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EMBEDDING THEOREMS OF FUNCTION CLASSES,III
B.SIMONOV AND S.TIKHONOV
Abstract.
In this paper we obtain the necessary and su±cient con-
ditions for embedding results of di®erent function classes.The main
result is a criterion for embedding theorems for the so-called general-
ized Weyl-Nikol'skii class and the generalized Lipschitz class.To de-
¯ne the Weyl-Nikol'skii class,we use the concept of a (¸;¯)-derivative,
which is a generalization of the derivative in the sense of Weyl.As
corollaries,we give estimates of norms and moduli of smoothness of
transformed Fourier series.
1.Introduction
History of the question.One of the main problems of constructive ap-
proximation
1
is in ¯nding a relationship between di®erential properties of
a function and its structural or constructive characteristics.This topic
started to develop more than a century ago and in many cases the research
was conducted as follows;authors considered a given functional class and
by investigating the properties of its elements obtained embedding theorems
with other functional classes.We recommend the articles by A.Pinkus [Pi]
and V.V.Zhuk and G.I.Natanson [Zh-Na] for a historical review.
The following three classical results gave rise to development of new areas
within the approximation theory:
(A) f
(r)
2 Lip® ()E
n
(f) = O
µ
1
n
r+®

(0 < ® < 1;r 2 Z
+
);
(B) f
(r)
2 Lip® ()!
r+1
³
f;
1
n
´
= O
µ
1
n
r+®

(0 < ® < 1;r 2 Z
+
);
(C) f 2 Lip® =)
e
f 2 Lip® (0 < ® < 1):
2000 Mathematics Subject Classi¯cation.46E35,26A16,26A33,42A45.
Key words and phrases.Embedding theorems,Lipschitz class,Weyl-Nikol'skii class,
(¸;¯)-derivative,Moduli of smoothness of fractional order.
1
The concept became well known through Bersntein's paper [Be,S,4,V.2,p.295-300;
p.349-360].
1
2 B.SIMONOV AND S.TIKHONOV
Result (A) was proved
2
by D.Jackson (1911,[Ja]) in the necessity part,
and by S.Bernstein (1912,[Be,S,1],[Be,S,2]) and Ch.de la Vall¶ee-Poussin
(1919,[Va]) in the su±ciency part.
The theorems of this type are called direct and inverse theorems for
trigonometric approximation.Direct theorems for L
p
;1 · p · 1 (see the
review [Zh-Na]) are written as follows:
E
n
(f)
p
· C(k)!
k
³
f;
1
n +1
´
p
;k;n 2 N;(1)
E
n
(f)
p
·
C(k)
n
r
!
k
³
f
(r)
;
1
n +1
´
p
;k;n;r 2 N:(2)
Inverse theorems for L
p
;1 · p · 1(see the review [Zh-Na]):
!
k
³
f;
1
n +1
´
p
·
C(k)
(n +1)
k
n
X
º=0
(º +1)
k¡1
E
º
(f)
p
;k;n 2 N;(3)
!
k
³
f
(r)
;
1
n +1
´
p
· (4)
C(k)
Ã
1
(n +1)
k
n
X
º=0
(º +1)
k+r¡1
E
º
(f)
p
+
1
X
º=n+1
(º +1)
r¡1
E
º
(f)
p
!
;
k;n 2 N:
Here and further,the best trigonometric approximation E
n
(f)
p
and the
modulus of smoothness!
k
(f;±)
p
are de¯ned by
E
n
(f)
p
= min
³
kf ¡Tk
p
;T 2 T
n
´
;
T
n
= spanfcos mx;sinmx:jmj · ng
and
!
k
(f;±)
p
= sup
jhj·±
°
°
4
k
h
f(x)
°
°
p
;(5)
4
k
h
f(x) = 4
k¡1
h
(4
h
f(x)) and 4
h
f(x) = f(x +h) ¡f(x)
respectively.
Note that the theorems on existence of the r-th derivative of f from a
given space have been initiated by Bernstein [Be,S,1].He proved that the
condition
1
P
º=1
º
r¡1
E
º
(f)
1
< 1 implies f
(r)
2 C.Later,for L
p
(1 · p ·
1),the following results were obtained (see the review [Zh-Na] and the
2
See [Be,S,3] for a detailed review of the question before the 30-ths XX century.
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 3
paper by O.V.Besov [Be,O]).For convenience,we write these results in
terms of the Besov space B
r
p;µ
and the Sobolev space W
r
p
:
B
r
p;1
½ W
r
p
½ B
r
p;1
p = 1;1;
B
r
p;p
½ W
r
p
½ B
r
p;2
1 < p · 2;
B
r
p;2
½ W
r
p
½ B
r
p;p
2 · p < 1:
Result (B) was proved by A.Zygmund (1945,[Zy,1]).He was one of the
¯rst to use the modulus of smoothness concept of integer order introduced
by S.Bernstein in 1912 ([Be,S,1]).At present,the moduli of smoothness
properties are well-studied ([Jo],[Zh-Na]) and the result (B) follows from
inequalities (see [De-Lo,Chapters 2 and 6],[Jo-Sc]):
!
k+r
³
f;
1
n
´
p
·
C(k;r)
n
r
!
k
³
f
(r)
;
1
n
´
p
;k;r;n 2 N (6)
!
k
³
f
(r)
;
1
n
´
p
· C(k;r)
1
X
º=n+1
º
r¡1
!
k+r
³
f;
1
º
´
p
;k;r;n 2 N:(7)
Comparing the last two inequalities and inequalities (2) and (4) we see that
from (6) and (7),using (1) and (3),it is easy to get (2) and (4).
Result (C) was proved by I.I.Privalov (1919,[Pr]).The most complete
version of the inequality,from which embedding (C) follows,was obtained
by A.Zygmund ([Zy,1]) and N.K.Bary and S.B.Stechkin ([Ba-St]) and is
the following one (p = 1;1)
!
k
³
e
f
(r)
;
1
n
´
p
· C(k;r)
Ã
n
¡k
n
X
º=1
º
k+r¡1
!
k
³
f;
1
º
´
p
+
1
X
º=n+1
º
r¡1
!
k
³
f;
1
º
´
p
!
;
(8)
k;n 2 N;r 2 Z
+
:
Finally,we note the paper by G.H.Hardy and J.S.Littlewood [Ha-Li]
in which seemingly,for the ¯rst time,some problems were formulated and
solved in the same setting as in the present paper.For the historical aspects
of this approach see also [Og].
Embedding theorems for functional classes.The results (A) - (C) as well
as their generalizations mentioned above can be written as the embedding
theorems of the following functional classes:
W
r
p
=
n
f 2 L
p
:f
(r)
2 L
p
o
;
f
W
r
p
=
n
f 2 L
p
:
e
f
(r)
2 L
p
o
;
W
r
p
H
®
['] =
n
f 2 W
r
p
:!
®
³
f
(r)

´
p
= O['(±)]
o
;
4 B.SIMONOV AND S.TIKHONOV
f
W
r
p
H
®
[Á] =
½
f 2
f
W
r
p
:!
®
³
e
f
(r)

´
p
= O[Á(±)]
¾
;
W
r
p
E[»] =
n
f 2 W
r
p
:E
n
³
f
(r)
´
p
= O[»(1=n)]
o
:
We will study more general classes for which W
r
p
,
f
W
r
p
,W
r
p
H
®
['],
f
W
r
p
H
®
[Á],
W
r
E
p
[»] are particular cases.
Transformed Fourier series.Let L
p
= L
p
[0;2¼] (1 · p < 1) be a space of
2¼-periodic measurable functions for which jfj
p
is integrable,and L
1
´
C[0;2¼] be the space of 2¼-periodic continuous functions with kfk
1
=
maxfjf(x)j;0 · x · 2¼g:
Let summable function f(x) have the Fourier series
f(x) » ¾(f):=
a
0
(f)
2
+
1
X
º=1
(a
º
(f) cos ºx +b
º
(f) sinºx) ´
1
X
º=0
A
º
(f;x):
(9)
By the transformed Fourier series of (9) we mean the series
¾(f;¸;¯):=
1
X
º=1
¸
º
·
a
º
cos
µ
ºx +
¼¯
2

+b
º
sin
µ
ºx +
¼¯
2
¶¸
;
where ¯ 2 R and ¸ = f¸
n
g is a given sequence of positive numbers.
Studies of the transformed Fourier series are naturally related to the
problems of Fourier multipliers theory (see [Be-LÄo],[Be-Il-Ni],[Zy,2,Vol 1,
Chapter III]),summability methods (see [Bu-Ne,Chapter 1.2],[Zy,2,Vol
1,Chapter III]) and
3
so-called the fractional Sobolev classes or the Weyl
classes.
We call the class
W
¸;¯
p
=
n
f 2 L
p
:9 g 2 L
p
;g(x) » ¾(f;¸;¯)
o
the Weyl class.It is because ¸
n
= n
r
;r > 0 and ¯ = r the class W
¸;¯
p
co-
incides with the class W
r
p
,which is de¯ned in the term of fractional deriv-
atives f
(r)
in the Weyl sense ([Zy,2,Vol.2,Chapter XII]).In the case
¸
n
= n
r
;r > 0 and ¯ = r +1 the class W
¸;¯
p
coincides with the class
f
W
r
p
.
We call the function g(x) » ¾(f;¸;¯) the (¸;¯)-derivative of the function
f(x) and denote it by f
(¸;¯)
(x).
Generalized Weyl-Nikolskii class.In the de¯nition of this functional class we
use the modulus of smoothness concept!
®
(f;±)
p
of fractional
4
order of a
3
See also references to [Ba,2,x13,Chapter II].
4
The term"fractional"can be found in earlier papers ([Bu-Dy-GÄo-St] and [Ta]) which
used this de¯nition.As in the case of fractional derivatives,the positive number ® that
de¯nes the modulus order is not necessarily rational.
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 5
function f(x) 2 L
p
,i.e.,
!
®
(f;±)
p
= sup
jhj·±
k4
®
h
f(x)k
p
;
where
4
®
h
f(x) =
1
X
º=0
(¡1)
º
µ
®
º

f(x +(® ¡º)h);® > 0
is the ®-th di®erence
5
of a function f with step h at the point x.It is clear
that for ® 2 N this de¯nition is the same as (5).
Let ©
®
(® > 0) be the class of functions'(±),de¯ned and non-negative on
(0;¼] such that
1.'(±)!0 (±!0),
2.'(±) is non-decreasing,
3.±
¡®
'(±) is non-increasing.
For such ® > 0;'2 ©
®
and ¸ = f¸
n
g the generalized Weyl-Nikolskii class
is de¯ned similarly to the classes W
r
p
H
®
['] and
f
W
r
p
H
®
[Á]:
W
¸;¯
p
H
®
['] =
n
f 2 W
¸;¯
p
:!
®
³
f
(¸;¯)

´
p
= O['(±)];±!+0
o
:
It is clear that if ¸
n
= n
r
;r > 0 and ¯ = r,then W
¸;¯
p
H
®
['] ´ W
r
p
H
®
[']
and if ¸
n
= n
r
;r > 0 and ¯ = r +1,then W
¸;¯
p
H
®
['] ´
f
W
r
p
H
®
['].
In the case ¸
n
´ 1 and ¯ = 0 the class W
¸;¯
p
H
®
['] coincides with the
generalized Lipschitz class H
'
®
,i.e.,
H
p
®
['] =
n
f 2 L
p
:!
®
(f;±)
p
= O['(±)] ±!+0
o
:
In particular,
Lip(°;L
p
) ´ H
p
1

°
] =
n
f 2 L
p
:!
1
(f;±)
p
= O[±
°
] ±!+0
o
:
The problem setting and the structure of the paper.In this paper,we obtain
embedding theorems for the Weyl class W
¸;¯
p
,for the generalized Weyl-
Nikolskii class W
¸;¯
p
H
®
['] and for the generalized Lipschitz class H
p
°
[!].
We show a relation between the parameters ® and ° depending on the
behavior of the sequence f¸
n
g and on the metric L
p
.
The remainder of the paper is organized as follows.In section 2 we
present the main theorem.Sections 3 and 4 contain the proofs of the suf-
¯ciency and necessity parts of the main theorem respectively.In section
5 we provide several corollaries.In particular,we describe the di®erence
in results for metrics L
p
;1 < p < 1 and L
p
;p = 1;1.The estimates
5
As usual,
￿
¯
º
￿
=
¯(¯¡1)¢¢¢(¯¡º+1)
º!
for º > 1,
￿
¯
º
￿
= ¯ for º = 1,and
￿
¯
º
￿
= 1 for
º = 0.
6 B.SIMONOV AND S.TIKHONOV
!
°
(f
(r)
;±)
p
and!
°
(
e
f
(r)
;±)
p
are written in terms of!
¯
(f;±)
p
for di®erent
values of r;° and ¯.The concluding remarks are given in section 6.
2.Embedding theorems for generalized Lipschitz and
Weyl-Nikolskii classes
For ¸ = f¸
n
g
n2N
we de¯ne 4¸
n
:= ¸
n
¡¸
n+1
;4
2
¸
n
:= 4(4¸
n
).
Theorem 1.
Let µ = min(2;p),® 2 R
+
;¯ 2 R,and ¸ = f¸
n
g be a non-
decreasing sequence of positive numbers.Let ½ be a non-negative number
such that the sequence fn
¡½
¸
n
g is non-increasing.
I.Let 1 < p < 1.Then
H
p
®+½
[!] ½ W
¸;¯
p
()
1
X
n=1
¡
¸
µ
n+1
¡¸
µ
n
¢
!
µ
µ
1
n

< 1;(10)
H
p
®+½
[!] ½ W
¸;¯
p
H
®
['] ()
½
n
¡®µ
n
X
º=1
º
(½+®)µ
¡
º
¡½µ
¸
µ
º
¡(º +1)
¡½µ
¸
µ
º+1
¢
!
µ
µ
1
º

+
1
X
º=n+2
¡
¸
µ
º+1
¡¸
µ
º
¢
!
µ
µ
1
º


µ
n+1
!
µ
µ
1
n +1
¶¾
1
µ
= O
·
'
µ
1
n +1
¶¸
;(11)
W
¸;¯
p
½ H
p
®+½
[!] ()
1
¸
n
= O
·
!
µ
1
n
¶¸
;(12)
W
¸;¯
p
H
®
['] ½ H
p
®+½
[!] ()
'
¡
1
n
¢
¸
n
= O
·
!
µ
1
n
¶¸
:(13)
II.Let p = 1 or p = 1.
(a) If 4
2
¸
n
¸ 0 or 4
2
¸
n
· 0,then
H
p
®+½
[!] ½ W
¸;¯
p
() j cos
¯¼
2
j
1
X
n=1

n+1
¡¸
n
)!
µ
1
n

+ j sin
¯¼
2
j
1
X
n=1
¸
n
!
¡
1
n
¢
n
< 1;(14)
and if,additionally,for some ¿ > 0 the following inequality holds,
4
2
µ
¸
n
n
r

¸ 0 with r = ½ +¿ sign
¯
¯
¯
¯
sin
(¯ ¡½)¼
2
¯
¯
¯
¯
;
then
H
p
®+r
[!] ½ W
¸;¯
p
H
®
['] ()
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 7
() n
¡®
n
X
º=1
º
r+®
¡
º
¡r
¸
º
¡(º +1)
¡r
¸
º+1
¢
!
µ
1
º

+ j cos
¯¼
2
j
1
X
º=n+2

º+1
¡¸
º
)!
µ
1
º

+ j sin
¯¼
2
j
1
X
º=n+2
¸
º
!
¡
1
º
¢
º

n+1
!
µ
1
n +1

= O
·
'
µ
1
n +1
¶¸
:(15)
(b) If for ¯ = 2k;k 2 Z,the condition 4
2
(1=¸
n
) ¸ 0 holds,and for
¯ 6= 2k;k 2 Z;conditions 4
2
(1=¸
n
) ¸ 0 and
1
P
º=n+1
1
º¸
º
·
C
¸
n
are ful¯lled,
then
W
¸;¯
p
½ H
p
®+½
[!] ()
1
¸
n
= O
·
!
µ
1
n
¶¸
;(16)
and if,additionally,for some ¿ > 0 the following inequality holds,
4
2
µ
n
r
¸
n

¸ 0 or 4
2
µ
n
r
¸
n

· 0 with r = ½ +¿ sign
¯
¯
¯
¯
sin
(¯ ¡½)¼
2
¯
¯
¯
¯
;
then
W
¸;¯
p
H
®
['] ½ H
p
®+r
[!] ()
'
¡
1
n
¢
¸
n
= O
·
!
µ
1
n
¶¸
:(17)
3.Proof of sufficiency in Theorem 1.
We will use the following notations.
Let a function f(x) 2 L have Fourier series (9).Then S
n
(f) denotes the
n-th partial sum of (9),V
n
(f) denotes the de la Vall¶ee-Poussin mean and
K
n
(x) is the Fej¶er kernel,i.e.,
S
n
(f) =
n
P
º=0
A
º
(x);V
n
(f) =
1
n
2n¡1
P
º=n
S
º
(f);
K
n
(x) =
1
n+1
n
P
º=0
µ
1
2
+
º
P
m=1
cos mx

:
The following lemmas play the central role in the proof of Theorem 1.
Lemma 3.1.
Let f(x) 2 L
p
;1 · p · 1,and ® > 0.Then
C
1
(p;®)!
®
³
f;
1
n
´
p
·
µ
n
¡®
°
°
°
V
(®)
n
(f;x))
°
°
°
p
+kf(x) ¡V
n
(f;x)k
p

·C
2
(p;®)!
®
³
f;
1
n
´
p
:(18)
8 B.SIMONOV AND S.TIKHONOV
If f(x) 2 L
p
;1 < p < 1,then
C
1
(p;®)!
®
³
f;
1
n
´
p
·
µ
n
¡®
°
°
°
S
(®)
n
(f;x))
°
°
°
p
+kf(x) ¡S
n
(f;x)k
p

· C
2
(p;®)!
®
³
f;
1
n
´
p
:(19)
Proof of Lemma 3.1.The estimate from above!
®
³
f;
1
n
´
p
follows from
the inequality (see [Bu-Dy-GÄo-St])!
®
³
T
n
;
1
n
´
p
· C(p;®)n
¡®
°
°
°
T
(®)
n
°
°
°
p
,where
T
n
is a trigonometric polynomial of order n.We get
!
®
³
f;
1
n
´
p
· C(p;®)
µ
!
®
³
T
n
;
1
n
´
p
+kf ¡T
n
k
p

· C(p;®)
µ
n
¡®
°
°
°
T
(®)
n
°
°
°
p
+kf ¡T
n
k
p

:
Now we will estimate!
®
³
f;
1
n
´
p
from below.We need the generalized
Nikol'skii-Stechkin inequality (see [Ta]) n
¡®
°
°
°
T
(®)
n
°
°
°
p
· C(p;®)!
®
³
T
n
;
1
n
´
p
and the generalized Jackson inequality (see [Bu-Dy-GÄo-St])
E
n
(f)
p
· C(®)!
®
µ
f;
1
n +1

p
:(20)
It is well known that the Vall¶ee-Poussin mean is the nearly best approxi-
mant,i.e.,
kf ¡V
n
(f)k
p
· CE
n
(f)
p
:(21)
Then,
n
¡®
°
°
°
V
(®)
n
(f;x)
°
°
°
p
+kf(x) ¡V
n
(f;x)k
p
· C(p;®)
µ
!
®
³
V
n
;
1
n
´
p
+E
n
(f)
p

· C(p;®)
µ
!
®
³
f;
1
n
´
p
+!
®
³
f ¡V
n
;
1
n
´
p

· C(p;®)!
®
³
f;
1
n
´
p
;
and (18) is proved.Using
kf ¡S
n
(f)k
p
· C(p) E
n
(f)
p
(22)
for 1 < p < 1,we will have (19) similarly.This completes the proof of
Lemma 3.1.
We note that (18) and (19) are realization results (see paper [Di-Hr-Iv]
by Z.Ditzian,V.H.Hristov,K.G.Ivanov).
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 9
Lemma 3.2.
([St,S]).Let f(x) 2 L
p
;p = 1;1,and let
1
P
k=1
k
¡1
E
k
(f)
p
< 1
be true.Then
~
f(x) 2 L
p
and
E
n
(
~
f)
p
· C
Ã
E
n
(f)
p
+
1
X
k=n+1
k
¡1
E
k
(f)
p
!
;n 2 N:
Lemma 3.3.
Let p = 1;1 and f¸
n
g be monotonic concave (or convex)
sequence.Let
T
n
(x) =
n
X
º=0
a
º
cos ºx +b
º
sinºx;
T
n
(¸;x) =
n
X
º=0
¸
º
(a
º
cos ºx +b
º
sinºx):
Then for M > N ¸ 0 one has
kT
M
(¸;x)¡T
N
(¸;x)k
p
· ¹(M;N) kT
M
(x)¡T
N
(x)k
p
;
where ¹(M;N) =
8
>
<
>
:
2M(¸
M
¡¸
M¡1
)+¸
N+1
¡(N +1)(¸
N+2
¡¸
N+1
);if ¸
n
"(n"),4
2
¸
n
¸ 0;

M
+(N +1)(¸
N+2
¡¸
N+1
)¡¸
N+1
;if ¸
n
"(n"),4
2
¸
n
· 0;
(N +1)(¸
N+1
¡¸
N+2
) +¸
N+1
;if ¸
n
#(n"),4
2
¸
n
¸ 0:
Proof of Lemma 3.3.Applying two times Abel's transformation we write
kT
M
(¸;x)¡T
N
(¸;x)k
p
=
1
¼
°
°
°
¼
Z
¡¼
(T
M
¡T
N
) (x +u)
M
X
º=N+1
¸
º
cos ºudu
°
°
°
p
=
=
1
¼
°
°
°
¼
Z
¡¼
(T
M
¡T
N
) (x +u)
n
M¡2
X
º=N+1

º
¡2¸
º+1

º+2
) (º +1)K
º
(u) +
+(¸
N+2
¡¸
N+1
) (N +1)K
N
(u) +
+(¸
M¡1
¡¸
M
) MK
M¡1
(u)
o
du +¸
M
(T
M
¡T
N
)
°
°
°
p
·
· kT
M
(x) ¡T
N
(x)k
p
¢
¢
n
M¡2
X
º=N+1

º
¡2¸
º+1

º+2
j (º +1) +j¸
M¡1
¡¸
M
j M +¸
M
o
=:kT
M
(x) ¡T
N
(x)k
p
I(M;N):
10 B.SIMONOV AND S.TIKHONOV
First we estimate I(M;N) in the case ¸
n
"(n"),4
2
¸
n
¸ 0.We have
I(M;N) =
M¡2
X
º=N+1

º
¡2¸
º+1

º+2
) (º+1)+(¸
M
¡¸
M¡1
) M+¸
M
= ¡(N +1) (¸
N+2
¡¸
N+1
) +(¸
N+1
¡¸
M¡1
)

M
+(2M ¡1) (¸
M
¡¸
M¡1
)
= ¡(N +1) (¸
N+2
¡¸
N+1
) +¸
N+1
+2M(¸
M
¡¸
M¡1
):
If ¸
n
"(n"),4
2
¸
n
· 0,then
I(M;N) = (N +1) (¸
N+1
¡¸
N+2
) +(¸
M¡1
¡¸
N+1
) +¸
M
+(¸
M
¡¸
M¡1
):
Finally,if ¸
n
#(n"),4
2
¸
n
¸ 0,then
I(M;N) = (N +1)(¸
N+1
¡¸
N+2
) +¸
N+1
:
This completes the proof of Lemma 3.3.
Lemma 3.4.
Suppose p = 1;1.
Let T
2
n
;2
n+1
(x) =
2
n+1
P
º=2
n
³
c
º
cos ºx +d
º
sinºx
´
;then
C
1
°
°
°
e
T
2
n
;2
n+1
(x)
°
°
°
p
·
°
°
T
2
n
;2
n+1
(x)
°
°
p
· C
2
°
°
°
e
T
2
n
;2
n+1
(x)
°
°
°
p
:(23)
Proof of Lemma 3.4.We rewrite T
2
n
;2
n+1
(x) in the following way
T
2
n
;2
n+1
(x) =
2
n+1
X
º=2
n
1
º
³
ºc
º
cos ºx +ºd
º
sinºx
´
:
Applying Lemma 3.3 and the Bernstein inequality we have
°
°
T
2
n
;2
n+1
(x)
°
°
p
· C
1
2
n
°
°
°
°
°
°
2
n+1
X
º=2
n
³
ºc
º
cos ºx +ºd
º
sinºx
´
°
°
°
°
°
°
p
= C
1
2
n
°
°
°
°
°
°
0
@
2
n+1
X
º=2
n
¡d
º
cos ºx +c
º
sinºx
1
A
0
°
°
°
°
°
°
p
· C
°
°
°
e
T
2
n
;2
n+1
(x)
°
°
°
p
:
The same reasoning for
e
T
2
n
;2
n+1
(x) works for the left-hand inequality of
(23).The proof is now complete.
Su±ciency in (10) - (17).
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 11
I.1 < p < 1.In this case,if ¸
n
´ 1,the Riesz inequality ([Zy,2,V.1,
p.253]) k
~
fk
p
· C(p)kfk
p
implies
kf
(¸;¯)
k
p
· C(p;¯)kfk
p
:(24)
Henceforth,C(s;t;¢ ¢ ¢ ) will be positive constants that are dependent only
on s;t;¢ ¢ ¢ and may be di®erent in di®erent formulas.
Let the series in the right part of (10) be convergent and f 2 H
p
®+½
[!]:
We will use the following representation
¸
µ
2
n
= ¸
µ
1
+
n+1
X
º=2
¡
¸
µ
2
º¡1
¡¸
µ
2
º¡2
¢
:
Applying the Minkowski's inequality,we get
³
here and further 4
1
:=
A
1
(f;x);4
n+2
:=
2
n+1
P
º=2
n
+1
A
º
(f;x),where A
º
(f;x) is from (9)
´
I
1
:=
8
<
:

Z
0
"
1
X
n=1
¸
2
2
n¡1
4
2
n
#
p
2
dx
9
=
;
µ
p
· C(p)
0
B
@
¸
µ
1
8
<
:

Z
0
"
1
X
n=1
4
2
n
#
p
2
dx
9
=
;
µ
p
+
+
1
X
s=2
¡
¸
µ
2
s¡1
¡¸
µ
2
s¡2
¢
8
<
:

Z
0
"
1
X
n=s
4
2
n
#
p
2
dx
9
=
;
µ
p
1
C
A
1
µ
:(25)
By the Littlewood-Paley theorem ([Zy,2,V.II,p.233]) and (22),we write
I
1
· C(p)
(
¸
µ
1
kfk
µ
p
+
1
X
s=1
¡
¸
µ
2
s
¡¸
µ
2
s¡1
¢
E
µ
2
s¡1
(f)
p
)
1
µ
:(26)
Then,both the generalized Jackson inequality (20) and f 2 H
p
®+½
[!] imply
I
1
< 1:Thus,there exists a function g 2 L
p
with Fourier series
1
X
n=1
¸
2
n¡1
4
n
;(27)
and also kgk
p
· C(p)I
1
.We rewrite series (27) in the formof
1
P
n=1
°
n
A
n
(f;x),
where °
i
:= ¸
i
;i = 1;2 and °
º
:= ¸
2
n
for 2
n¡1
+1 · º · 2
n
(n = 2;3;¢ ¢ ¢ ):
12 B.SIMONOV AND S.TIKHONOV
Further,we write the series
1
X
n=1
¸
n
A
n
(f;x) =
1
X
n=1
°
n
¤
n
A
n
(f;x);(28)
where ¤
1
= ¤
2
= 1,¤
º
:= ¸
º

n
= ¸
º

2
n
for 2
n¡1
+ 1 · º · 2
n
(n =
2;3;¢ ¢ ¢ ):The sequence f¤
n
g satis¯es the conditions of the Marcinkiewicz
multiplier theorem ([Zy,2,V.II,p.232]),i.e.,series (28) is the Fourier series
of a function f
(¸;0)
2 L
p
,kf
(¸;0)
k
p
· C(p)kgk
p
:Then,inequalities (20),
(24) and (26) imply
kf
(¸;¯)
k
p
· C(p;¯)
(
¸
µ
1
kfk
µ
p
+
1
X
s=1
E
µ
2
s¡1
(f)
p
2
s
¡1
X
n=2
s¡1
¡
¸
µ
n+1
¡¸
µ
n
¢
)
1
µ
(29)
· C(p;¯;®;½)
(
¸
µ
1
kfk
µ
p
+
1
X
n=1
¡
¸
µ
n+1
¡¸
µ
n
¢
!
µ
®+½
µ
f;
1
n

p
)
1
µ
;
i.e.,the su±ciency in (10) has been proved.
Let the inequality in (11) hold,and f 2 H
p
®+½
[!].Let us prove f 2
W
¸;¯
p
H
®
[']:First we estimate!
®
(f
(¸;¯)
;
1
n
)
p
:By Lemma 3.1,
!
®
µ
f
(¸;¯)
;
1
n

p
· C(p;®)
³
kf
(¸;¯)
¡S
n
(f
(¸;¯)
)k
p
+n
¡®
kS
(®)
n
(f
(¸;¯)
)k
p
´
:
(30)
Using (29) for the function (f ¡S
n
),we have ([a] is the integer part of a)
kf
(¸;¯)
¡S
n
(f
(¸;¯)
)k
p
·
· C(p;¯;®;½)
(
¸
µ
n+1
kf ¡S
n
k
µ
p
+ E
µ
[
n
2
]
(f)
p
2n
X
s=1
¡
¸
µ
s+1
¡¸
µ
s
¢
+
1
X
s=n+1
¡
¸
µ
s+1
¡¸
µ
s
¢
E
µ
[
s
2
]
(f)
p
)
1
µ
· C(p;¯;®;½)
(
¸
µ
n+1
!
µ
®+½
µ
f;
1
n

p
+
1
X
º=n+1
¡
¸
µ
º+1
¡¸
µ
º
¢
!
µ
®+½
µ
f;
1
º

p
)
1
µ
:
(31)
Further,we estimate the second item of (30).Let m be an integer such
that 2
m
· n +1 < 2
m+1
:We will use here the representation
2
¡s½µ
¸
µ
2
s
= 2
¡(m+1)½µ
¸
µ
2
m+1
+
m
X
º=s
³
2
¡º½µ
¸
µ
2
º
¡2
¡(º+1)½µ
¸
µ
2
º+1
´
:
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 13
Then,using Lemmas 3.1,3.3,we can follow the way of proof in (25)-(29).
Then,one can obtain
n
¡®
kS
(®)
n
³
f
(¸;¯)
´
k
p
·
· C(p;¯;®;½)
(
¸
µ
n+1
!
µ
®+½
(f;
1
n
)
p
+ n
¡®µ
n
X
º=1
¡
º
¡½µ
¸
µ
º
¡(º +1)
¡½µ
¸
µ
º+1
¢
º
(½+®)µ
!
µ
®+½
µ
f;
1
º

p
)
1
µ
:(32)
We use (31),(32) and the right part of (11) to obtain f 2 W
¸;¯
p
H
®
[']:
Now we prove that conditions
1
¸
n
= O
£
!
¡
1
n
¢¤
and
'
(
1
n
)
¸
n
= O
£
!
¡
1
n
¢¤
are
su±cient for W
¸;¯
p
½ H
p
®+½
[!] and W
¸;¯
p
H
®
['] ½ H
p
®+½
[!],respectively.
From the properties of the sequence f¸
n
g,using the Littlewood-Paley
and the Marcinkiewicz multiplier theorem,we get
!
®+½
µ
f;
1
n

p
·
· C(p;®;½)
³
kf ¡S
n
(f)k
p
+n
¡(®+½)
kS
(®+½)
n
(f)k
p
´
· C(p;¯;®;½)
³
¸
¡1
n
kf
(¸;¯)
¡S
n
³
f
(¸;¯)
´
k
p

¡1
n
n
¡®
kS
(®)
n
(f
(¸;¯)
)k
p
´
:
Then,by Lemma 3.1,the following inequalities are true
!
®+½
µ
f;
1
n

p
· C(p;¯;®;½)¸
¡1
n
!
®
µ
f
(¸;¯)
;
1
n

p
·
· C(p;¯;®;½)¸
¡1
n
kf
(¸;¯)
k
p
:
Thus,the ¯rst part implies su±ciency in (13) and the second implies su±-
ciency in (12).
II.p = 1 or p = 1.Let the series in (14) be convergent,and let
f 2 H
p
®+½
[!].Consider the series
cos
¼¯
2
V
1
(¸;f) ¡sin
¼¯
2
f
V
1
(¸;f) (33)
+
1
X
n=1
½
cos
¼¯
2
(V
2
n
(¸;f)¡V
2
n¡1
(¸;f))¡sin
¼¯
2
³
g
V
2
n
(¸;f)¡
]
V
2
n¡1
(¸;f)
´
¾
;
where V
1
(¸;f):= ¸
1
A
1
(f;x);
V
n
(¸;f):= ¾(¸;V
n
(f)) =
14 B.SIMONOV AND S.TIKHONOV
=
n
X
m=1
¸
m
A
m
(f;x) +
2n¡1
X
m=n+1
¸
m
µ
1 ¡
m¡n
n

A
m
(f;x) (n ¸ 2):
Let M > N > 0.From the inequality kf ¡V
n
(f)k
p
· CE
n
(f)
p
and the
Jackson inequality (20),and using the properties of f¸
n
g and the outline of
Lemma 3.3,we get
A:=
°
°
°
°
°
M
X
n=N
·
cos
¼¯
2
(V
2
n+1
(¸;f) ¡V
2
n
(¸;f)) ¡sin
¼¯
2
³
^
V
2
n+1
(¸;f) ¡
g
V
2
n
(¸;f)
´
¸
°
°
°
°
°
p
·
M
X
n=N
2
4
j cos
¼¯
2
j kV
2
n+1
(f) ¡V
2
n
(f)k
p
0
@
2
n+2
¡1
X
m=2
n
j4
2
¸
m
j(m+1) +2
n+2
j4¸
2
n+2
j
1
A
+j sin
¼¯
2
j
°
°
°
^
V
2
n+1
(f) ¡
g
V
2
n
(f)
°
°
°
p
0
@
2
n+2
¡1
X
m=2
n
j4
2
¸
m
j(m+1) +2
n+2
j4¸
2
n+2
¡1
j
1
A
3
5
+j cos
¼¯
2
j
°
°
°
°
°
M
X
n=N
¸
2
n+2
(V
2
n+1
¡V
2
n
) (f)
°
°
°
°
°
p
+j sin
¼¯
2
j
°
°
°
°
°
M
X
n=N
¸
2
n+2
³
^
V
2
n+1
¡
g
V
2
n
´
(f)
°
°
°
°
°
p
· C
(
¸
2
N
µ
j cos
¼¯
2
jE
2
N
(f)
p
+j sin
¼¯
2
jE
2
N
(
~
f)
p

+
1
X
n=2
N
¡1

n
+1
¡¸
n
)
Ã
j cos
¼¯
2
j!
®
+
½
µ
f;
1
n

p
+j sin
¼¯
2
j!
®
+
½
µ
~
f;
1
n

p
!)
:(34)
To complete the proof of the su±ciency part in (14),we apply Lemma 3.2,
inequality (3) (see [Ta] for the case k > 0),and inequality (20).Then
the convergence of series in (14) and f 2 H
p
®+½
[!] imply that there exists
'2 L
p
such that series (33) converges to'in L
p
.Let us show that
¾(') = ¾(f
(¸;¯)
):If F
n
is the n-th partial sum of (33),then,say for cosine
coe±cients,
a
n
(') = a
n
('¡F
N+n
) +a
n
(F
N+n
) = a
n
('¡F
N+n
) +a
n
(f
(¸;¯)
);
and
a
n
('¡F
N+n
) =
1
¼
¼
Z
¡¼
('¡F
N+n
) (x) cos nxdx
· C(p) k'¡F
N+n
k
p
¡!0 (N!1):
This completes the proof of the su±ciency part of (14).
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 15
Let f 2 H
p
®+r
[!] and the condition in the right part of (15) hold.We
will estimate from above!
®
¡
f
(¸;¯)
;
1
n
¢
p
.By Lemma 3.1,
!
®
µ
f
(¸;¯)
;
1
n

p
· C(®)
µ
°
°
°
f
(¸;¯)
¡V
n
(f
(¸;¯)
)
°
°
°
p
+n
¡®
°
°
°
V
(®)
n
(f
(¸;¯)
)
°
°
°
p

:
Let us show that
°
°
°
f
(¸;¯)
¡V
n
(f
(¸;¯)
)
°
°
°
p
·
· C(¯;®;r)
Ã
¸
n
!
®+r
µ
f;
1
n

p
+j cos
¼¯
2
j
1
X
º=n+1

º+1
¡¸
º
)!
®+r
µ
f;
1
º

p
+ j sin
¼¯
2
j
1
X
º=n+1
¸
º
º
!
®+r
µ
f;
1
º

p
!
:(35)
We have already shown that A =
°
°
°
°
M
P
n=N
¡
V
2
n+1
(f
(¸;¯)
)¡V
2
n
(f
(¸;¯)
)
¢
°
°
°
°
p
:Then
for 2
m
· n < 2
m+1
°
°
°
f
(¸;¯)
¡V
n
(f
(¸;¯)
)
°
°
°
p
·
°
°
°
V
n
(f
(¸;¯)
) ¡V
2
m+1
(f
(¸;¯)
)
°
°
°
p
+
°
°
°
°
°
1
X
º=m+1
³
V
2
º
(f
(¸;¯)
) ¡V
2
º+1
(f
(¸;¯)
)
´
°
°
°
°
°
p
=:I
1
+I
2
:
By Lemma 3.4,we have
I
1
=
°
°
°
°
cos
¼¯
2
(V
n
(¸;f) ¡V
2
m+1
(¸;f))¡sin
¼¯
2
³
f
V
n
(¸;f)¡
]
V
2
m+1
(¸;f)
´
°
°
°
°
p
· kV
n
(¸;f) ¡V
2
m+1
(¸;f)k
p
+
°
°
°
f
V
n
(¸;f) ¡
^
V
2
m+1
(¸;f)
°
°
°
p
· CkV
n
(¸;f) ¡V
2
m+1
(¸;f)k
p
and,by Lemma 3.3,we get I
1
· ¸
n
!
®+r
¡
f;
1
n
¢
p
.
Secondly
I
2
· j cos
¼¯
2
j
°
°
°
°
°
1
X
º=m+1
(V
2
º
(¸;f) ¡V
2
º+1
(¸;f))
°
°
°
°
°
p
+ j sin
¼¯
2
j
°
°
°
°
°
1
X
º=m+1
³
g
V
2
º
(¸;f) ¡
]
V
2
º+1
(¸;f)
´
°
°
°
°
°
p
:
16 B.SIMONOV AND S.TIKHONOV
As in (34),we have
°
°
°
°
°
M
X
º=N
(V
2
º+1
¡V
2
º
) (¸;f)
°
°
°
°
°
p
· C
Ã
¸
2
N
!
®+r
µ
f;
1
2
N

p
+
+
1
X
º=2
N
+1

º+1
¡¸
º
)!
®+r
µ
f;
1
º

p
1
A
;
°
°
°
°
°
M
X
º=N
³
g
V
2
º
(¸;f) ¡
]
V
2
º+1
(¸;f)
´
°
°
°
°
°
p
·
°
°
°
°
°
M
X
º=N
¸
2
º+2
³
g
V
2
º
(f) ¡
]
V
2
º+1
(f)
´
°
°
°
°
°
p
+
M
X
º=N
°
°
°
g
V
2
º
(f) ¡
]
V
2
º+1
(f)
°
°
°
p
0
@
2
º+2
¡1
X
m=2
º
j4
2
¸
m
j(m+1) +2
º+2
j4¸
2
º+2
j
1
A
=:I
21
+I
22
:
By Lemma 3.2,
I
21
· ¸
2
N+1
°
°
°
^
V
2
M+1
(f) ¡
g
V
2
N
(f)
°
°
°
p
+
M
X
º=N

2
º+2
¡¸
2
º+1
)
°
°
°
]
V
2
º+1
(f) ¡
g
V
2
º
(f)
°
°
°
p
· C
Ã
¸
2
N+1
1
X
º=N
E
2
º
(f)
p
+
1
X
º=N

2
º+2
¡¸
2
º+1
)
1
X
s=º
E
2
s
(f)
p
!
· C
1
X
º=2
N
¸
º
º
!
®+r
µ
f;
1
º

p
;
I
22
· C
M
X
º=N

2
º+3
¡¸
2
º¡1
) E
2
º
(
~
f)
p
· C
1
X
º=2
N
¸
º
º
!
®+r
µ
f;
1
º

p
;
and (35) follows.
Repeating the argument in (34),we estimate n
¡®
°
°
°
V
(®)
n
(f
(¸;¯)
)
°
°
°
p
.By
Lemma 3.3 and by inequalities (20) and (21),we write
°
°
°
V
(®)
n
(f
(¸;¯)
)
°
°
°
p
· C(¯;®;r)
Ã
n
®
¸
n
!
®+r
µ
f;
1
n

p
+ (36)
+
n
X
º=1
µ
¸
º
º
r
¡
¸
º+1
(º +1)
r

º
®+r
!
®+r
µ
f;
1
º

p
!
:
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 17
By means of (35) and (36) we get f 2 W
¸;¯
p
H
®
['];because of the condition
in (15).
Let us prove (16).Let f 2 W
¸;¯
p
:To establish f 2 H
p
®+½
(!),we shall
¯rst estimate E
n
(f)
p
for the case sin
¼¯
2
6= 0.Repeating the reasoning in
(34),by (21),we get ( 2
m
· n < 2
m+1
)
E
n
(f)
p
· C
1
X
º=m
1
¸
2
º
°
°
°
¡
V
[2
º¡1
]
¡V
2
º
¢
(f
(¸;¯)
)
°
°
°
p
· CE
[2
m¡1
]
(f
(¸;¯)
)
p
1
X
º=m
1
¸
2
º
· C(½;p)
kf
(¸;¯)
k
p
¸
n
:
If sin
¼¯
2
= 0,it is easy to see that
E
n
(f)
p
·
C
¸
n
E
n
(f
(¸;¯)
)
p
·
C
¸
n
kf
(¸;¯)
k
p
:
Substituting the bound for E
n
(f)
p
into (3) and using the fact that n
½
¸
¡1
n
"
(n"),we can write
!
®+½
µ
f;
1
n

p
· C(®;½)
1
n
®+½
n
X
º=0
º
®+½¡1
E
º¡1
(f)
p
·
C(®;½)
¸
n
= O
·
!
µ
1
n
¶¸
;
i.e.,f 2 H
p
®+½
(!):This completes the proof of the su±ciency part in (16).
Let the right part of (17) be true,and f 2 H
p
®+½
[!].First let us prove
that
°
°
°
V
(®+r)
n
(f)
°
°
°
p
· C(®;r)
n
r
¸
n
°
°
°
V
(®)
2n
(f
(¸;¯)
)
°
°
°
p
:(37)
If ¯ = ½ +2m,and therefore,r = ½,then V
(®+r)
n
(f) = V
(®)
n
³
f
º
r
¸
º
g;f
(¸;¯)
´
and,by Lemma 3.3
°
°
°
V
(®+r)
n
(f)
°
°
°
p
· C(®;r)
n
r
¸
n
°
°
°
V
(®)
n
(f
(¸;¯)
)
°
°
°
p
· C(®;r)
n
r
¸
n
°
°
°
V
(®)
2n
(f
(¸;¯)
)
°
°
°
p
:
If ¯ 6= ½ +2m,and therefore,r > ½,then by Lemma 3.4
°
°
°
V
(®+r)
2
n
(f)
°
°
°
p
·
·
n
X
º=0
°
°
°
V
(®+r)
2
º
(f) ¡V
(®+r)
2
º¡1
(f)
°
°
°
p
+
°
°
°
V
(®+r)
1
(f)
°
°
°
p
· C
n
X
º=0
2
ºr
¸
2
º
°
°
°
V
(®)
2
º
(f
(¸;¯)
) ¡V
(®)
2
º¡1
(f
(¸;¯)
)
°
°
°
p
+
1
¸
1
°
°
°
V
(®)
1
(f
(¸;¯)
)
°
°
°
p
18 B.SIMONOV AND S.TIKHONOV
· C
°
°
°
V
(®)
2
n+1
(f
(¸;¯)
)
°
°
°
p
Ã
n
X
º=0
2
ºr
¸
2
º
+
1
¸
1
!
· C(®;r)
2
nr
¸
2
n
°
°
°
V
(®)
2
n+1
(f
(¸;¯)
)
°
°
°
p
:
Thus,by (37) and the estimate E
n
(f)
p
·
C
¸
n
E
[
n
4
]
(f
(¸;¯)
)
p
;we can write
!
®+r
µ
f;
1
n

p
· C(®;r)
µ
n
¡(®+r)
°
°
°
V
(®+r)
n
(f)
°
°
°
p
+E
n
(f)
p

·
C(®;r)
¸
n
µ
n
¡®
°
°
°
V
(®)
n
(f
(¸;¯)
)
°
°
°
p
+E
[
n
4
]
(f
(¸;¯)
)
p

·
C(®;r)
¸
n
!
®
µ
f
(¸;¯)
;
1
n

p
= O
"
'
¡
1
n
¢
¸
n
#
= O
·
!
µ
1
n
¶¸
:
Thus,
'
(
1
n
)
¸
n
= O
£
!
¡
1
n
¢¤
is su±cient for W
¸;¯
p
H
®
['] ½ H
p
®+r
[!].
4.Proof of necessity in Theorem 1.
First we de¯ne the trigonometric polynomials ¿
n+1
(x):
¿
n+1
(x) =
n+1
X
j=1
®
n
j
sinjx;where ®
n
j
=
(
j
n+2
;1 · j ·
n+2
2
1 ¡
j
n+2
;
n+2
2
· j · n +1:
We will use the following lemmas as well as Lemmas 3.1-3.2.
Lemma 4.1.
([Te]).Let f(x) 2 L
1
have the Fourier series (9).Then
E
n
(f)
1
¸ C
¯
¯
¯
¯
¯
1
X
º=n+1
b
º
º
¯
¯
¯
¯
¯
:
Lemma 4.2.
([Zy,2,V.1,p.215]) Let 1 · p < 1.
(a) If a function f(x) 2 L
p
has the Fourier series
1
P
º=1
(a
º
cos 2
º
x+b
º
sin2
º
x),
then
(
1
X
º=1
¡
a
2
º
+b
2
º
¢
)
1
2
· Ckfk
p
:
(b) Let a
n
;b
n
(n 2 N) be real numbers such that
1
P
º=1
¡
a
2
º
+b
2
º
¢
< 1:Then
1
P
º=1
(a
º
cos 2
º
x + b
º
sin2
º
x) be the Fourier series of a function f(x) 2 L
p
,
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 19
and
kfk
p
· C
(
1
X
º=1
¡
a
2
º
+b
2
º
¢
)
1
2
:
Lemma 4.3.
([Ba,2,Vol.2,p.269]).Let f(x) 2 L
1
have the Fourier
series
1
P
º=1
(a
º
cos 2
º
x +b
º
sin2
º
x);where a
º
;b
º
¸ 0.Then
C
1
1
X
»=n
(a
»
+b
»
) · E
2
n
¡1
(f)
1
· C
2
1
X
»=n
(a
»
+b
»
):
We need the following de¯nitions.Let!(¢) 2 ©
®
.
We call à a Q
®;µ
(!)-sequence if
0 < Ã
n
· n
®
!
µ
1
n


n
"(n") (38)
C
1
!
µ
1
n

·
(
1
X
º=n
º
¡®µ¡1
Ã
µ
º
)
1
µ
· C
2
!
µ
1
n

:(39)
We call"a q
®;µ
(!)-sequence if
0 <"
n
·!
µ
1
n +1

;"
n
#(n") (40)
C
1
!
µ
1
n +1

·
(
(n +1)
¡®µ
n+1
X
º=1
º
®µ¡1
"
µ
º
)
1
µ
· C
2
!
µ
1
n +1

:(41)
Necessity in (10) - (17).
We prove the necessity part by constructing corresponding examples.The
proof is in eight steps.
I.1 < p < 1.Step 1.Let us show the necessity part in (10).
Let!(¢) 2 ©
®+½
and µ = min(2;p).We will construct a sequence à that
will be a Q
®+½;µ
(!)-sequence.
Let us assume that we have chosen integers 1 = n
1
< n
2
< ¢ ¢ ¢ < n
s
.Then,
we de¯ne n
s+1
as minimum of integers N > n
s
such that
!
µ
1
N

<
1
2
!
µ
1
n
s

·!
µ
1
N ¡1

:
We set
Ã
n
=
(
n
½+®
s
!
³
1
n
s
´
;if n
s
· n < n
s+1
;
0;if n = 0:
It is easy to see that this sequence is required.
20 B.SIMONOV AND S.TIKHONOV
Let H
p
®+½
[!] ½ W
¸;¯
p
,and let the series in (10) be divergent.By means
of properties of sequence fÃ
n
g,we have
1 =
1
X
º=1
¡
¸
µ
º+1
¡¸
µ
º
¢
!
µ
µ
1
º

· C(®;½;µ)
1
X
º=1
¡
¸
µ
º+1
¡¸
µ
º
¢
1
X
m=º
m
¡(®+½)µ¡1
Ã
µ
m
· C(®;½;µ)
1
X
º=1
¸
µ
º
º
¡(®+½)µ¡1
Ã
µ
º
:
Step 1(a):2 · p < 1.We consider the series
1
X
º=1
2
¡º(®+½)
¡
Ã
2
2
º
¡Ã
2
2
º¡1
¢
1
2
cos 2
º
x:(42)
Since
1
X
º=1
2
¡2º(®+½)
¡
Ã
2
2
º
¡Ã
2
2
º¡1
¢
·
1
X
º=1
¡
Ã
2
2
º
¡Ã
2
2
º¡1
¢
1
X
»=º
2
¡2»(®+½)
·
1
X
º=1
2
¡2º(®+½)
Ã
2
2
º
· C!
2
(1);(43)
then,by Zygmund's Lemma 4.2,series (42) is the Fourier series of a function
f
1
(x) 2 L
p
.By Lemmas 3.1 and 4.2,
C(®;½)!
®+½
³
f
1
;
1
2
n
´
p
·
· 2
¡n(®+½)
Ã
n
X
º=1
a
2
º
2
2(®+½)º
!
1
2
+
Ã
1
X
º=n+1
a
2
º
!
1
2
=:I
1
+I
2
;
where a
º
= 2
¡(®+½)
¡
Ã
2
2
º
¡Ã
2
2
º¡1
¢
1
2
:Repeating the argument in (43),we
get
I
1
· 2
¡n(®+½)
Ã
2
n
·!
³
1
2
n
´
;
because of (38) and
I
2
· C(®;½)
Ã
1
X
º=n+1
2
¡2º(®+½)
Ã
2
2
º
!
1
2
· C(®;½)!
³
1
2
n
´
;
because of (39).
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 21
Thus,f
1
(x) 2 H
p
®+½
[!]:Then from our assumption,f
1
(x) 2 W
¸;¯
p
.On
the other hand,
°
°
°
f
(¸;¯)
1
°
°
°
p
¸ C(®;½;µ)
Ã
1
X
º=1
¸
2
º
º
¡2(®+½)¡1
Ã
2
º
!
1
2
= 1:
This contradiction proves the convergence of series in (10).
Step 1(b):1 < p · 2.Consider series
6
Ã
1
cos x +
1
X
º=1
2
¡º(®+½)
2
º(
1
p
¡1)
¡
Ã
p
2
º
¡Ã
p
2
º¡1
¢
1
p
2
º
X
¹=2
º¡1
+1
cos ¹x:(44)
Using Jensen inequality
µ
1
P
n=1
a
®
n

1=®
·
µ
1
P
n=1
a
¯
n

1=¯
(a
n
¸ 0 and 0 < ¯ ·
® < 1),we write

Z
0
2
6
4
1
X
º=1
0
@
2
¡º(®+½)
2
º(
1
p
¡1)
¡
Ã
p
2
º
¡Ã
p
2
º¡1
¢
1
p
2
º
X
¹=2
º¡1
+1
cos ¹x
1
A
2
3
7
5
p
2
dx
·

Z
0
2
4
1
X
º=1
2
¡ºp(®+½)
2
º(1¡p)
¡
Ã
p
2
º
¡Ã
p
2
º¡1
¢
¯
¯
¯
¯
¯
¯
2
º
X
¹=2
º¡1
+1
cos ¹x
¯
¯
¯
¯
¯
¯
p
3
5
dx
· C(p)
1
X
º=1
¡
Ã
p
2
º
¡Ã
p
2
º¡1
¢
2
¡ºp(®+½)
· C(p)!
p
(1);
because of C
1
(p)2
º(p¡1)
·
°
°
°
°
°
2
º
P
¹=2
º¡1
+1
cos ¹x
°
°
°
°
°
p
p
· C
2
(p)2
º(p¡1)
.By the
Littlewood-Paley theorem,there exists a function f
2
2 L
p
with Fourier
series (44).One can see that f
2
2 H
p
®+½
[!]:Then f
2
2 W
¸;¯
p
.On the other
hand,Paley's theorem on Fourier coe±cients [Zy,2,V.2,p.121] implies
that for f
2
2 L
p
°
°
°
f
(¸;¯)
2
°
°
°
p
p
¸
¸ C(p)
1
X
º=1
2
¡ºp(®+½)
2
º(1¡p)
¡
Ã
p
2
º
¡Ã
p
2
º¡1
¢
2
º
X
¹=2
º¡1
+1
¸
p
¹
¹
p¡2
¸ C(®;½;p)
1
X
º=1
¡
Ã
p
2
º
¡Ã
p
2
º¡1
¢
1
X
»=º
³
2
¡»(®+½)p
¸
p
2
»
¡2
¡(»+1)(®+½)p
¸
p
2
»+1
´
6
Series of this type was considered in [Ti,M,1].
22 B.SIMONOV AND S.TIKHONOV
¸ C
1
(®;½;p)
1
X
º=1
Ã
p
º
¸
p
º
º
¡p(®+½)¡1
¡C
2
(®;½;p)Ã
p
1
¸
p
2
2
¡p(®+½)¡1
= 1:
This contradiction shows that the series in the right part of (10) converges.
This completes the proof of the necessity part of (10).
Step 2.Let us prove the necessity in (11) for 2 · p < 1:
We notice that by Lemmas 3.1 and 4.2,we have for f(x) »
1
P
º=1
(a
º
cos 2
º
x+
b
º
sin2
º
x)
!
®
µ
f;
1
2
m

p
³
Ã
2
¡2m®
m
X
º=1
(a
2
º
+b
2
º
)2
2º®
!
1
2
+
Ã
1
X
º=m+1
(a
2
º
+b
2
º
)
!
1
2
:(45)
Let!(¢) 2 ©
®+½
.One can construct
7
a sequence"such that"is a q
®+½;µ
(!)-
sequence.We consider for this case
"
0
+
¡
"
2
1
¡"
2
2
¢
1
2
cos x +
1
X
º=1
¡
"
2
2
º
¡"
2
2
º+1
¢
1
2
cos 2
º
x:(46)
Repeating the argument for series (42) we obtain that series (46) is the
Fourier series of a function f
3
2 L
p
.Since E
2
n¡1
(f
3
)
p
· C(p)"
2
n
,then
by (3),(41) implies f
3
2 H
p
®+½
[!].We de¯ne f
13
:= f
1
+ f
3
:We have
f
13
2 H
p
®+½
[!] ½ W
¸;¯
p
H
®
['].It is easy to see from (45) that
C(®;¯)!
®
³
f
(¸;¯)
13
;
1
n +1
´
p
¸!
®
³
f
(¸;¯)
1
;
1
n +1
´
p
+!
®
³
f
(¸;¯)
3
;
1
n +1
´
p
:
(47)
Let us estimate!
®
³
f
(¸;¯)
1
;
1
n+1
´
p
:By (45),using the properties of the
sequence fÃ
º
g,we have (2
m
· n +1 < 2
m+1
)
!
2
®
µ
f
(¸;¯)
1
;
1
n +1

p
¸
¸ C(®;¯)
1
X
º=m
Ã
2
2
º
2
¡2º(r+®)
Ã
º
X
k=m

2
2
k
¡¸
2
2
k¡1
) +¸
2
2
m¡1
!
¸ C(®;¯)
Ã
¸
2
2
m
!
2
µ
1
2
m

+
1
X
k=m

2
2
k
¡¸
2
2
k¡1
)!
2
µ
1
2
k

!
¸ C(®;¯)
Ã
¸
2
n+1
!
2
µ
1
n +1

+
1
X
º=n+2
¡
¸
2
º+1
¡¸
2
º
¢
!
2
µ
1
º

!
:(48)
7
See,for example,[Ge].
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 23
Now we estimate!
®
³
f
(¸;¯)
3
;
1
n+1
´
p
:By (45),we have (2
m
· n + 1 <
2
m+1
):
!
2
®
µ
f
(¸;¯)
3
;
1
n +1

p
¸
¸ C(®;¯)2
¡2m®
m
X
º=0
2
2º®
¸
2

¡
"
2
2
º
¡"
2
2
º+1
¢
¸ C
1
(®;¯)2
¡2m®
m
X
º=0
2
2º®
¸
2
2
º
"
2
2
º
¡C
2
(®;p)¸
2
2
m+1
"
2
2
m+1
:(49)
The Jackson inequality implies
!
2
®
µ
f
(¸;¯)
3
;
1
n +1

p
¸ C(®)E
2
2
m
¡1
³
f
(¸;¯)
3
´
p
¸ C(®;¯)
1
X
º=m
¸
2

¡
"
2
2
º
¡"
2
2
º+1
¢
¸ C(®;¯)¸
2
2
m
"
2
2
m
:(50)
Both estimates (49) and (50) imply
!
®
µ
f
(¸;¯)
3
;
1
n +1

p
¸ C(®;¯)
Ã
(n +1)
¡2®
n+1
X
º=1
¸
2
º
º
2®¡1
"
2
º
!
1
2
:(51)
Using (41) and º
¡½
¸
º
#,we get
¸
2
n+1
!
2
µ
1
n +1

+
+ (n +1)
¡2®
n+1
X
º=1
º
2(½+®)
!
2
µ
1
º

¡
¸
2
º
º
¡2½
¡¸
2
º+1
(º +1)
¡2½
¢
· C(®;½)(n +1)
¡2®
n+1
X
º=1
º
2®¡1
¸
2
º
"
2
º
:(52)
Combining estimates (47),(48),(51),(52),and!
®
³
f
(¸;¯)
13
;
1
n+1
´
p
=
O
£
'
¡
1
n
¢¤
,we arrive at the condition in the right part of (11).
Step 3.Let us prove the necessity in (11) for 1 < p < 2:The proof
for 1 < p · 2 is similar to 2 · p < 1.The only di®erence is that we use
Paley's theorem on Fourier coe±cients instead of Zygmund's theorem.In
24 B.SIMONOV AND S.TIKHONOV
this case we consider the sum of f
2
(x) and the following function
"
0
+("
p
1
¡"
p
2
)
1
p
cos x +
1
X
º=0
2
º
(
1
p
¡1
)
¡
"
p
2
º+1
¡"
p
2
º+2
¢
1
p
2
º+1
X
¹=2
º
+1
cos ¹x:(53)
Step 4.To prove the necessity in (12) and (16),we consider the general
case 1 · p · 1:Let © be the class of all decreasing null-sequences.It is
clear that
1
¸
n
= O
·
!
µ
1
n
¶¸
() 8 ° = f°
n
g 2 ©
°
n
¸
n
= O
·
!
µ
1
n
¶¸
:
Let us assume that
°
n
¸
n
= O
£
!
¡
1
n
¢¤
does not hold for all ° 2 © and W
¸;¯
p
½
H
p
®+½
[!].Then there exist ° = f°
n
g 2 © and fC
n
"1g such that
°
m
n
¸
m
n
¸
C
n
!
³
1
m
n
´
:Further,we choose a subsequence fm
n
k
g such that
m
n
k+1
m
n
k
¸ 2
and °
m
n
k
· 2
¡k
:Consider the series
1
X
k=0
°
m
n
k
¸
m
n
k
cos(m
n
k
+1)x:(54)
Since
1
P
k=0
°
m
n
k
¸
m
n
k
·
1
¸
m
n
0
1
P
k=0
1
2
k
< 1;there exists a function f
4
2 L
p
with
Fourier series (54).Because
1
P
k=0
°
m
n
k
·
1
P
k=0
1
2
k
< 1;we have f
(¸;¯)
4
2 L
p
,
i.e.,f
4
2 W
¸;¯
p
.
On the other hand,by (20) and E
n¡1
(f)
p
¸ C(ja
n
j +jb
n
j),
!
®+½
µ
f
4
;
1
m
n
k

p
¸ C(®;½)E
m
n
k
(f
4
)
p
¸ C(®;½)
°
m
n
k
¸
m
n
k
¸ C(®;½)C
n
k
!
µ
1
m
n
k

;
i.e.,f
4
=2 H
p
®+½
[!]:This contradiction implies that the condition
1
¸
n
=
O
£
!
¡
1
n
¢¤
is necessary for W
¸;¯
p
½ H
p
®+½
[!].
Step 5.To prove the necessity in (13) and (17),we verify that for any
½ > 0 and for any 1 · p · 1,
W
¸;¯
p
H
®
['] ½ H
p
®+½
[!] =)
'
¡
1
n
¢
¸
n
= O
·
!
µ
1
n
¶¸
:(55)
First we remark that
'
¡
1
n
¢
¸
n
= O
·
!
µ
1
n
¶¸
()8 ° = f°
n
g 2 ©
°
n
'
¡
1
n
¢
¸
n
= O
·
!
µ
1
n
¶¸
:
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 25
Assume that the relation in the right hand side of (55) does not hold.Then
there exist ° = f°
n
g 2 © and fC
n
"1g such that
°
m
n
'
(
1
m
n
)
¸
m
n
¸ C
n
!
³
1
m
n
´
:
We choose a subsequence fm
n
k
g such that
m
n
k+1
m
n
k
¸ 2 and °
m
n
k
· 2
¡k
:
Because
1
P
k=0
°
m
n
k
'
³
1
m
n
´
·'
³
1
m
0
´
1
P
k=0
1
2
k
< 1;there exists a function
f
5
2 L
p
with Fourier series
1
X
k=0
°
m
n
k
'
µ
1
m
n
k

cos(m
n
k
+1)x:(56)
For m
n
k
· n < m
n
k+1
,by Lemmas 3.1 and 4.3,we have
!
®
µ
f
5
;
1
n

p
· C!
®
µ
f
5
;
1
n

1
· C
Ã
n
¡®
k
X
s=0
°
m
n
s
'
µ
1
m
n
s

m
®
n
s
+
1
X
s=k+1
°
m
n
s
'
µ
1
m
n
s

!
· C
Ã
'
µ
1
n

k
X
s=0
°
m
n
s
+'
µ
1
n

1
X
s=k+1
°
m
n
s
!
· C'
µ
1
n

:
Therefore,f
5
2 H
p
®
['],i.e.,setting
1
¸
:=
n
1
¸
1
;
1
¸
2
;
1
¸
3
;¢ ¢ ¢
o
,we have
f
(
1
¸
;¡¯)
5
2 W
¸;¯
p
H
®
[']:
However,we have
!
®+½
µ
f
(
1
¸
;¡¯)
5
;
1
m
n
k

p
¸ CE
m
n
k
(f
(
1
¸
;¡¯)
5
)
p
¸ C
°
m
n
k
'
³
1
m
n
k
´
¸
m
n
k
¸ CC
n
k
!
µ
1
m
n
k

;
i.e.,f
(
1
¸
;
¡
¯
)
5
=2 H
p
®+½
[!]:This contradicts our assumption.The proof of the
necessity part in (12)-(13) and (16)-(17) is now complete.
II.p = 1 or p = 1.Step 6.Let us prove the necessity in (14).Let
H
p
®+½
[!] ½ W
¸;¯
p
and the series in (14) be divergent.
Step 6(a):sin
¯¼
2
6= 0.Then the divergence of the series in (14) is equiv-
alent to the divergence of the series
1
P
n=1
¸
n
n
!
¡
1
n
¢
:
Let p = 1.We take a sequence"which is a q
®+½;1
(!)-sequence and
consider the series
1
X
º=1
("
º
¡"
º+1
) K
º
(x):(57)
26 B.SIMONOV AND S.TIKHONOV
This series is convergent in L
1
(see [Ge]) to a function f
6
(x),E
n
(f
6
)
1
=
O("
n
).By means of (3),(41),we get f
6
2 H
p
®+½
[!] ½ W
¸;¯
p
.One can
rewrite (57) in the following form
1
X
º=1
a
º
cos ºx;where a
º
="
º
¡º
1
X
j=º
"
j
¡"
j+1
j +1
:
By Lemma 4.1,
°
°
°
f
(¸;¯)
6
°
°
°
1
¸ C(¯)
1
X
º=1
¸
º
º
a
º
= C(¯)
0
@
1
X
º=1
¸
º
º
"
º
¡
1
X
º=1
¸
º
1
X
j=º
"
j
¡"
j+1
j +1
1
A
= C(¯)
Ã
1
X
º=1
¸
º
º
"
º
¡
1
X
º=1
(a
º
¡a
º+1

º
!
¸ C
1
(¯)
1
X
º=1
¸
º
º
"
º
¡C
2
(¯)
Ã
¸
1
a
1
+
1
X
n=1

n+1
¡¸
n
)a
n
!
:
Using (40) and (41),we get
1
X
º=1
¸
º
º
"
º
· C(½)
1
X
º=1
¸
º
º
!
µ
1
º

· C(½)
1
X
º=1
¸
º
º
¡(®+½)¡1
º
X
m=1
m
®+½¡1
"
m
= C(½)
1
X
m=1
m
®+½¡1
"
m
1
X
º=m
¸
º
º
½
º
¡®¡1
· C(½;®)
1
X
m=1
¸
m
m
"
m
:
Therefore,
°
°
°
f
(¸;¯)
6
°
°
°
1
¸ (58)
¸ C
1
(®;½;¯)
1
X
º=1
¸
º
º
¡1
!
µ
1
º

¡C
2
(®;½;¯)
Ã
¸
1
a
1
+
1
X
º=1

º+1
¡¸
º
)a
º
!
:
On the other hand,using monotonicity of fa
º
g and Lemma 4.1,we have
C(¯;½)
°
°
°
f
(¸;¯)
6
°
°
°
1
¸ C(½)
Ã
¸
1
a
1
+
1
X
º=1
¸
º
a
º
º
!
¸ C(½)
Ã
¸
1
a
1
+
1
X
º=0
¸
2
º
a
2
º
!
¸ ¸
1
a
1
+
1
X
º=0
a
2
º

2
º+1
¡¸
2
º
)
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 27
¸ ¸
1
a
1
+
1
X
º=1

º+1
¡¸
º
)a
º
:(59)
From (58) and (59) we get
°
°
°
f
(¸;¯)
6
°
°
°
1
¸ C(®;½;¯)
1
X
º=1
¸
º
º
¡1
!
µ
1
º

= 1:
This contradiction implies the convergence of series in (14).
Let nowp = 1.De¯ne the function (see [Ba,1]) f
7
(x) =
1
P
º=1
"
º
º
¡1
sinºx;
where"is a q
®+½;1
(!)-sequence.We have E
n
(f
7
)
1
· C"
n+1
.Using (3)
and (41),we get f
7
2 H
p
®+½
[!] ½ W
¸;¯
p
.On the other hand,
1=
°
°
°
f
(¸;¯)
7
°
°
°
1
¸ C(¯)
1
X
º=0
2
º(®+½)
"
2
º
¸
2
º
2
º(®+½)
¸ C(®;½;¯)
1
X
º=0
2
º(®+½)
"
2
º
1
X
m=º
¸
2
m
2
m(®+½)
¸ C(®;½;¯)
1
X
º=1
¸
º
º
¡1
!
µ
1
º

:
This implies the convergence of the series in (14).
Step 6(b):sin
¯¼
2
= 0.Let the series in (14) be divergent.We will
consider only the non-trivial case ½ > 0:Let"be a q
®+½;1
(!)-sequence.By
means of the properties f¸
n
g,we have
1
X
º=2

º+1
¡¸
º
)!
µ
1
º

· C(®;½)
1
X
s=0
("
2
s
¡"
2
s+1
)
"
s
X
m=0
2
m(®+½)
s
X
º=m
2
¡º(®+½)

2
º
¡¸
2
º¡1
)
+
s
X
m=0
2
m(®+½)
1
X
º=s+1
2
¡º(®+½)

2
º
¡¸
2
º¡1
)
#
· C(®;½)
1
X
º=0
("
º
¡"
º+1
) ¸
º
:(60)
Let p = 1.The series
1
X
º=1
("
º
¡"
º+1
) ¿
º
(x) (61)
28 B.SIMONOV AND S.TIKHONOV
converges ([Ge]) to a f
8
2 L
1
,and E
n
(f
8
)
1
· C"
n+1
.Then f
8
2 H
p
®+½
[!] ½
W
¸;¯
p
.We rewrite (61) in the following way
1
X
º=1
b
º
sinºx;where
b
º
=
2º¡2
X
j=º
µ
1 ¡
º
j +1

("
º
¡"
º+1
) +
1
X
j=2º¡1
º
j +1
("
º
¡"
º+1
):
By Lemma 4.1,we write
°
°
°
f
(¸;¯)
8
°
°
°
1
¸ C(¯)
1
X
º=1
¸
º
b
º
º
¸ C(¯)
1
X
º=0
¸
º
("
º
¡"
º+1
):
This contradicts the divergence of the series in (60).Thus,the series in (14)
converges.
Let p = 1.Let à be a Q
®+½;1
-sequence.Then,we de¯ne
f
9
(x) = Ã
1
cos x +
1
X
º=1
2
¡º(½+®)

2
º
¡Ã
2
º¡1
) cos 2
º
x:
Similarly,as for f
1
in the case 2 · p < 1;it is easy to see that f
9
2
H
p
®+½
[!] ½ W
¸;¯
p
.By Lemma 4.3,we have
°
°
°
f
(¸;¯)
9
°
°
°
1
¸ C(¯)
Ã
¸
1
Ã
1
+
1
X
º=1
¸
2
º
2
¡º(½+®)

2
º
¡Ã
2
º¡1
)
!
¸ C(¯)
1
X
º=1
¸
º
º
¡(½+®)¡1
Ã
º
¸ C(¯)
1
X
º
=1

º+1
¡¸
º
)!(1=º) = 1;
this contradicts f
9
2 W
¸;¯
p
.This completes the proof of the necessity in
(14).
Step 7.We will prove the necessity in (15) for the case sin
¼¯
2
6= 0.Let
H
p
®+r
[!] ½ W
¸;¯
p
H
®
['].Let"be a q
®+r;1
(!)-sequence.So,(41) holds for
® +r instead of ® and 1 instead of µ.Since sin
¼¯
2
6= 0,
J:= ¸
n+1
!
µ
1
n +1

+n
¡®
n
X
º=1
º
r+®
¡
º
¡r
¸
º
¡(º +1)
¡r
¸
º+1
¢
!
µ
1
º

+ j cos
¯¼
2
j
1
X
º=n+2

º+1
¡¸
º
)!
µ
1
º

+j sin
¯¼
2
j
1
X
º=n+2
¸
º
!
¡
1
º
¢
º
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 29
·C(®;¯;r)
Ã
1
X
º=n+1
¸
º
"
º
º
+n
¡®
n
X
º=1
¸
º
"
º
º
®¡1
!
=:C(®;¯;r) (J
1
+J
2
):(62)
We will use several times the following evident inequalities:
!
®
³
f;
1
n
´
p
³!
®
³
f
+
;
1
n
´
p
+!
®
³
f
¡
;
1
n
´
p
;where f
§
(x):=
f(x) §f(¡x)
2
:
(63)
Step 7(a):p = 1and cos
¼®
2
6= 0.Then by Lemma 3.1,we write
³
f
(¸;¯)
7+
:=
¡
f
(¸;¯)
7
¢
+
´
!
®
³
f
(¸;¯)
7
;
1
n
´
p
¸!
®
³
f
(¸;¯)
7+
;
1
n
´
p
¸ C(®)n
¡®
°
°
°
V
(®)
n
(f
(¸;¯)
7+
)(¢)
°
°
°
p
¸ C(®)n
¡®
¯
¯
¯
V
(®)
n
(f
(¸;¯)
7+
)(0)
¯
¯
¯
p
¸ C(®;¯)J
2
:(64)
By (20) and
1
P
k=2n
a
k
· 4E
n
(f)
1
(see [Ba,1]),we have
!
®
³
f
(¸;¯)
7
;
1
n
´
p
¸ C(®)E
[
n
2
]
(f
(¸;¯)
7+
)
p
¸ C(®;¯)J
1
:(65)
We have already proved that f
7
2 H
p
®+r
[!] ½ W
¸;¯
p
H
®
[']:Collecting in-
equalities (64),(65),and (62),we arrive at the right-hand side of (15).
Step 7(b):p = 1and cos
¼®
2
= 0.If cos
¼¯
2
6= 0;then we use (65) and
!
®
³
f
(¸;¯)
7
;
1
n
´
p
¸ C(®)n
¡®
°
°
°
V
(®)
n
(f
(¸;¯)

)(¢)
°
°
°
p
¸ C(®;¯)J
2
:
If cos
¼¯
2
= 0;then f
7
= §f
7+
and
!
®
³
f
(¸;¯)
7
;
1
n
´
p
¸ C(®)E
[
n
2
]
(f
(¸;¯)
7+
)
p
¸ C(®;¯)J
1
:
To obtain the estimate of J
2
,we de¯ne
f
10
(x) =
"
0
2
+("
1
¡"
2
) cos x +
1
X
º=1
("
2
º
¡"
2
º+1
) cos 2
º
x:
It is clear that E
n
(f
10
)
p
·"
n+1
.Then,by (41),we have f
10
2 H
p
®+r
[!] ½
W
¸;¯
p
H
®
['].Using!
®
¡
f;
1
n
¢
p
¸ C(®)n
¡®
°
°
°
V
(®)
n
(f)
°
°
°
p
and Lemma 4.3,we
write (2
m
· n +1 < 2
m+1
)
!
®
µ
f
(¸;¯)
10
;
1
2
m

p
¸
30 B.SIMONOV AND S.TIKHONOV
¸ C(®;¯)2
¡m®
m
X
º=0
¸
2
º
2
º®
("
2
º
¡"
2
º+1
)
¸ C
1
(®;¯;r)2
¡m®
m
X
º=0
¸
2
º
2
º®
"
2
º
¡C
2
(®;¯;r)¸
2
m+1
"
2
m+1
:(66)
Concurrently,!
®
³
f
(¸;¯)
10
;
1
2
m
´
p
¸ C(®;¯)¸
2
m+1
"
2
m+1
:Then,we get
!
®
µ
f
(¸;¯)
10
;
1
n

p
¸ C(®;¯;r)J
2
;(67)
and by (63),
C(®;¯;r)
h
J
1
+J
2
i
·!
®
µ
f
(¸;¯)
10
;
1
2
m

p
+!
®
µ
f
(¸;¯)
7
;
1
2
m

p
³!
®
µ
(f
7
+f
10
)
(¸;¯)
;
1
n

p
= O
·
'
µ
1
n
¶¸
:
The necessity in (15) follows.
Step 7(c):p = 1 and cos
¼®
2
6= 0.We use the function f
6
.We have f
6
2
H
p
®+r
[!] and by Lemma 4.1,
!
®
³
f
(¸;¯)
6
;
1
n
´
1
¸ C(®)E
n
(f
(¸;¯)

)
p
¸ C(®;¯)
1
X
º=n+1
¸
º
a
º
º
¸ C(®;¯)
Ã
1
X
º=n+1
¸
º
"
º
º
¡a
n+1
¸
n
¡
1
X
º=n+1

º
¡¸
º¡1
) a
º
!
:
On the other hand,
!
®
³
f
(¸;¯)
6
;
1
n
´
1
¸ C(®;¯;r)
Ã
a
n+1
¸
n
+
1
X
º=n+1

º
¡¸
º¡1
) a
º
!
:
The last two inequalities imply!
®
¡
f
(¸;¯)
6
;
1
n
¢
1
¸ C(®;¯;r)J
1
.Also,
!
®
³
f
(¸;¯)
6
;
1
n
´
p
¸ C(®)n
¡®
°
°
°
V
(®)
n
(f
(¸;¯)

)(¢)
°
°
°
p
¸ C(®;¯)J
2
:(68)
Step 7(d):p = 1 and cos
¼®
2
= 0.If cos
¼¯
2
6= 0;then we use!
®
¡
f
(¸;¯)
6
;
1
n
¢
1
¸
C(®;¯;r)J
1
and
!
®
³
f
(¸;¯)
6
;
1
n
´
p
¸ C(®)n
¡®
°
°
°
V
(®)
n
(f
(¸;¯)
6+
)(¢)
°
°
°
p
¸ C(®;¯)J
2
:
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 31
If cos
¼¯
2
= 0;we consider f
6
+f
8
.Using Lemmas 3.1 and 4.1,we have
!
®
µ
f
(¸;¯)
8
;
1
n

1
¸ C(®;¯)n
¡®
n
X
º=1
¸
º
º
®¡1
b
º
¸ C
1
(®;¯;r)n
¡®
n
X
º=1
¸
º
º
®¡1
"
º¡1
¡C
2
(®;¯;r)¸
n
"
n
:
Since!
®
³
f
(¸;¯)
8
;
1
n
´
1
¸ C(®;¯;r)¸
n
"
n
,then!
®
³
f
(¸;¯)
8
;
1
n
´
1
¸ C(®;¯;r)J
2
.
Thus
C(®;¯;r)
h
J
1
+J
2
i
·!
®
µ
(f
6
+f
8
)
(¸;¯)
;
1
n

p
= O
·
'
µ
1
n
¶¸
;
i.e.,the necessity in (15) follows.
Step 8.We will prove the necessity in (15) for the case sin
¼¯
2
= 0.Let
H
p
®+r
[!] ½ W
¸;¯
p
H
®
['] and"be a q
®+r;1
(!)-sequence.Since sin
¼¯
2
= 0,we
have from (41)
¸
n+1
!
µ
1
n +1

+ n
¡®
n
X
º=1
º
r+®
¡
º
¡r
¸
º
¡(º +1)
¡r
¸
º+1
¢
!
µ
1
º

+ j cos
¯¼
2
j
1
X
º=n+2

º+1
¡¸
º
)!
µ
1
º

· C(®;¯;r)
Ã
1
X
º=n+1
¸
º
("
º
¡"
º+1
)+n
¡®
n
X
º=1
¸
º
"
º
º
®¡1
!
=:C(®;¯;r) (J
3
+J
4
):(69)
Step 8(a):p = 1 and cos
¼®
2
6= 0.By the Jackson inequality and Lemma
4.3,we have
!
®
µ
f
(¸;¯)
10
;
1
2
m

p
¸ C(®;¯)
1
X
º=m
¸
2
º
("
2
º
¡"
2
º+1
):(70)
We also note that by Lemma (4.3),(67) holds for all ® > 0.This and
(70) give!
®
³
f
(¸;¯)
10
;
1
n+1
´
p
¸ C(®;¯) (J
3
+J
4
).Using condition (69) and
f
10
2 H
p
®+r
[!] ½ W
¸;¯
p
H
®
['],we arrive at the relation in the right part of
(15).
Step 8(b):p = 1 and cos
¼®
2
= 0.Then we consider f
10
and f
11
:=
f
f
10
.It is clear that f
11
2 L
p
and f
10
+ f
11
2 H
p
®+r
[!].Nevertheless,
!
®
³
f
(¸;¯)
10
;
1
n+1
´
p
¸ C(®;¯)J
3
,!
®
³
f
(¸;¯)
11
;
1
n+1
´
p
¸ C(®;¯)J
4
,and
32 B.SIMONOV AND S.TIKHONOV
!
®
³
f
(¸;¯)
10
;
1
n+1
´
p
+!
®
³
f
(¸;¯)
11
;
1
n+1
´
p
³!
®
³
(f
10
+f
11
)
(¸;¯)
;
1
n+1
´
p
.
Step 8(c):p = 1 and cos
¼®
2
6= 0.Since f
(¸;¯)
8
(x) » §
1
P
º=1
¸
º
b
º
sinºx;by
Lemmas 3.1 and 4.1,we write (see Step 7(d))
!
®
³
f
(¸;¯)
8
;
1
n
´
p
¸ C(®)n
¡®
°
°
°
V
(®)
n
(f
(¸;¯)

)(¢)
°
°
°
p
¸ C(®;¯)J
4
:(71)
Also,by Lemma 4.1 and the Jackson inequality (20),we have
!
®
µ
f
(¸;¯)
8
;
1
n

1
¸ C(®;¯)
1
X
º=n+1
¸
º
b
º
º
¸ C(®;¯)
1
X
º=n+1
¸
º
1
X
j=2º¡1
"
j
¡"
j+1
j +1
¸ C(®;¯;r)
1
X
j=4n¡1
("
j
¡"
j+1
) ¸
º
:
Using the properties of modulus of smoothness,we get (15).
Step 8(d):p = 1 and cos
¼®
2
= 0.We use that f
12
:=
e
f
8
2 L
1
and E
n
(f
12
)
p
·
C"
n+1
(see [Ge]).Then f
8
+f
12
2 H
p
®+r
[!] and!
®
³
f
(¸;¯)
8
;
1
n
´
1
¸ C(®;¯)J
3
,
!
®
³
f
(¸;¯)
12
;
1
n
´
1
¸ C(®;¯)J
4
.This completes the proof of Theorem 1.
5.Corollaries.Estimates of transformed Fourier series.
Theorem 1 actually provides estimates of the norms and moduli of
smoothness of the transformed Fourier series,i.e.,the estimates of k'k
p
and
!
®
(';±)
p
,where'» ¾(f;¸),by!
°
(f;±)
p
.Analyzing Theorem 1,one can
see that the following two conditions play a crucial role.The ¯rst is the be-
havior of the transforming sequence f¸
n
g and the second is the alternation
between L
p
;1 < p < 1 and L
p
;p = 1;1.
We will investigate in detail some important examples for L
p
;1 < p < 1
and for L
p
;p = 1;1,separately.
1.The case 1 < p < 1.
Theorem 2.
Let 1 < p < 1,µ = min(2;p) ¿ = max(2;p),® 2 R
+
;and
¸ = f¸
n
g be a non-decreasing sequence of positive numbers.Let ½ be a
non-negative number such that the sequence fn
¡½
¸
n
g is non-increasing.
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 33
I.If for f 2 L
0
p
the series
1
X
n=1
¡
¸
µ
n+1
¡¸
µ
n
¢
!
µ
®+½
µ
f;
1
n

p
converges,then there exists a function'2 L
0
p
with Fourier series ¾(f;¸),
and
k'k
p
·
· C(p;¸;®;½)
(
¸
µ
1
kfk
µ
p
+
1
X
n=1
¡
¸
µ
n+1
¡¸
µ
n
¢
!
µ
®+½
µ
f;
1
n

p
)
1
µ
;(72)
!
®
µ
';
1
n +1

p
·
· C(p;¸;®;½)
(
n
¡®µ
n
X
º=1
º
(½+®)µ
¡
º
¡½µ
¸
µ
º
¡(º +1)
¡½µ
¸
µ
º+1
¢
!
µ
®+½
µ
f;
1
º

p
+
1
X
º=n+2
¡
¸
µ
n+1
¡¸
µ
n
¢
!
µ
®+½
µ
f;
1
º

p
+ ¸
µ
n+1
!
µ
®+½
µ
f;
1
n

p
)
1
µ
:(73)
II.If for f 2 L
0
p
there exists a function'2 L
p
with Fourier series ¾(f;¸),
then
(
¸
¿
1
kfk
¿
p
+
1
X
n=1
¡
¸
¿
n+1
¡¸
¿
n
¢
!
¿
®+½
µ
f;
1
n

p
)
1
¿
· C(p;¸;®;½)k'k
p
;(74)
(
n
¡®¿
n
X
º=1
º
(½+®)¿
¡
º
¡½¿
¸
¿
º
¡(º +1)
¡½¿
¸
¿
º+1
¢
!
¿
®+½
µ
f;
1
º

p
+ (75)
1
X
º=n+2
¡
¸
¿
n+1
¡¸
¿
n
¢
!
¿
®+½
µ
f;
1
º

p
+ ¸
¿
n+1
!
¿
®+½
µ
f;
1
n

p
)
1
¿
· C(p;¸;®;½)!
®
µ
';
1
n +1

p
;
!
®+½
µ
f;
1
n

p
· C(p;¸;®;½)
k'k
p
¸
n
;(76)
!
®+½
µ
f;
1
n

p
· C(p;¸;®;½)
!
®
¡
';
1
n
¢
p
¸
n
:(77)
34 B.SIMONOV AND S.TIKHONOV
Inequalities (72)-(73) and (76)-(77) were proved in Theorem 1 (see the suf-
¯ciency in part I).To prove (74)-(75),we use similar reasoning,applied
to the theorems by Littlewood-Paley,Marcinkiewicz,and the Minkowski's
inequality.
As an important corollary of Theorem 2,
Corollary 1.
Let 1 < p < 1,µ = min(2;p) ¿ = max(2;p).Then,for any
k;r > 0,
C
1
(
1
X
º=n+1
º
r¿¡1
!
¿
k+r
µ
f;
1
º

p
)
1
¿
·!
k
µ
f
(r)
;
1
n

p
(78)
· C
2
(
1
X
º=n+1
º
rµ¡1
!
µ
k+r
µ
f;
1
º

p
)
1
µ
;
where C
1
= C
1
(p;k;r);C
2
= C
2
(p;k;r);n 2 N:
The last two inequalities are an improvement compared to (6) and (7).
Indeed,by properties of the modulus of smoothness and by the Jensen
inequality,
n
r
!
k+r
µ
f;
1
n

p
· C(k;r)
(
1
X
º=n+1
º
r¿¡1
!
¿
k+r
µ
f;
1
º

p
)
1
¿
;
(
1
X
º=n+1
º
rµ¡1
!
µ
k+r
µ
f;
1
º

p
)
1
µ
· C(k;r)
1
X
º=n+1
º
r¡1
!
k+r
µ
f;
1
º

p
:
Example.Let Ã(t) = t
r
ln
¡A
(1=t) and 2 · p < 1;
1
2
< A < 1:For
!
k+r
(f;t)
p
³ Ã(t),(6) and (7) give only Cln
¡A
(1=t) ·!
k
¡
f
(r)
;t
¢
p
,and
(78) gives C
1
ln
¡A+1=p
(1=t) ·!
k
¡
f
(r)
;t
¢
p
· C
2
ln
¡A+1=2
(1=t),which is
superior.
Proof of Corollary 1 is obvious from (73) and (75) with r = ½,because if
f 2 L
p
;1 < p < 1,then f
(r)
» ¾(f;¸) for f¸
n
= n
r
g.
From (6),(7) and (78),one can see that it is natural to estimate
!
®
¡
f
(°)

¢
p
by!
®+r
(f;t)
p
.Further analysis allows the distinction be-
tween three di®erent types of such estimates.It will be convenient for us to
write inequalities in integral form:
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 35
1.° = r (see Corollary 1)
8
8
<
:
±
Z
0
t
¡r¿¡1
!
¿
r+®
(f;t)
p
dt
9
=
;
1
¿
¿!
®
(f
(r)
;±)
p
¿
8
<
:
±
Z
0
t
¡rµ¡1
!
µ
r+®
(f;t)
p
dt
9
=
;
1
µ
;
(79)
2.° = r ¡";0 <"< r (see Theorem 2 for ½ = r and ¸
n
= n
r¡"
):
8
<
:
±
Z
0
t
¡(r¡")¿¡1
!
¿
r+®
(f;t)
p
dt+±
®¿
1
Z
±
t
¡(r¡"+®)¿¡1
!
¿
r+®
(f;t)
p
dt
9
=
;
1
¿
¿
¿!
®
(f
(r¡")
;±)
p
;(80)
!
®
(f
(r¡")
;±)
p
¿
¿
8
<
:
±
Z
0
t
¡(r¡")µ¡1
!
µ
r+®
(f;t)
p
dt+±
®µ
1
Z
±
t
¡(r¡"+®)µ¡1
!
µ
r+®
(f;t)
p
dt
9
=
;
1
µ
;(81)
3.° = r +";0 <"< ®;(see [Po-Si-Ti,1]):
8
<
:
±
Z
0
t
¡(r+")¿¡1
!
¿
r+®
(f;t)
p
dt
9
=
;
1
¿
¿±
®¡"
8
<
:
1
Z
±
t
¡(®¡")µ¡1
!
µ
®
(f
(r+")
;t)
p
dt
9
=
;
1
µ
;
(82)
±
®¡"
8
<
:
1
Z
±
t
¡(®¡")¿¡1
!
¿
®
(f
(r+")
;t)
p
dt
9
=
;
1
¿
¿
8
<
:
±
Z
0
t
¡(r+")µ¡1
!
µ
r+®
(f;t)
p
dt
9
=
;
1
µ
:
(83)
One can consider a more general sequence than f¸
n
= n
°
g,° > 0.Let us
de¯ne f¤
n
(s):= ¤(s;
1
n
)g,where
¤(s;t) = ¤(s;r;t) =
0
@
1
Z
t
»(u)du +t
¡rs
t
Z
0
u
rs
»(u)du
1
A
1
s
;(84)
and non-negative function »(u) on [0;1] is such that u
rs
»(u) is summable.
The sequence f¤
n
= ¤
n
(s)g is often considered a transforming sequence.
Example.1.f¤
n
= n
°
g,0 < ° < r;2.f¤
n
= n
r
ln
¡A
ng,A > 0;3.

n
= ln
A
ng,A > 0.
8
Here and further ¿ = max(2;p);µ = min(2;p).We will write A
1
¿A
2
,if A
1
· CA
2
,
C ¸ 1.Also,if A
1
¿A
2
and A
2
¿A
1
,then A
1
³ A
2
.
36 B.SIMONOV AND S.TIKHONOV
Properties of ¤(s;t):¤(s;t) is non-increasing on t,¤(s;r;t)t
r
is non-
decreasing on t.By these properties and Theorem 2,one can obtain the fol-
lowing estimates,which are more general than (80){(81) (see also [Po-Si,1]).
Theorem 3.
Let 1 < p < 1,µ = min(2;p) ¿ = max(2;p),®;r > 0,and
¤ = f¤
n
(s):= ¤(s;
1
n
)g,where ¤(s;t) and »(t) were denoted above.
I.If for f 2 L
p
the integral
I
1
:=
8
<
:
1
Z
0
»(t)!
s
®+r
(f;t)
p
dt
9
=
;
1
s
is ¯nite for s = µ,then there exists a function'2 L
p
with Fourier series
¾(f;¤),k'k
p
· C(p;»;®;r;s)I
1
,and
!
®
(';±)
p
· C(p;»;®;r;s)
8
<
:
±
Z
0
t
¡rs¡1
t
Z
0
u
rs
»(u) du!
s
r+®
(f;t) dt
+ ±
®s
1
Z
±
t
®s¡1
1
Z
t
»(u) du!
s
r+®
(f;t)
p
dt
9
=
;
1
s
=:C I
2
:(85)
II.If for f 2 L
p
there exists a function'2 L
p
with Fourier series ¾(f;¤),
then for s = ¿ k'k
p
¸ C(p;»;®;r;s)I
1
,and!
®
(';±)
p
¸ C(p;»;®;r;s)I
2
:
In a similar way we can generalize inequalities (82){(83).
Theorem 4.
Let 1 < p < 1,µ = min(2;p) ¿ = max(2;p),®;r > 0,and
¤ = f¤
n
(s):= ¤(s;
1
n
)g,where ¤(s;t) and »(t) were denoted above.
I.If for f 2 L
p
the integral
I
3
:=
8
<
:
1
Z
0
»(t)!
s
®+r
(f;t)
p
dt
9
=
;
1
s
is ¯nite for s = µ,then there exists a function'2 L
p
with Fourier series
¾(f;¤),k'k
p
· C(p;»;®;r;s)I
3
,and for any ®
1
> ®
I
4
(¿;s) =:
=:
8
<
:
±
(r+®)¿
¤
¿
(s;r;±)
1
Z
±
t
¡(r+®)¿¡1
¤
¿
(s;r;t)
!
¿
r+®
1
(';t)
p
dt
9
=
;
1
¿
· C(p;»;®;®
1
;r;s)
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 37
£
2
6
4
¤(s;r;±)!
r+®
(f;±)
p
+
8
<
:
±
Z
0
t
¡rµ¡1
t
Z
0
u
r¿
»(u)du!
µ
r+®
(f;t)
p
dt
9
=
;
1
µ
3
7
5
=:CI
5
(µ) (86)
II.If for f 2 L
p
there exists a function'2 L
p
with Fourier series
¾(f;¤),then for s = ¿ and for any ®
1
> ® k'k
p
¸ C(p;¸;®;r;s)I
3
and
I
4
(s;µ) ¸ C(p;»;®;®
1
;r;s)I
5
(s).
2.The case p = 1;1.
Estimates!
®
(';t)
p
by!
r+®
(f;t)
p
for this case follow from Theorem 1 (see
part II).We will write only the commonly used estimates of!
®
(f
(r)
;t)
p
and!
®
(
~
f
(r)
;t)
p
by!
r+®
(f;t)
p
.
Corollary 2.
If p = 1;1,then (6),(7) are true for any k;r > 0:
As a corollary of Theorem 1 for f¸
n
= n
½
g,½ ¸ 0 and ¯ = ½ +1,
Corollary 3.
Let p = 1;1.One has
H
p
®+½
[!] ½
f
W
½
p
()
1
X
n=1
n
½¡1
!
µ
1
n

< 1:
For p = 1;½ = 0 and ® = 1 Corollary 3 gives the answer for the question
by F.Moricz [Mo].Also,we mention the papers [Be,S,1],[Ha-Sh],[St,S],
where the embedding results were obtained in the necessity part.
Corollary 4.
Let p = 1;1,and r;®;"> 0.
I.If for f 2 L
p
the series
1
P
º=1
º
r¡1
!
r+®+"
¡
f;
1
º
¢
p
converges,then there
exists
~
f
(r)
2 L
p
,and
!
®
³
e
f
(r)
;
1
n
´
p
·
· C(r;®;")
Ã
n
¡®
n
X
º=1
º
r+®¡1
!
r+®+"
³
f;
1
º
´
p
+
1
X
º=n+1
º
r¡1
!
r+®+"
³
f;
1
º
´
p
!
;
n 2 N:
II.If for f 2 L
p
there exists
~
f
(r)
2 L
p
,then
!
r+®+"
µ
f;
1
n

p
·
C(r;®;")
n
r
!
®
µ
~
f
(r)
;
1
n

p
;n 2 N:(87)
Using direct and inverse approximation theorems,we can rewrite inequal-
ity in I in the following equivalent form (compare to [Ba-St]):
38 B.SIMONOV AND S.TIKHONOV
Corollary 5.
Let p = 1;1,and r;® > 0.If for f 2 L
p
the series
1
P
º=1
º
r¡1
E
º
(f)
p
converges,then there exists
~
f
(r)
2 L
p
,and
!
®
³
e
f
(r)
;
1
n
´
p
· C(r;®;")
Ã
n
¡®
n
X
º=1
º
r+®¡1
E
º
(f)
p
+
1
X
º=n+1
º
r¡1
E
º
(f)
p
!
;
n 2 N:
6.Final remarks
1.The Weyl class W
¸;¯
p
coincides with the class of functions fromL(0;2¼)
such that their Fourier series can be presented in the following form
a
0
(f)
2
+
1
X
º=1
1
¼¸
º

Z
0
Ã(x ¡t) cos
³
ºt ¡
¼¯
2
´
dt;Ã(x) 2 L
0
:
Further,consider the case when
1
X
º=1
1
¸
º
cos
³
ºt ¡
¼¯
2
´
is the Fourier series of a summable function D
¸;¯
(t).For example,it is
so if f¸
º
"1g (n"),and
1
P
º=1
1
º¸
º
< 1.Then the elements of W
¸;¯
p
can
di®er only by the mean value from functions f,which have the following
representation by convolution,
f(x) =
1
¼

Z
0
Ã(x ¡t)D
¸;¯
(t) dt;Ã(x) 2 L
0
:
Here,Ã coincides almost everywhere with f
(¸;¯)
:See,for example,[Bu-Ne,
Ch.11].A representation by convolution was ¯rst considered in [Sz].
2.The generalization of W
r
p
E[»] is the class
W
¸;¯
p
E[»] =
n
f 2 W
¸;¯
p
:E
n
³
f
(¸;¯)
´
p
= O[»(1=n)]
o
:
We do not consider in this paper the embedding results between W
¸;¯
p
E[»],
W
¸;¯
p
H
®
['] and E['] ´ W
f1g;0
p
E['].We only notice that some results of
such types followfromdirect and inverse theorems (1)-(4).Also,we mention
the papers [Ha-Li],[Og],[Si-Ti],and [St,A].
3.The results of sections 2 and 5 can be generalized to a multidimensional
case.We only write here the following estimates for the mixed modulus of
smoothness!
®
1

2
(f;±
1

2
)
p
of a function f (in the L
p
metric) of orders ®
1
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 39
and ®
2

1

2
> 0) with respect to the variables x
1
and x
2
,respectively
(see,for example,[Po-Si-Ti,1]).
Theorem 5.
Let f(x
1
;x
2
) 2 L
0
p
,1 < p < 1,µ = min(2;p);¿ = max(2;p),
and let ®
1

2
,r
1
;r
2
> 0,
I.If
J
1
(µ):=
0
@
1
Z
0
1
Z
0
t
¡r
1
µ¡1
1
t
¡r
2
µ¡1
2
!
µ
r
1

1
;r
2

2
(f;t
1
;t
2
)
p
dt
1
dt
2
1
A
1
µ
< 1;
then f has a mixed derivative in the sense of Weyl f
(r
1
;r
2
)
2 L
0
p
,
kf
(r
1
;r
2
)
k
p
· C(p;r
1
;r
2
)J
1
(µ);
and
!
®
1

2
(f
(r
1
;r
2
)

1

2
)
p
·
· C(p;®
1

2
;r
1
;r
2
)
0
@
±
1
Z
0
±
2
Z
0
t
¡r
1
µ¡1
1
t
¡r
2
µ¡1
2
!
µ
r
1

1
;r
2

2
(f;t
1
;t
2
)
p
dt
1
dt
2
1
A
1
µ
=:CJ
2
(µ):
II.If f has a mixed derivative in the sense of Weyl f
(r
1
;r
2
)
2 L
0
p
,then
J
1
(¿) · C(p;r
1
;r
2
)kf
(r
1
;r
2
)
k
p
;
and
J
2
(¿) · C(p;®
1

2
;r
1
;r
2
)!
®
1

2
(f
(r
1
;r
2
)

1

2
)
p
:
Theorem 6.
Let f(x
1
;x
2
) 2 L
0
p
,p = 1;1,and let ®
1

2
;r
1
;r
2
> 0.
I.If
J
3
:=
1
Z
0
1
Z
0
t
¡r
1
¡1
1
t
¡r
2
¡1
2
!
r
1

1
;r
2

2
(f;t
1
;t
2
)
p
dt
1
dt
2
< 1;
then f has a mixed derivative in the sense of Weyl f
(r
1
;r
2
)
2 L
0
p
,
kf
(r
1
;r
2
)
k
p
· C(r
1
;r
2
)J
3
;
and
!
®
1

2
(f
(r
1
;r
2
)

1

2
)
p
·
· C(®
1

2
;r
1
;r
2
)
±
1
Z
0
±
2
Z
0
t
¡r
1
¡1
1
t
¡r
2
¡1
2
!
r
1

1
;r
2

2
(f;t
1
;t
2
)
p
dt
1
dt
2
:
40 B.SIMONOV AND S.TIKHONOV
II.If f has a mixed derivative in the sense of Weyl f
(r
1
;r
2
)
2 L
0
p
,then
!
r
1

1
;r
2

2
(f;±
1

2
)
p
· C(®
1

2
;r
1
;r
2

r
1
1
±
r
2
2
kf
(r
1
;r
2
)
k
p
;
and
!
r
1

1
;r
2

2
(f;±
1

2
)
p
· C(®
1

2
;r
1
;r
2

r
1
1
±
r
2
2
!
®
1

2
(f
(r
1
;r
2
)

1

2
)
p
:
See [Po-Si-Ti,1] for the estimates of transformed series in a multidimen-
sional case.
4.In view of inequalities (79) and (80) - (83),the problem of ¯nding the
estimates of!
®
(';t)
p
by!
®+r
(f;t)
p
arises such as in the case'» ¾(f;¸)
with ¸
n
= n
r
ln
A
n.If A < 0 (that is the analogue of the case ¸
n
= n
r¡"
),
then estimates!
®
(';t)
p
follow from Theorem 3.For example,if p = 2 and
'» ¾(f;n
r
ln
A
n);A < 0,then
!
2
¯
(';±)
2
³
±
Z
0
t
¡2r¡1
ln
2jAj
¡
2
t
¢
!
2
r+¯
(f;t)
2
dt +±

1
Z
±
t
¡2(r+¯)¡1
ln
1+2jAj
¡
2
t
¢
!
2
r+¯
(f;t)
2
dt:
(88)
Note that the only di®erence between (88) and (80)-(81) is related to the
replacement of n
¡"
by ln
A
n.
The case A > 0 (that is the analogue of the case ¸
n
= n
r+"
) is interesting.
For p = 2 and'» ¾(f;n
r
ln
A
n);A > 0 we have
±

ln
2A
µ
2
±

1
Z
±
t
¡2¯¡1
ln
1+2A
¡
2
t
¢
!
2
¯
(';t)
2
dt +!
2
¯
(';±)
2
³
³
±
Z
0
t
¡2r¡1
ln
2A
µ
2
t

!
2
r+¯
(f;t)
2
dt:(89)
Comparing this with the estimates (82)-(83) one can remark that the new
item!
2
¯
(';±)
2
has appeared in (89).Thus,this case has essential distinc-
tions.See for detail [Po-Si-Ti,2],[Ti,S,1].
5.De¯ning the class W
¸;¯
p
H
®
['] we assumed that'2 ©
®
.This restric-
tion is natural for a majorant of the modulus of smoothness of order ® (see
[Ti,S,3]).
6.In Theorem 1 (item II) we used the inequality
1
P
º=n+1
1
º¸
º
·
C
¸
n
.We re-
mark that it is equivalent to the following condition:there exists"> 0 such
that the sequence fn
¡"
¸
n
g is almost increasing,i.e.,n
¡"
¸
n
· Cm
¡"
¸
m
,
C ¸ 1,n · m:This and other equivalence results can be found in [Ba-St]
and [Ti,S,2].
EMBEDDING THEOREMS OF FUNCTION CLASSES,III 41
7.Acknowledgements
The ¯rst named author acknowledges ¯nancial support from Russian
Fund of Basic Research (project 03-01-00080).The second named author
acknowledges ¯nancial support from the European Commission (Contract
MIF1-CT-2004-509465).The paper was written while they were staying
at the Centre de Recerca Matematica (CRM) in Barcelona,Spain.The
authors are grateful to the sta® of CRM and in particular to its director,
Manuel Castellet,for their hospitality.
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BORIS SIMONOV
VOLGOGRAD STATE TECHNICAL UNIVERSITY
VOLGOGRAD,400131
RUSSIA
e-mail:htf@vstu.ru
SERGEY TIKHONOV
CENTRE DE RECERCA MATEMATICA (CRM)
BELLATERRA (BARCELONA) E-08193,SPAIN
e-mail:stikhonov@crm.es