces over \ (X,{B},{\bf\mu}),\ in particular, Orlicz spaces or Grand Lebesgue ones. We set ourselves the goal to estimate of the correspondent operator norms

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2. 2. Grand Lebesgue Spaces (GLS).

Let \ (X,{B},{\bf\mu})\ be again the source probability space. Let also $\psi=\psi(p),\ p\in[1,b),\ b=\rm const\in(1,\infty]$ be certain bounded from below: $\inf\psi(p)>0$ continuous inside the semi - open interval $p\in[1,b)$ numerical valued function. We can and will suppose without loss of generality

[TABLE]

and $\ b=\sup\{p,\ \psi(p)<\infty\},$ so that $\rm supp\ \psi=[1,b)$ or $\rm supp\ \psi=[1,b].$ The set of all such a functions will be denoted by $\ \Psi(b)=\{\psi(\cdot)\};\ \Psi:=\Psi(\infty).$

By definition, the (Banach) Grand Lebesgue Space (GLS) $\ G\psi=G\psi(b)$ consists on all the real (or complex) numerical valued measurable functions (random variables, r.v.) \ f:X\to R\ defined on our probability space and having a finite norm

[TABLE]

The function \ \psi=\psi(p)\ is said to be *generating function * for this space.

Furthermore, let now $\eta=\eta(z),\ z\in S$ be arbitrary family of random variables defined on any set \ z\in S\ such that

[TABLE]

The function $p\to\psi_{S}(p)$ is named as a natural function for the family of random variables $S.$ Obviously,

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The family \ S\ may consists on the unique r.v., say \ \Delta:\

[TABLE]

if of course the last function is finite for some value \ p=p_{0}>1.\

Note that the last condition is satisfied if for instance the r.v. \ \Delta\ satisfies the so-called Kramer’s condition; the inverse proposition is not true.

The generating \ \psi(\cdot)\ function in (1.2) may be introduced for instance as natural one for some famoly of a functions.

These spaces are Banach functional space, are complete, and rearrangement invariant in the classical sense, see [2], chapters 1, 2; and were investigated in particular in many works, see e.g. [4], [5], [6], [14], [16], [18], [22], chapters 1,2; [23], [24] etc. We refer here some used in the sequel facts about these spaces and supplement more.

The so-called tail function \ T_{f}(y),\ y\geq 0\ for arbitrary (measurable) numerical valued function \ f\ is defined as usually

[TABLE]

It is known that

[TABLE]

and if $\ f\in G\psi,\ f\neq 0,$ then

[TABLE]

where

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Here and in the sequel the operator (non - linear) \ f\to f^{*}\ will denote the famous Young-Fenchel transform

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Conversely, the last inequality may be reversed in the following version: if

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and if the auxiliary function $\ v(p)=v_{\psi}(p)$ is positive, finite for all the values $\ p\in[1,\infty),$ continuous, convex and such that

[TABLE]

then \ \zeta\in G(\psi)\ and besides $\ ||\zeta||\leq C(\psi)\cdot K.$

Let us consider the so-called exponential Orlicz space $L(M)$ builded over source probability space with correspondent Young-Orlicz function

[TABLE]

The exponentiality implies in particular that the Orlicz space \ L(M)\ is not separable as long as the correspondent Young-Orlicz function \ M(y)=M[\psi](y)\ does not satisfy the \ \Delta_{2}\ condition.

The Orlicz \ ||\cdot||L(M)=||\cdot||L(M[\psi](\cdot))\ and \ ||\cdot||G\psi\ norms are quite equivalent:

[TABLE]

[TABLE]

Furthermore, let now $\eta=\eta(z),\ z\in W$ be arbitrary family of measurable functions (random variables) defined on any set \ W\ such that

[TABLE]

The function $p\to\psi_{W}(p)$ is named as a natural function for the family of random variables $W.$ Obviously,

[TABLE]

The family \ W\ may consists on the unique r.v., say \ \Delta:\

[TABLE]

if of course the last function is finite for some value \ p=p_{0}>1.\

Note that the last condition is satisfied if for instance the r.v. \ \zeta\ satisfies the so-called Kramer’s condition; the inverse proposition is not true.

Example 2.0.  Let us consider also the so - called degenerate \ \Psi\ -\ function \ \psi_{(r)}(p),\ where $r=\rm const\in[1,\infty):$

[TABLE]

so that the corresponent value $b=b(r)$ is equal to $r.$ One can extrapolate formally this function onto the whole semi-axis $R^{1}_{+}:$

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The classical Lebesgue-Riesz $L_{r}$ norm for the r.v. $\eta$ is quite equal to the GLS norm $||\eta||G\psi_{(r)}:$

[TABLE]

Thus, the ordinary Lebesgue-Riesz spaces are particular, more precisely, extremal cases of the Grand-Lebesgue ones.

Example 2.1. For instance, let $\psi$ function has a form

[TABLE]

The function \ f:X\to R\ belongs to the space \ G\psi_{m}:\

[TABLE]

if and only if the correspondent tail estimate is follow:

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The correspondent Young-Orlicz function for the space \ G\psi_{m}\ has a form

[TABLE]

There holds for arbitrary function \ f\

[TABLE]

if of course as a capasity of the value \ V=V(m)\ we understand its minimal positive value from the relation (2.7).

The case \ m=2\ correspondent to the so-called subgaussian case, i.e. when

[TABLE]

It is presumes as a rule in addition that the function \ f(\cdot)\ has a mean zero: \ \int_{X}f(x)\ \mu(dx)=0.\ More examples may be found in [4], [16], [22].

We bring a more general example, see [17]. Let \ m=\rm const>1\ and define \ q=m^{\prime}=m/(m-1).\ Let also \ L=L(y),\ y>0\ be positive continuous differentiable *slowly varying * at infinity function such that

[TABLE]

Introduce a following \ \psi\ -\ function

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and a correspondent exponential tail function

[TABLE]

The following implication holds true:

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A particular cases: $\ L(y)=\ln^{r}(y+e),\ r=\rm const,\ y\geq 0;$ then the correspondent generating functions have a form

[TABLE]

and correspondingly the tail function

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Example 2.2. Bounded support of generating function.

Introduce the following tail function

[TABLE]

where as before \ L=L(x),\ x\geq 1\ is positive continuous slowly varying function as \ x\to\infty,\ and

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Introduce also the following (correspondent!) \ \Psi(b)\ function

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Let the measurable function \ f(\cdot)\ be such that

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then

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or equivalently

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Conversely, if the estimate (2.14) holds true, then

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or equally

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Notice that there is a logarithmic “gap” as \ y\to\infty\ between the estimations (2.15) and (2.16). Wherein all the estimates (2.14) and (2.16) are non - improvable, see [17], [18], [24].

Remark 2.1.  These GLS spaces are used for obtaining of an exponential estimates for sums of independent random variables and fields, estimations for non-linear functionals from random fields, theory of Fourier series and transform, theory of operators etc., see e.g. [4], [14], [18], [22], sections 1.6, 2.1 - 2.5.

3. 3. Main result. The case of equal powers.

We consider in this section the case when in the relations (1.1) - (1.2) \ \nu=\lambda=\rm const>0,\ i.e.

[TABLE]

This relation holds true if for example in the relation (1.1) \ f\in\ G\psi;\ one can assume without loss of generality for simplicity \ ||f||G\psi=1,\ Z=1,\ so that \ |f|_{p}\leq\psi(p),\ 1\leq p<b.\

Let us introduce some auxiliary constructions. Let the function \ \psi=\psi(p),\ \psi\in\Psi(b),\ b=\rm const\in(1,\infty]\ be a given. Let also \ q\ be some fixed number inside the set \ (1,b):\ 1<q<b.\ Suppose the (measurable) function \ g=g(x)\ satisfies the inequality (3.1). We apply the Lyapunov’s inequality: \ p\in[1,q]\Rightarrow|g|_{p}\leq|g|_{q},\ hence

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We retain the value of the function \ \psi(\cdot)\ on the additional set:

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Let us introduce the following \ \psi\ -\ function $\ \tilde{\psi}(p)=$

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so that

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Here and further \ I(p\in A)\ denotes the indicator function of the set $A.$

So, we have eliminated the possible singularity at the point \ p\to 1+0.\

Let us prove now that

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We conclude taking into account the restriction \ \psi(p)\geq 1\

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[TABLE]

Further, let us denote

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then \ K_{\lambda}[\psi,b]\in[1,\infty);\ and we derive the following estimate

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We get due to proper choice of the parameter \ q:\

Proposition 3.1. We propose under formulated above notations and conditions, in particular, condition (3.1)

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One can give a very simple upper estimate for the value \ K_{\lambda}[\psi,b];\ indeed, we choose in (3.6) \ q=2\ in the case when \ b>2\ and \ q=(b+1)/2\ if \ b\in(1,2];\ we get

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As a slight consequence: if \ b<\infty,\ then

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where \ C(b,\lambda,\psi)\ is continuous bounded function relative the variable \ b\ in arbitrary finite segment $\ 1<b\leq V,\ V=\rm const<\infty.$

Example 3.a. Let

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We obtain after simple calculations

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[TABLE]

In particular,

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Note by the way \ \forall\lambda>0\ \Rightarrow\lim_{m\to\infty}K_{m}(\lambda)=1.\

Example 3.b. Let $\ b=\rm const>1;\ \beta=\rm const>0.$ Define the following tail function

[TABLE]

and the following \ \Psi(b)\ function with bounded support

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The tail inequality of the form

[TABLE]

entails the inclusion \ \eta\in G\psi[b,\beta].\ The inverse conclusion is not true.

We find after come computations

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Example 3.c. Let now \ \psi(p)=\psi_{(r)}(p),\ r=\rm const>1.\ It is easily to calculate

[TABLE]

Let us return to the theory of operators, see (1.1), (1.2). Namely, assume the operator \ Q\ satisfies the inequality (1.1) or more generally (1.2). It follows immediately from proposition (3.1) the following statement.

Theorem 3.1. Suppose the function \ f(\cdot)\ belongs to the space \ G\psi\ for some generating function \ \psi\ from the set \ \Psi(b),\ 1<b\leq\infty.\ Our statement: the function \ g=Q[f]\ from the relations (1.1) (or (1.2)) belongs to at the same Grand Lebesgue Space \ G\psi,\ or equivalently to the correspondent exponential Orlicz space $\ L\left(M_{\psi}\right):$

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or equally in the terms of exponential Orlicz spaces

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Remark 3.1. The statement of theorem (3.1) may be reformulated as follows. Under at the same conditions: \ f\in G\psi\ etc.

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or equally

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Remark 3.2. Note that the considered here Young-Orlicz function \ M_{\psi}(y)\ does not satisfy the \ \Delta_{2}\ condition, in contradiction to the considered ones in the book [19], section 12.

Example 3.1. Suppose the function \ f(\cdot)\ from the estimate (1.1) belongs to the space $\ G\psi_{m},\ m=\rm const>0:$

[TABLE]

or equivalently

[TABLE]

Then there exists a positive finite constant \ C_{3}=C_{3}(m,\lambda)\ for which

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or equivalently

[TABLE]

More generally, let \ L=L(y),\ y>0\ be the positive continuous differentiable *slowly varying * at infinity function such that

[TABLE]

i.e. as in the example 2.1. Recall the following notation for \ \psi\ -\ function

[TABLE]

and the correspondent exponential tail function

[TABLE]

where \ m=\rm const>1,\ q=m/(m-1).\

Suppose the function \ f(\cdot)\ from the estimate (1.1) belongs to the space $\ G\psi_{m,L},\ m=\rm const>0:$

[TABLE]

or equivalently

[TABLE]

Then there exists a positive constant \ C_{3}=C_{3}(m,L,\lambda)\ for which

[TABLE]

or equivalently

[TABLE]

Example 3.2. The case of bounded support.

This case is more complicated. Recall the following notation for tail function

[TABLE]

where as before \ L=L(x),\ x\geq 1\ is the positive continuous slowly varying function as \ x\to\infty,\ and let as before

[TABLE]

and recall also notation for the following correspondent \ \Psi(b)\ function

[TABLE]

Let the source (measurable) function \ f(\cdot)\ be such that

[TABLE]

then also ||g||\in G\psi^{<b,\gamma,L>}\ and moreover

[TABLE]

But if we assume the following tail restriction on the function \ f\

[TABLE]

then we conclude only

[TABLE]

or equally

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*Open question:  * what is the ultimate value instead \ ``\gamma+1^{\prime\prime}\ in the last estimate?

4. 4. Main result. The case of different powers.

Let as before some function \ \psi=\psi(p),\ p\in[1,b),\ b=\rm const\in(1,\infty]\ from the set \ \Psi(b)\ be a given.  Suppose in this section that in the inequalities (1.1) or (1.2) \ f\in G\psi,\ ||f||G\psi<\infty,\ \lambda>\nu\geq 0,\ and denote \ \Delta=\lambda-\nu;\ (\Delta>0),\

[TABLE]

Obviously, \ \zeta(\cdot)\in\Psi(b).\

Let for beginning \ ||f||G\psi=1,\ then \ |f|_{p}\leq\psi(p),\ p\in[1,b).\ We deduce from the inequality (1.1) taking into account the estimate \ |f|_{p}\leq\psi(p),\ p\in[1,b)\ alike the foregoing section denoting \ g=Q[f]\

[TABLE]

It follows immediately from proposition (3.1) or theorem 3.1

[TABLE]

We proved in fact the following result.

Theorem 4.1. Suppose as above that the function \ f(\cdot)\ belongs to the space \ G\psi\ for some generating function \ \psi\ from the set \ \Psi(b),\ 1<b\leq\infty.\ Let in (1.1) \ \lambda>\nu\geq 0.\ Our statement: the function \ g=Q[f]\ from the relations (1.1) belongs to the other certain Grand Lebesgue Space \ G\zeta,\ or equivalently to the correspondent exponential Orlicz space $\ L\left(M_{\zeta}\right):$

[TABLE]

or equally in the terms of exponential Orlicz spaces

[TABLE]

Remark 4.1. In the case \ b<\infty\ the estimate (4.3) may be simplified as follows. As long as in this case \ p^{\Delta}\ \psi(p)\ \leq b^{\Delta}\ \psi(p),\ we conclude that the operator \ Q\ acts from the space \ G\psi\ into at the same space:

[TABLE]

5. 5. Convergence in the Grand Lebesgue and non-separable Orlicz spaces.

Let us consider here the sequence of the form

[TABLE]

such that for some non - negative constants \ \lambda,\nu;\ \lambda\geq\nu\

[TABLE]

for certain positive continuous function $\ \psi=\psi(p),\ p\in[1,b),\ b=\rm const\in(1,\infty]$ from the set \ \Psi(b).\

It follows from theorem 4.1 that

[TABLE]

where in the case \ \lambda=\nu\ \Rightarrow\zeta(p)=\psi(p).\

*We suppose in addition to (5.1) (or following (5.2) ) that the sequence * \ \{g_{n}(\cdot)\}\ *converges in all the norms * \ L_{p}(X,\mu),\ p\in[1,b);\

[TABLE]

such that

[TABLE]

Our claim in this section is investigation under formulated before condition the problem of convergence \ g_{n}\to g_{\infty}\ in more strong norms, concrete: in the CLS sense or correspondingly in Orlicz spaces norms.

The simplest example of (5.2)-(5.3a), (5.3b) give us the theory of martingales. It makes sense to dwell on this in more detail.

This approach may be used for instance in the martingale theory, see [8], where \ \lambda=\nu=1,\ and

[TABLE]

where \ \{f_{i}\}\ is a centered martingale (or semi-martingale) sequence relative certain filtration \ \{F_{i}\}:\

[TABLE]

if of course the right-hand side is finite.

J. Neveu proved in [19], pp. 209-220 that if the Orlicz space \ L(M)\ builded over our probability space with correspondent Young-Orlicz function \ M(\cdot)\ satisfying the \ \Delta_{2}\ condition, or equvalently if the space \ L(M)\ is separable,

[TABLE]

and

[TABLE]

then there esists almost ewerywhere a limit

[TABLE]

and the convergence in (4.2) take place also in the \ L(M)\ norm:

[TABLE]

We must first of all recall some definitions and facts about comparison of GLS from an article [24]. Let \ \psi,\nu\ be two functions from the set \ G\psi(b),\ b\in(1,\infty].\ We will write \ \psi<<\nu,\ or equally \ \nu>>\psi,\ iff

[TABLE]

[TABLE]

There exists an equivalent version (and notion) for Young-Orlicz function, see [26], chapters 2,3.

Theorem 4.1. Assume the formulated above notations, conditions (5.3a), (5.3b) remains true. Our statement: for arbitrary \ \Psi(b)\ function \ \tau=\tau(p),\ 1\leq p<b\ such that \ \tau<<\zeta\

[TABLE]

i.e. the sequence \ f_{n}\ converges not only almost surely but also in arbitrary \ G\tau\ norm for which \ \tau<<\zeta.\

Proof is very simple.  The needed convergense \ f_{n},\ n\to\infty\ in the \ G\tau\ norm follows immediately from one of the main results of the article [24], p. 238.

6. 6. Concluding remarks.

A. One can consider a more general case as in (1.1), (1.2):

[TABLE]

or equally $\ g(x):=Q[f](x),$

[TABLE]

where \ W=W(p),\ p\in(1,b)\ is any measurable function, not necessary to be continuous or bounded.

Indeed, let as above \ q\ be arbitrary number from the open interval \ (1,b):\ q\in(1,b).\ We have using again the Lyapunov’s inequality

[TABLE]

following

[TABLE]

Thus, if we denote \ \upsilon(p)=\upsilon[W,\psi](p):=\

[TABLE]

**Proposition 6.1.  **

[TABLE]

B.   In the case of martingales the condition of almost surely convergence follows from the boundedness of its moment: \ \sup_{n}|f_{n}|_{p}<\infty,\ \exists p\geq 1,\ by virtue of the famous theorem of J.Doob.

C.   Lower bounds for the norm of considered operators.

Assume in addition to the estimates (1.1), or (1.2), (3.1), that

[TABLE]

The inequality (6.3) is true for example for every maximal operators, in the J.Doob’s inequality for martingales etc.

Suppose that the function \ \psi(\cdot)\ is a natural function for appropriate function \ f_{0}:X\to R:\ \psi(p)=|f_{0}|_{p},\ such that \ \forall p<b\ \Rightarrow\psi(p)<\infty.\ Then the relation (6.4) takes the form

[TABLE]

hence

[TABLE]

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