A Remark on the Inequalities of Bernstein - Markov Type in Exponential % Orlicz and Lorentz Spaces.
E.Ostrovsky
% We prove in this article the generalizations on the exponential Orlicz and Lorentz % spaces inequalities of Bernstein - Markov type for algebraic polynomials and rational % functions. %
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\begin{document}
%\begin{frontmatter}
%\title{A Remark on the  Inequalities of Bernstein - Markov Type in Exponential 
% Orlicz and Lorentz Spaces.}
%\author[Ostrovsky]{E.Ostrovsky}
%\address[Ostrovsky]{Department of Mathematic and Computer Science, 84105, Ben - Gurion 
%University of the Negev, Beer - Sheva, Ben Gurion street, 2; P.O. BOX 61; Israel; \\
% e - mail: galaostr@cs.bgu.ac.il; \ \ galo@list.ru}

%\begin{abstract} 
% We prove in this article the generalizations on the exponential Orlicz and Lorentz
% spaces  inequalities of Bernstein - Markov type for algebraic polynomials and rational 
% functions.
%\end{abstract}

%\begin{keyword}
% Polynomials, Exponential Orlicz Spaces, Markov's and Bernstein's Inequalities, 
% Equivalent Norms.
%\end{keyword}

%\end{frontmatter}

 \large

\begin{center}

{\sc A REMARK ON THE INEQUALITIES OF BERNSTEIN - MARKOV TYPE 
IN EXPONENTIAL ORLICZ AND LORENTZ SPACES.}\\
\vspace{1mm}
{\bf E.I.OSTROVSKY.} \footnote{ Department of Mathematics and Computer science, 
Ben - Gurion University, Beer - Sheva, Israel. E - mail: galaostr@cs.bgu.ac.il} \\
\vspace{1mm}

\end{center}

{\bf Abstract.} {\it We prove in this article the generalizations on the exponential 
Orlicz spaces Markov's - Bernstein's inequalities for  algebraic
polynomials and rational functions.}\\

\vspace{2mm}

{\bf Key Words.} Polynomials, Exponential Orlicz Spaces, Markov's and Bernstein's
 Inequalities, equivalent Norms.\\

\vspace{2mm}

{\bf AMS (MOS) subject classification. } Primary 34C11, 34B40, 47E05; Secondary 30xx, 26Bxx, 46xx.\\

\section {Introduction. Statement of problem.}\par
 
 It is well - known the application to the approximation theory Markov's (or Bernstein's) 
inequalities, for example, for inverse theorems of approximation theory; 
 see for instance ( [7], p. 208;  [13], [14])  etc. \par

 Let us denote by $ {\bf A} $ 
the set of all algebraical polynomials defined on the $ x \in X, \ X = [-1,1]; 
$ by $ {\bf T } $ the set of all 
trigonometrical polynomials on the set $ X = [0, 2 \pi], $ and by $ {\bf R} $ the set of all 
rational functions defined on the  $ X = [-1,1] $ without poles on $ X; $ for $ Q \in {\bf A} 
\cup {\bf T} $ the symbol $ \deg Q $ will denote the usually degree of $ Q; $ for 
irreducible fraction 
$ Q(x) = Q_1(x)/Q_2(x), \ Q_2(x) \ne 0, \ x \in [-1,1], Q_{1,2} \in {\bf A} $ we define 
$ \deg Q = \max(\deg Q_1, \deg Q_2). $  The space $ X $ is equipped usually {\it normed }
Lebesgue measure $ \mu(dx)= Cdx; \ \mu(X) = 1. $  \par

{\bf T}. For all $ Q \in {\bf T} $ 
 hold the famous generalized Bernstein's - Zygmund's inequalities: for all rearrangement 
invariant space $ S $ on the set $ X $
$$
||dQ/dx||S \le \deg Q \cdot ||Q||S, \eqno(1.1)
$$
([12], p. 36;  [20])  and for $ p \in (0,1) $
$$
||dQ/dx||_p \le \deg Q \cdot ||Q||_p,
$$
([7], p. 104),  where by definition for all values $ p > 0 $

$$
||f||_p \ = \ I^{1/p}(|f|^p), \ \ I(f) \stackrel{def}{=} \int_X f(x) \ \mu(dx).
$$

{\bf A}. For all $ Q \in {\bf A}, \ p > 0 $

$$
||dQ/dx||_p \le C^{1 +1/p} \cdot (\deg Q)^2 \cdot ||Q||_p,  
$$
where the symbol $ C $ will denote  (here and further) 
 some  {\it absolute } constant ([4], p.406). In the case $ p > 2 $ 

$$
||dQ/dx||_p \le K(p) \ (\deg Q)^2 \cdot || Q||_p, 
$$

$$
K(p) = \left[ 4 \pi (p+3)^2 /(p \sin(2 \pi/p)) \right]^{1/p}. \eqno(1.2)
$$
 (see [1]).
 Note than  $ p \ge 4 \ \Rightarrow K(p) \le K(4) = (49 \pi)^{1/4} \approx 3.52238228\ldots \ . $ \par
{\bf R}. Let now $ r = 1,2,\ldots, \ p \in (0,\infty), \gamma = \gamma(p,r) =
p/(pr+1), \ Q \in {\bf R}. $ Then
$$
||Q^{(r)}||_{\gamma} \le D_1(p, r) \cdot (\deg Q)^r \cdot ||Q||_p, \eqno(1.3)
$$
where at $ \ p \ge 4 \ \Rightarrow $
$$
D_1(p,r) \le D(r) \stackrel{def}{=} \exp(1/e) \cdot r! \cdot (4/3)^{r+1/4} \cdot (r+1/4)^{r+1/4}, 
$$
see [13], [14]; in [12], p. 300 - 301 was proved that this estimation (1.3) is exact in 
different senses. \par
 For the nonnegative General Algebraic Polynomials (GAP), i.e. for the functions of a view 
$$
Q(x) = \prod_{j=1}^m |x-z(j)|^{r(i)}, \ r(j) \ge 1, \  z(j) = a(j) + b(j)i, \ i = \sqrt{-1},
$$
$$
 \deg Q \stackrel{def}{=} \sum_{j=1}^m r(j), \ x \in [-1,1],
$$
in [4], p.402 - 406 was proved the inequality:

$$
||dQ/dx||_p \le C^{1+1/p}  \cdot (\deg Q)^2 \cdot ||Q||_p, \ p > 0. \eqno(1.4)
$$

 There are many generalizations of this inequalities on the Muntz polynomials [10], spline 
functions [15] and so one ([2], [3], [9], [10], [11], [19]). \par
{\bf Our goal is generalization of inequalities (1.2) and (1.3) on the exponential Orlicz 
spaces, i.e. when in the left and right sides of inequalities (1.2), (1.3) 
instead $ L_p $ norms are the norms on some Orlicz spaces. } \par

\section { Description of using Orlicz spaces.}\par
 We will consider here a so - called exponential Orlicz spaces on the  $ X $ with usually 
 {\it normed} Lebesgue measure $ \mu. $  Recall here  that  if the function 
$ N = N(u),  u \in R^1  $ is some $ N \ - $ Orlicz function (even, continuous, 
downwards convex, $ N(u) \ge 0, \ N(u) = 0 \ \Leftrightarrow u = 0, $
strictly increasing in the self - line  $ R^1_+ $), then the Orlicz norm $ ||f||L(N) $
in the Orlicz's space $ L(N) $
of a (measurable) function $ f: R \to R $ relative to the $ N \ - $ Orlicz function $ N = N(u) $
may be defined by the formula
$$
||f||L(N) = \inf \{l, \ l > 0, \ I(|f|/l) \le 1 \}.
$$
As a particular case, if $ N(u) = |u|^p, \ p = const \ge 1 $
we obtain the classical $ L_p = L_p(R) \ $ spaces with norm $ ||f||_p. $ More information about 
Orlicz spaces see in the books [16], [17]. \par
 A very important class of $ N \ - $ Orlicz functions are the so - called 
$ EOF \ = $ Exponential Orlicz Functions  and correspondent Exponential Orlicz Spaces $ EOS $ 
(like to the terminology of articles [6], [8]) will be considered.  We give here a more general
definition of this spaces.\par
 Let $ \varphi = \varphi(z), \ z \ge 0 $ be some continuous  function such that $ \varphi(z) = 0 \
\Leftrightarrow z = 0, $ and the function $ h(y) := \varphi(\exp y), 
\ y \in [-\infty, \infty) $ is strictly increasing, downward 
convex and 
$$
 \sum_{k \ge 3} \exp(h(k) - h(k+1)) < \infty. 
$$
 The set of all those function we will denote $ \Phi; \ \Phi = \{\varphi \}. $
Let us define the following $ N \ - $ Orlicz function $ N = N(u) = N(\varphi;u): $ at $ |u| \le C_1 $
$$
\ N(\varphi;u) = C_2 \ |u|, \ C_1, C_2 = C_{1,2}(\varphi(\cdot)) \in (0,\infty)  \eqno(2.1)
$$
and for $ |u| > C_1 $

$$
 N(\varphi; u) = \exp \varphi(|u|). \eqno(2.2)
$$
and will denote the correspondent Orlicz's space and norm $ B(\varphi) \stackrel{def}{=} 
L(N(\varphi, \cdot)), \\ ||\cdot||B(\varphi): $  \par
$$
||f||B(\varphi) = ||f||L(N(\varphi; \cdot)).
$$
 It is very simple to prove the existence for all $ \varphi \in \Phi $
the values $ C_1= C_1(\varphi), C_2= C_2(\varphi) $ such that $ N(\varphi;u) $ is some $ N \ - $ 
Orlicz's function.\par
 We denote also $ C_3 = C_3(\varphi) = \max(1,C_1, 1/C_2), $ 

$$
k_0=\max \{ 4+ \max(\log C_1,1), h_+^{*/}(1) \},
$$

$$
C_4(\varphi) = e^2 \left[ N(\varphi; \exp(k_0 -2)) + \sum_{k \ge k_0} \exp(h(k-1)- h(k)) \right].
$$


 Let us consider some  examples. Put $ \varphi(z) = \varphi_{m,r}(z) = z^m \  
\log^{-mr} [(\exp(m + |r|) + z], $
$ z \ge 0, \ m = const \in (0, \infty); $  
or $ \varphi(z) = \varphi_{\nu}(z) = \log^{1+ \nu}( 1 + z), z \ge 0, \ \nu = const > 0.  $ 
Then $  \varphi_{m,r}(\cdot) \in \Phi, \ \varphi_{\nu} (\cdot) \in \Phi. $ We will denote the 
norm in this spaces as
$$
 ||f||B(m,r) = ||f||B(\varphi_{m,r}(\cdot)), \ \ ||f||B(m) = ||f||B(m,0).
$$

\section{Main results.}

 We will prove that the direct generalization of inequality (1.2)
is true for exponential Orlicz spaces.  Denote for $ \varphi \in \Phi $
$$
W(\varphi;n) = \sup_{Q \in A, Q \ne 0} \frac{||dQ/dx||B(\varphi)}{||Q||B(\varphi)}, \eqno(3.1)
$$
where  $ "\sup" $ is calculated over all the algebraic polynomials $ Q; \ Q \ne 0, 
\deg Q = n. $ \\ 

{\bf Theorem 1.} {\it For all} $ \ \varphi(\cdot) \in \Phi $ {\it there exists } $ C_5(\varphi) 
\in (0,\infty) $ such that 
$$
C_5(\varphi) \ n^2 \le W(\varphi,n) \le n^2 \cdot K(4) \cdot \max(1, \psi(4))  \cdot C_{4} \cdot C_3.
\eqno(3.2)
$$
 
 Note than since $ \mu(X) = 1, $ the inequalities (3.2) hold for all the Orlicz's spaces with the 
$ N \ - $ functions which are equivalent to $ N(\varphi; u). $ \par
  In order to formulate an other result, we introduce some new notations.
 For  $ \varphi \in \Phi $  we denote
$$
\psi(p) = \psi(\varphi; p) = \exp(h^*(p)/p),
$$
 where  $ h^*(p) $  denotes the classical Young - Fenchel, or Legendre transform:

$$
h^*(p) = \sup_{y \in (-\infty, \infty) } (py - h(y)),
$$
 We  define for  $ r = 1,2,\ldots, \ \varphi \in \Phi $ a new quasinorm

$$
||f||V(\varphi;r) \stackrel{def}{=} \sup_{\beta \in (4/(4r+1), 1/r)} ||f||_{\beta}/
\psi(\beta/(1-r\beta))
$$
and the correspondent space of measurable  functions $ V(\varphi; r) $ with finite norm 
$ ||f||V(\varphi;r) < \infty. $ \\

{\bf Theorem 2. } $ \forall \varphi \in \Phi, \ \forall Q \in {\bf R} $

$$
 ||Q^{(r)}||V(\varphi;r) \ \le \ C_4(\varphi) \cdot D(r) \cdot (\deg Q)^r \cdot
||Q||B(\varphi). \eqno(3.3)
$$

\section{ Auxiliary result.}
  
 Let us introduce a {\it new } Banach space $ G( \varphi), \ \varphi \in \Phi, $ 
as a set of all measurable functions $ f:X \to R  $ with finite norm

$$
||f||G( \varphi) \stackrel{def}{=} \sup_{p \ge 1} |f|_p /\psi(p) < \infty.
$$

 Note than by virtue of Iensen - Lyapunov inequality 
$$
\sup_{p \ge 4} ||f||_p/\psi(p) \le ||f||G(\varphi) \le \max(1, \psi(4)) \ 
\sup_{p \ge 4} ||f||_p /\psi(p).
$$

{\bf Theorem 3.} 
 {\it We propose that the norms } $ || \cdot ||B( \varphi) $ {\it and } 
 $ || \cdot ||G(\varphi) $ {\it are equivalent: } 

$$
C_3^{-1} \ ||f||G( \varphi) \le ||f||B( \varphi) \le C_4 \ ||f||G( \varphi). \eqno(4.1)
$$

{\bf Proof of theorem 3.} 
 Assume at first $  ||f||B( \varphi) < \infty. $ Without loss of generality we 
can suppose  

$$
I (N(\varphi; |f|) = 1. 
$$
 Let us introduce the function

$$
g(p) = \sup_{z > 0} z^p / N( \varphi; z).
$$
 We have for all the values $ p \ge 1: $ 
$$
g(p) \le  \max \left[\max_{z \in (0, C_1] } C_2^{-1}  z^{p-1},  \ \sup_{z \ge C_1} z^p \ 
\exp (- \varphi(z)) \right]  \le
$$

$$
\max \left[ C_2^{-1} \max(C_1,1)^{p-1}, \ \exp( \sup_{z \ge C_1} (p \log z - \varphi(z))) \right] <
$$
$$
 \max \left[ C_2^{-1} \max(C_1,1)^{p-1}, \ \exp(\sup_{v \in (-\infty,\infty)} 
(p v - h(y))) \right] \le
$$

$$
 \max \left[C_3^{p}, \  \exp h^*(p) \right] \le C_3^p \exp h^*(p).
$$

 Following, for the values $ p \ge 1 $ we have: $ z \ge 0 \ \Rightarrow $

$$
z^p \le g(p) \ N(\varphi; z) \le C_3^p \ \psi^p(p) \ N(\varphi; z). 
$$

Therefore
$$
 |f|^p \le C_3^p \ \psi^p(p) \ N(\varphi; h^*(|f|)), \ \ 
|f|_p \le C_3( \varphi) \ \psi(p),
$$

$$
||f||G( \varphi) \le C_3( \varphi(\cdot)) < \infty.
$$

 Inverse, suppose 

$$
|f|_p^p  \le \exp \left(h^*(p) \right), \ p \ge 1.
$$
 We have by virtue of Chebyshev's inequality for the values $ w \ge e^2:$

$$
T(|f|,w) \stackrel{def}{=} \mu \ \{x: |f(x)| > w \} \le \exp \left(h^*(p) - p \log w \right),
$$
and after the minimization over $ p: $
$$
T(|f|,w) \le \exp \left( -\sup_{p \ge 1} (p \log w - h^*(p) \right) \le 
$$
$$
\exp \left(- \sup_p (p \log w - h^*(p) \right) =
$$ 

$$
\exp \left(-h^{**}(\log w) \right) = \exp \left( -h( \log w) \right)
$$
by virtue of theorem Fenchel - Moraux. \par
 We conclude for 
the value  $ \varepsilon = \exp(-2), $ choosing $ W(k) = \exp(k) $ and denoting

$$
U(k) = U(|f|,k) = \{x: \ W(k) \le |f(x)| < W(k+1) \}: 
$$

$$
I(\exp(N(\varphi; \varepsilon |f|))) =\int_{ \{x: |f(x)| \le \exp(k_0) \} } N(\varepsilon |f(x)|)
 \ d \mu \ +
$$

$$
\int_{ \{x: |f(x)| > \exp(k_0) \} } N(\varphi;\varepsilon |f(x)|) \ d \ \mu \le 
N(\varphi;\exp(k_0-2)) \ +
$$

$$ 
\sum_{k \ge k_0} \int_{U(k)}\exp(\varphi(\varepsilon |f(x)|) ) \ dx \le 
$$

$$
N(\varphi; \exp(k_0-2) +\sum_{k \ge k_0} \exp 
\left[\left( h(\varepsilon \  W(k+1) \right) \right] \ \cdot \ \left[ T(|f|, W(k)) \right] \le
$$

$$
N(\varphi; \exp(k_0-2)) + \sum_{k \ge k_0} \exp(h(k) - h(k+1)) =C_4 \ e^{-2} < \infty.
$$
This completes the proof of theorem 3.\par
 Note than this result is some generalization of [5], p. 309 - 314, [6], [8], [17], p. 305.\par
 For example, let $ N(u) = N_{m,r}(u)= N(\varphi_{m,r}(\cdot);u). \  $  
It follows from theorem 3 that 
$$
 ||f||L(B(m,r))  < \infty \ \Longleftrightarrow \ \sup_{p \ge 4} |f|_p \ \cdot 
 \left(p^{-1/m} \log^{m r} p \right) < \infty,
$$
 or equally

$$
\exists \varepsilon > 0, \ I(N(\varphi_{m,r}; \varepsilon |f|) < \infty \ \Longleftrightarrow
\ \sup_{p \ge 4} |f|_p \ \cdot \left( p^{-1/m} \ \log^{mr} p \right) < \infty.
$$
{\bf Notice.} Let us introduce the {\it weight Lorentz } norm:

$$
||f||^*_b G(\varphi) = \sup_{p \ge 1} ||f||_{p,b}/\psi(p),
$$
where $ ||f||_{p,b} $ is the Lorentz norm (more exactly, seminorm):

$$
||f||_{p,b} = \left[\int_0^{\infty} T^{p/b}(|f|,x) \ dx^b \right]^{1/b}, 
$$
$ p \in [1, \infty), \ b \in [1,\infty], $ where in the case $ b = +\infty $ 
$$ 
||f||_{p,\infty} = \sup_{x \ge 0} \left( x \ T^{1/p}(|f|,x) \right).
$$
 Using the embedding theorem for Lorentz spaces it is easy to prove as well as by 
proving of theorem 4 that all the following norms are equivalent
$$
||\cdot||B( \varphi) \ \sim \ ||\cdot|| G( \varphi) \ \sim \ ||\cdot||^*_b G(\varphi)
$$ 
 with constants does not depending on $ b. $ Therefore, it is easy to formulate theorems 
1,2 in the terms of those spaces.\par

\section{ Proof of the main results.}
{\bf Proof of theorem 1.} The low bound in (3.2) is attained, for instance, on the 
so - called Jacobi ultraspherical polynomials $ Q = P_n^{2,2}: $
$$
P_n^{2,2}(x) = \frac{(-1)^n (1-x^2)^{-2}}{2^n \ n!} \left( \frac{d}{dx} \right)^n 
\left[(1-x^2)^{n+2} \right]:
$$

$$
||(d/dx) \ P_n^{2,2}||_p \ge C \ n^2 \ ||P_n^{2,2}||_p, \ n = \deg P_n^{2,2}. \eqno(5.1)
$$

see [18], p. 66, 165;  [11]. The upper bound may be simple prove by virtue of theorem 3. 
Namely, suppose $ Q \in A, \ Q \ne 0, \ \deg Q = n \ge 2, \ ||Q||B(\varphi) = 1.$ Then 

$$
||Q||G(\varphi) \le C_3||Q||B(\varphi) = C_3 < \infty;
$$

$$
 \sup_{p \ge 4} \ \left( ||Q||_p / \psi(p) \right) \le C_3; \ \Rightarrow  ||Q||_p \le 
C_3 \psi(p) 
$$
for all the values $ p \ge 4.$ It follows from [1] that
$$
||dQ/dx||_p \le n^2 \cdot K(4) \cdot \psi(p) \cdot C_3. 
$$
 Therefore 
$$
||dQ/dx||G(\psi) \le n^2 \ K(4) \ \max(1, \psi(4)) \cdot C_3, 
$$
and finally

$$
||dQ/dx||B(\varphi) \le n^2 \cdot K(4) \cdot C_3 \cdot C_4 \cdot \max(1, \psi(4)) \cdot 
||Q||B(\varphi).
$$

{\bf Proof of theorem 2.} Let $ Q \in R, \ \deg Q = n, \ ||Q||B(\varphi) = 1. $ By virtue of 
theorem 3 we receive:

$$
||Q||G(\varphi) \le C_4; \ \Rightarrow ||Q||_p \le C_4 \ \psi(p), \ p \ge 4.
$$
 From (1.3) it follows
$$
||Q^{(r)}||_{\gamma} \le n^r \ D(r) \ C_4(\varphi) \ \psi(p).
$$
 Since $ p \ge 4, \ \gamma \in [4/(4r+1),1/r). $ After the substitution 
$ \beta = p/(pr+1) \in [4/(4r+1), 1/r), \ $ or $ p= \beta/(1-\beta \ r), $ we obtain
that for all $ \beta \in [4/(4r+1), 1/r) $
$$
||Q^{(r)}||_{\beta} \le n^r \ C_4(\varphi) \ D(r) \ \psi(\beta/(1-\beta \ r)),
$$
which is equivalent to the statement (3.3) of theorem 2. \par 

\section{Concluding Remark.} 
The spaces $ V(\varphi, r) $ are only Frech\'et spaces. Now we 
investigate  the connection between the norm $ || f ||V(\varphi_{m,0};r) $ 
in those spaces ant tail behavior of distribution $ T(|f|,u). $ At first assume that 
$ ||f||V(\varphi_{m,0},r) = 1 $  for some $ m > 0. $
then for all values $ \beta \in [4/(4r+1), 1/r) $
$$
||f||_{\beta} \le \left( \frac{\beta}{1-r \beta} \right)^{1/m}, 
\ \ I \left(|f|^{\beta} \right) \le \left( \frac{\beta}{1-r\beta}  \right)^{\beta/m}.
$$
We obtain using the Chebyshev's inequality for sufficiently large values $ u 
\ge 3: $

$$
T(|f|, u) \le u^{-\beta} \ \left( \frac{\beta}{1-r \beta} \right)^{\beta/m}.
$$
We conclude after the minimization over $ \beta: $
$$
T(|f|,u) \le C_9(m,r) \ u^{-1/r} \ \left[\log u \right]^{1/(mr)}, \ u \ge 3. \eqno(6.1)
$$
 Inverse, if inequality (6.1) holds, then 

$$
I \left( |f|^{\beta} \right) \le C_{11}(\beta,r,m) \int_3^{\infty} x^{\beta-1-1/r} \ 
(\log x)^{1/mr} \ dx \le 
$$

$$
C_{12} (1/r - \beta)^{-(mr+1)/(mr)}; \ \ ||f||_{\beta} \le C_{13} (1/r - \beta)^{(mr+1)/m};
$$

$$
||f||V(\varphi_{m/(mr+1),0}; \ r) \le C_{14}(m,r) < \infty.
$$
\vspace{2mm}
{\bf Aknowledgements}. I am very grateful to prof. V.Fonf, M.Lin (Ben Gurion University, 
Beer - Sheva, Israel) for useful support of these investigations. \par
 This investigation was partially supported by ISF(Israel Science Foundation), grant 
$ N^o $ 139/03.\par

\newpage

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%\newpage

%\begin{center}

%{\bf  APPLICATION FORM FOR PARTICIPANTS }\\
%\end{center}

%{\bf First name:} Eugene \\

%{\bf Last (family) name:} Ostrovsky \\

%{\bf Institution:} University Ben Gurion, Beer - Sheva city, Israel, Ben Gurion street, 2;
%building 58, Department of Mathematic and Computer Science, PO BOX 61. \\

%{\bf Location:} Israel, 56209, Rehovot city, Shkolnik street, H.5 Ap. 8; \\
%Beer - Sheva, Ben - Gurion University, Dept. of Mathematic and Computer Science.\\

%{\bf Phone numbers.} Home: (972) - (08) - 9 - 45 - 16 - 13, \  Work: (972) -
%(08) - 647 - 78 - 49. \\

%{\bf E - mail address:} galaostr@cs.bgu.ac.il \\

%{\bf Title of report:} Correct Solvability of Non - Linear Ordinary Differential Equations 
% in Orlicz Spaces.\\

%{\bf References:} ??? publications. \\


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