Proceedings of the Steklov Institute of Mathematics, Vol. 227, 1999, pp. 50{69.
Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, Vol. 227, 1999, pp. 56{74.
Original Russian Text Copyright c 1999 by Besov.
English Translation Copyright c 1999 by maik \nAUKA/Interperiodica" (Russia).
On Spaces of Functions of Variable Smoothness
De ned by Pseudodierential Operators
1
O. V. Besov
Received January 1999
The function spaces Bps qa (Rn ) normed by the pseudodierential operator (PDO) a(x D) and
the series expansion that generalizes the Littlewood{Paley expansion were introduced and studied
by H.-G. Leopold 1{4]. These spaces are the interpolation spaces between the Lebesgue Lp(Rn )
and Sobolev Wpa (Rn ) spaces of variable order. In addition, they represent a generalization of the
spaces Bps q (Rn ) in which the norms can be dened both by the properties of the Littlewood{Paley
series expansions and (for s > 0) by the behavior of the Lp -module of smoothness of the functions
(see 14]). In 1], H.-G. Leopold established an equivalent normalization of the space Bps qa(Rn )
(where a is a hypoelliptic PDO of a certain class) by the Lp -norm of the dierence of a function
with variable step (i.e., with the step depending on the point x 2 Rn ).
In this paper, we establish another equivalent normalization of the spaces Bps qa(Rn ), s > 0.
When q = p, this normalization is based on the dierences of functions with the steps independent
of a point x 2 Rn that are estimated in the weight Lp -norm with the weight dened by the form of
the PDO a(x D). This demonstrates that the spaces Bps qa(Rn ) under consideration coincide with
the corresponding spaces of functions of variable smoothness that were introduced without any
relation to the PDOs (see 5, 6]). When q 6= p, the new normalization is based on the dierences of
a function with a piecewise constant step. The proofs widely employ the methods of 1{3]. At the
end of the paper, we study the interpolation properties of the spaces Bps pa (Rn ), s > 0, and establish
the embedding theorems.
A brief account of this work was presented in 16].
1. PSEUDODIFFERENTIAL OPERATORS
We begin with certain denitions and notations borrowed from 2, 3]. Below, we assume that
1 < p < 1, Lp = Lp (Rn ), and kukp = kujLp k.
Let p(x ) be a complex-valued polynomially bounded function dened on Rn Rn that is
innitely dierentiable with respect to x and . A PDO P (x D) with the symbol p(x ) is dened
by the formula
Z
P (x D)u(x) = (2)qn eix p(x )(Fu)() d
u 2 S (Rn )
P
where S (Rn ) is a Schwarz space of test functions, Fu isPthe Fourier transform of u, x = n1 xi ei ,
ei are the unit vectors of the standard basis, and x = n1 xi i.
1
Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 117966 Russia.
E-mail: [email protected]
50
ON SPACES OF FUNCTIONS OF VARIABLE SMOOTHNESS
51
Let h i = (1 + j j2 ) 12 , 2 N n0 , p(()) (x ) = @ Dx p(x ), @ = @ 1 1@:::@ n n , and Dx =
(yi)jj @x .
Henceforth, we assume that the number , 0 < 1, is xed and always the same.
De nition 1.1. The function p(x ) is said to belong to the class S1m , y1 < m < 1, 0 < 1, if, for any 2 N n0 , there exist constants c such that
j
p(()) (x ) c h imqjj+jj
j
for (x ) 2 Rn Rn :
The relation p 2 S1m is equivalent to the niteness of the norms
)
jpj((m
l k) =
max
sup p(()) (x ) h iqm+jjqjj
(1:1)
jj l jj k (x )
for all (l k) 2 N 0 N 0 .
We denote the symbol p(x ) of the PDO P (x D) also by P (x D)]D! .
Theorem 1.1 7]. Let N 2 N 0 and (L K ) 2 N 0 N 0 . Then, there exist a pair (L0 K 0) 2
N N and a constant c > 0 such that, for p 2 S1m1 and q 2 S1m2
P (x D)Q(x D)]D! =
in addition,
X 1 ()
! p (x )q() (x ) + rN (x )
(1:2)
jj<N
(m2 )
1 +m2 qjj(1q ))
1)
jp() q() j((m
cjpj((m
L K)
L+jj K ) jqj(L K +jj)
(1:3)
(m2 )
1 +m2 qN (1q ))
1)
jrN j((m
cjpj((m
L K)
L K )jqj(L K ) :
0
0
This theorem is taken from 1, 2]. Professor Leopold stated in 1, 2] and in private communication that his proof of Theorem 1.1 is based on the modication of Lemma 2.4 from 7] that involves
the norm (1.1).
Theorem 1.2. Let N be an even number and N > 32n + 1. Then, for any p 2 (1 1), there
exists a constant cp > 0 such that
kP ( D)ujLp k cp jpj(0)
(N 1) kujLp k
(1:4)
for any u 2 Lp.
This theorem is contained in 8]. A less general statement of this type was established earlier
in 9].
2. THE SPACE Bps qa(Rn )
Here, we present certain denitions and results from 1{3].
De nition 2.1. Let 0 < 1, 0 < m0 m. A function a(x ) 2 S1m is said to belong to
the class S (m m0 ) if there exists a constant Ra 0 such that
(1 )
ja(()) (x )j c ja(x )j h iqjj+jj
(2:1)
for x 2 Rn and j j Ra and
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(2 ) there exist constants c0 > 0 and c > 0 such that
c0 h im ja(x )j c him
(2:2)
for x 2 Rn and 2 Rn , j j Ra .
Lemma 2.1 1{3]. Let a 2 S (m m0 ) N 2 N J 2 N 0 and ja(x )j < 2J +1 for jj Ra .
Let 2 C 1(R1 ) 0 (t) 1 (t) = 1 for t 2J q1 (t) = 0 for t 2J '0 (x ) =
P
2N q 1
qk+N q1 ja(x )j) and 'j (x ) = (2qj qN ja(x )j) y (2qj +N q1 ja(x )j) for j 1.
k=1 (2
Then, the system of functions f'j (x )g1
j =0 possesses the following properties :
(i)
'j (x ) 2 C 1(Rn Rn )
'j (x ) 0
j 2 N0 0
(ii)
(iii)
supp 'j f(x ): ja(x )j < 2J +N +j g for 0 j N
supp 'j f(x ): 2J qN +j < ja(x )j < 2J +N +j g for j N + 1
sup supn '(j() ) (x ) c h iqjj+jj
j 0 x2R
1
X
(iv)
j =0
for 2 N n0 'j (x ) = cf'j g = 2N y 1 > 0:
For our purposes it is sucient to take N = 1.
De nition 2.2 1{3]. The symbols a b 2 S (m m0 ) are called equivalent, which is denoted
as a b, if there exist constants c1 > 0, c2 > 0, and R 0 such that
0 < c1 ab((xx )) c2 < 1 for (x ) 2 Rn Rn j j R:
Remark 2.1. If a 2 S (m m0 ), then there exists a symbol a~ 2 S (m m0 ) such that a~ a
and inf a~(x ) > 0. To construct a~, we can set
a~(x ) = ja(x )j
j j R
where R is suciently large, and appropriately dene a~ for j j < R.
De nition 2.3 1{3]. Let 1 < p < 1, 1 q 1, y1 < s < 1, and a 2 S (m m0 ). We
set
Bpa q = Bpa q (Rn ) = Bp1 qa (Rn )
9
8
01
1 1q
>
>
=
<
X
s
a
s
a
n
0
s
a
jsq
q
@
A
< 1>
Bp q = Bp q (R ) = >u : u 2 S kujBp q k =
2 k'j ( D)ujLp k
:
j =0
1
P
(for q = 1, ( 1 cq ) q , c 0, is understood as sup c ).
Theorem
(2.3)
j 0 j
2.1 1, 2]. Let a(x ) b(x ). Then, Bps qa = Bps qb up to the equivalence of norms.
was also shown that two spaces Bps qa constructed by dierent systems f'j g of the
j =0 j
j
In 1, 2], it
type described above coincide up to the equivalence of norms.
Theorem 2.2 1, Theorem 3.6]. When s > 0 Bps qa = Bp1 qjajs .
By virtue of this theorem, it is sucient to consider in what follows only the spaces Bpa q instead
of Bps qa, s > 0.
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3. A PDO WITH THE \ALMOST DECREASING{ALMOST INCREASING"
CONDITION
De nition 3.1 1]. We say that the symbol a 2 S (m m0 ) satises the \almost decreasing{
almost increasing" condition and denote this as2 a 2 S (m m0 #") if there exist constants c1 > 0,
c2 > 0, and r0 > 0 such that
j j m
c1 jj
0
jj m
ja(x )j
c2
ja(x )j
jj
for j j jj r0 :
(3:1)
Condition (3.1) implies that ja(x )j j jqm \almost increases" and ja(x )j j jqm \almost decreases" with respect to (j j r0 ).
Note that it is a 2 S (m m0 #") for which Leopold found in 1] the equivalent normalization
of Bps qa by the dierence of a function with variable step.
By a(x ) 2 S (m m0 #"), we construct a function a(x r), x 2 Rn , 0 r < 1, such that
a(x j j) a(x ) and, at the same time, a(x r) possesses certain properties of monotonicity with
respect to r.
Let r0 > 0 be a number that guarantees condition (3.1). We set
a1 (x r) = ja(x re1 )j for r r0 :
0
Obviously, after appropriate denition of a1 (x r) on 0 r r0 , we obtain
a1 (x jj) 2 S (m m0 )
a1 (x j j) a(x ):
Inequalities (3.1) imply the following estimate:
m
m
c1 rr2 aa1 ((xx rr2 )) c2 rr2
1
1
1
1
0
r0 r1 r2 :
(3:2)
The left inequality of (3.2) yields
a1 (x r) 12 a1 (x 2n0 r)
r r0
(3:3)
where n0 2 N is such that 2n0 m 2cq1 1 .
Let us construct the function a2 (x r), (x r) 2 Rn 0 +1). Let j0 2 N and 2n0 j0 r0 . We set
0
a2(x r) = r2qn0 j0 a1 (x 2n0 j0 ) for 0 r 2n0 j0
a2 (x 2n0 j ) = a1 (x 2n0 j ) for j = j0 + 1 j0 + 2 : : :
n0
qn0 j
qn0 j
a (x r) = a (x 2n0 j ) 2 y 2 r + a (x 2n0 (j +1) ) 2 r y 1
2
2
2n0 y 1
2
2n0 j r 2n0 (j +1)
(3.4)
2n0 y 1
(3.5)
for
j = j0 j0 + 1 j0 + 2 : : : :
For every xed x, the function a2 (x r), considered as a function of r 0, is a continuous,
piecewise linear, and strictly increasing function.
We construct the function a(x r), (x r) 2 Rn 0 +1), by a certain
R averaging of the function
a2(x r). Namely, let 2 C01(R1 ), 0, be an even function, (t)dt = 1, and (t) = 0 for
jtj 41 .
2
We slightly changed the notation as compared with that of 1] by introducing the symbol #".
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Let us dene the function a(x r) as follows.
De nition 3.2. When "j = 2n0 jq2, we set
Z
a(x r) = (t)a2 (x
Z
r + t2n0 j ) dt = 2qn0 j
Z v y r
a (x v) dv
2n0 j
for r 2 U"j0 (0)
2
for r 2 U"j (2n0 j )
a(x r) = (t)a2 (x jr + t2n0 j0 j) dt
a(x r) = a2 (x r) for r 2= U"j (2n0 j ) r 2= U"j0 (0):
(3:6)
j j 0
Lemma 3.1. Let a(x ) 2 S (m m0 #") and a(x r) be a function dened by (3:6). Then,
(1 ) a(x j j) 2 S (m m0 #")
(2 ) a(x j j) a(x ) a(x j j) 0
(3 ) there exists c0 > 0 such that
a(x r) > 0
@ a(x r) c rq1 a(x r)
0
@r
(3:7)
for r 1.
Proof. Assertion (2 ) follows from the construction of a. The property #" in (1 ) follows
from (2 ) and (3.1). To prove the fact thatS a 2 S (m m0 ), it is necessary to establish the
estimates of the type (2.1). For r > " , r 2=
U (2n0 j ), these estimates are obvious. When
r 2 U"j (2n0 j ),
j0
j j0 "j
they are deduced from the following estimates for the derivatives:
@ k @ a(x r) = 2qn0 j Z (y1)k 2qn0jk (k) v y r @ a (x v) dv
@rk x
2n0 j x 2
Z
q
n
j
(k+1)
(k) v y r
0
c 2
a (x v)vjj dv
2
2n0 j
Z
c2 rqkq1+jja2 (x r) (k) v ny0 jr dv
2
1
= c3 a2 (x r)rqk+jj c4 a(x r)rqk+jj :
(3.8)
Let us prove assertion (3 ). Let r 2 U"j (2n0 j ) (the notations from (3.6)). Then,
@ a(x r) = Z (t) @ a (x r + t2n0 j ) dt Z (t) 2qn0 j a (x 2n0 j ) dt
@r
@r 2
2n0 y 1 2
c1 2qn0 j a2 (x 2n0 j ) c2 rq1 a(x r):
For the convenience of references, we rewrite the assertion #" of the lemma as
r2 m
0 < c3 r
1
0
r2 m
a
(
x
r
2)
a(x r1 ) c4 r1
r0 r1 r2:
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4. EQUIVALENT NORMS IN Bpa q (Rn )
By the symbol M
i (t)u(x), we will denote the dierence of order M 2 N of the function u with
the step tei .
Theorem 4.1. Let a(x ) 2 S (m m0 #") 1 < p < 1 and m + < M 2 N . Then, the
following norms are equivalent to the norm (2:3) of the space Bpa p(Rn ):
kujBpa pk!
=
(X
1X
n
sup
qk p
ka( 2k e1 )M
i (t2 )ukp
k=0 i=1 0<jtj "
(X
) p1
1X
n
a
k
1
M
q
k
p
kujBp p k
=
ka( 2 e )i ("2 )ukp
k=0 i=1
) p1
+ kukp
+ kukp
where 0 < j"j " = " (a) 2 (0 1].
Remark 4.1. Obviously, Theorem 4.1 remains valid under the change of the weight factor
ja(x 2k e1 )j in the Lp -norm by any equivalent one. When proving this theorem, it is convenient to
use the weight factor a(x 2k ).
Remark 4.2. The case p = 2, a(x ) = his(x) of Theorem 4.1 in a similar form is known 15].
To formulate the following theorem, we need the system of functions fjl (x)gj l .
Suppose that
p
2 C 1 (0 1) 0 1 = 1 on 0 21
= 0 on 1 1)
(4:1)
aq1 (x t) is a function inverse to r ! a(x r),
j 0(x) = (aq1 (x 2j ))
jl(x) = (2qlq1 aq1 (x 2j )) y (2ql aq1 (x 2j ))
for l 2 N
j 2 N0 :
(4:2)
For more detail on the properties of the system fjl (x)g, see Section 5. Meanwhile, we note
that fsupp 'jl gj l , where 'jl (x ) = jl (x)'j (x ), is a covering of Rn Rn with the property
h i 2l
whereas
ja(x )j 2j for (x ) 2 supp 'jl
ja(x )j 2j for (x ) 2 supp 'j :
Lemma 4.1. Let a(x ) 2 S (m m0 #") 1 < p < 1 1 q 1 and y1 < s < 1. Then,
P 2j ' (x ) = P1 P1 2j (x)' (x ) and
(i) a(x ) a (x ) where a (x ) = 1
j
j
jk
j =0
j =0 k=0
(ii) the norm (2:3) of the space Bps qa(Rn ) is equivalent to the norm
1
81 "1
# pq 9
q
<
=
X
X
kujBps qa k(0) = :
k2js jk 'j ( D)ukpp :
j =0 k=0
Proof of this lemma is trivial.
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Theorem 4.2. Let a(x ) 2 S (m m0 #") 1 < p < 1 1 q 1 and m + < M 2 N .
Then, the norm (2:3) of the space Bpa q (Rn ) is equivalent to the norms
1
81 "1 n
# pq 9
q
=
<
X
X
X
qk p
kujBpa q k! = :
sup k2j jk M
i (t2 )ukp + kukp
j =0 k=0 i=1 jtj "
1
81 "1 n
# pq 9
q
<
=
X
X
X
qk p
kujBpa q k
= :
k2j jk M
i ("2 )ukp + kukp
j =0 k=0 i=1
where 0 < j"j " = " (a) 2 (0 1].
The proof of Theorem 4.2 will be carried out below in Sections 6 and 7. The assertion of
the theorem is contained in (6.12) and (7.14). Theorem 4.1 is a corollary to Theorem 4.2 and
Lemma 5.1.
Let us present an example of a PDO that satises the hypotheses of the theorem. (This example
was considered by several authors and is given, for example, in 1, 2].)
Suppose that a real function 2 S (Rn ), d > 0, (x) = d + (x), m0 = inf > 0, m = sup ,
and a(x ) = h i(x) . It can be easily veried that a(x ) 2 S (m m0 #") for any 2 (0 1).
There are also other examples of PDOs from S (m m0 #") in 1{3].
5. THE FAMILY fjl (x)gj l2N0
Suppose that a(x ) 2 S (m m0 #"), a(x r) is the same as that dened by (3.6) and the
system f'j (x )gj satises the assertions of Lemma 2.1.
Lemma 5.1. Let a(x ) 2 S (m m0 #"). Then, the functions of the system fjl (x)gj l2N0
in (4:2) satisfy the conditions
(1 ) jl 2 C 1(Rn ) j l 0
(2 ) for certain c1 c2 > 0 n0 2 N and any j l 2 N 0 x 2 supp jl
aq1 (x 2j ) 2 for l = 0
2lq1 aq1 (x 2j ) 2l+1 for l 1
2j a(x 1) for l = 0
c1 2j a(x 2l ) c2 2j for l 1
supp 'j (x ) f : j j < 2n0 g
for l = 0
l
q
n
l
+
n
supp 'j (x ) f : 2 0 < j j < 2 0 g for l 1
(3 ) for all 2 N n0 j l 2 N 0 and certain c independent of j and l
P
jDx jl (x)j c 2ljj (5.1)
(5.2)
(5.3)
(5:4)
(4 ) 1
l=0 jl (x) = 1 for all j 2 N 0 (5 ) the set fl : l 2 N 0 jl (x) 6= 0g contains at most three elements for every xed x and j .
For every xed x and l, the number of elements of the set fj : j 2 N jl (x) 6= 0g does not
exceed certain J0 independent of x and l.
Proof. Property (1 ) is obvious. Let us establish property (2 ). Estimates (5.1) follow from
(4.1) and (4.2). Applying the mapping r ! a(x r) to (5.1) and using (3.9), we obtain (5.2).
PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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To prove (5.3), we note that, for (x ) 2 supp 'j , the relation c3 2j < a(x j j) < c4 2j for certain
c3 c4 > 0 follows from assertion (ii) of Lemma 2.1 and assertion (2 ) of Lemma 3.1. Using (3.9),
we obtain (5.3) from this relation and (5.2).
Relations (2 ) for the case l = 0 are established similarly.
Let us prove (3 ). According to the rule of dierentiation of an implicit function, for r =
q
1
a (x 2j ), we have
@ aq1 (x 2j ) = y @x@ a(x r)
(5:5)
@
@x
@r a(x r)
so that, by virtue of assertion (1 ) of Lemma 3.1 and inequalities (3.7), we obtain
@ aq1 (x 2j ) cr+1
@x
r 1:
(5:6)
Therefore, by (5.1), we obtain the following inequalities for x 2 supp jl :
@ (x) c 2l :
@ aq1(x 2j ) c 2l(+1)
1
2
@x
@x jl
The case of an arbitrary 2 N n0 , j j > 1, is considered similarly with the use of (5.5).
Relation (4 ) immediately follows from (4.2). Assertion (5 ) follows from (5.1) and (5.2).
The lemma is proved.
Lemma 5.2. Let a(x ) 2 S (m m0 #"). Then, for certain n0 n1 2 N the functions of the
system fjl (x)gj l2N0 from (4:2) satisfy the condition
jk (x + y) n1
X
l: jlqkj n0
jl (x)
for jyj M 2qk
(5:7)
for any j k 2 N 0 .
Proof. It is clear from (4.2) that, to prove (5.7) for jyj < "M 2qk , " > 0, it is sucient to show
that, for certain n0 2 N and all x 2 supp jk ,
q1
y 2j ) 2n0
2qn0 a aq(x1 (y
x 2j )
for jyj < "M 2qk :
(5:8)
Let us set b(x) = (aq1 (x 2j ))q . Then, (5.8) is rewritten as
y) 2n0 for jyj < "M 2qk :
2qn0 b(xb(y
x)
(5:9)
We will assume that n0 1. By (5.6),
j grad b(x)j = aq1 (x 2j )qq1 j grad aq1 (x 2j )j c0
where c0 is independent of j and x. Then, taking into account (5.1), we obtain the following relation
for jyj < "M 2qk :
jb(x y y) y b(x)j c jyj(aq1 (x 2j ))q c "M 2qk 2q(kq1) = c "M 2 2qk(1q) 1
0
0
0
b(x)
2
for 0 < " = c0 M12+1 and all k 2 N 0 .
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Relation (5.10) entails (5.9) and inequality (5.7) with n1 = 1 and jyj < cq0 1 2qq1 2qk . Applying
this relation n1 times, where n1 1" , we obtain the assertion of the lemma.
Lemma 5.3. Suppose that a(x ) 2 S (m m0 #") and the function a(x + y r) is dened
by (3:6). Then, there exists c0 > 0 such that
sup a(x + y 2k ) c0 a(x 2k )
jyj
2 k
k
8x 2 Rn
8k 2 N 0 :
(5:11)
Proof. If 2j a(x 2k ) < 2j+1 and x 2 supp jl , then, by virtue of (5.2) and (3.9), jl y kj n
for certain n 2 N independent of j and k. Then, by (5.7), for every y, jyj 2qk , there exists
l 2 N 0 , jl y kj n + n0 , such that x + y 2 supp jl . Then, (5.2) and (3.9) imply (5.11).
Lemma 5.4. Let a(x ) 2 S (m m0 #") 2 f0 1 : : : M g and t 2 y1 1] 1 i n.
Then, there exists n 2 N 0 such that relations j0 j k0 k 2 N 0 and (supp j0 k0 ) \ (supp jk y
t2qk0 ei ) 6= ? imply the estimates
j0 j + n for k0 k
j0 y j m(k0 y k) + n for k0 k:
(5.12)
(5.13)
Proof. Let x(0) 2 supp j0l0 , x(0) + t2qk ei 2 supp jk . Then, x(0) 2 Sjlqkj
n0 supp jl
by
virtue of (5.7). Inequalities (5.2) and the monotonicity of a(x r) with respect to r > 0 imply that
there exist n n1 2 N 0 such that (5.12) holds and k0 k entails j0 j y n1 . Then, by (5.2) and
(3.9) we obtain (5.13).
Henceforth, we will use another two systems of functions, fl g and fel g.
Lemma 5.5. Let a(x ) 2 S (m m0 #") n0 2 N 0 be the same as in (5:4) e 2 C01(Rn )
supp f : 2qn0 q1 < j j < 2n0 +1 g
() = 1 for 2qn0 < j j < 2n0
e = 1 on supp supp e f : 2qn0 q2 < j j < 2n0 +2 g
l () = (2ql )
el () = e(2ql ) for l 2 N
0 e0 2 C01(Rn )
0 () = 1 if j0(x)'j (x ) 6= 0
j 2 N0
e0 = 1 on supp 0 :
Then,
(i) l ( ) = 1 if jl (x)'j (x ) 6= 0 j l 2 N 0 (ii) supp l f : 2lqn0 q1 < j j < 2l+n0 +1 g supp el f : 2lqn0 q2 < j j < 2l+n0 +2 g l 2 N (iii) el = 1 on supp l l 2 N 0 (iv) jl() ( )j c 2qljj and jel() ( )j c 2qljj for l 2 N 0 2 N n0 and 2 Rn .
Proof. Assertion (i) follows from (5.3). Assertions (ii){(iv) are obvious.
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6. ESTIMATES FOR THE NORMS OF DIFFERENCES
AND MODULES OF CONTINUITY
Let a(x ) 2 S (m m0 #"), f'j (x )g1
j =0 be the system from Lemma 2.1, and fjl (x)gj l be
the system from (4.2).
For u 2 Bpa p, we set uj (x) = c '1j 2j 'j (x D)u(x) and write
f
g
u(x) =
X
j
2qj uj (x)
this series converges to u in Lp (see (13) in 2]). Setting
vjl (x) = jl (x)uj (x)
by Lemma 5.1 we have
X p
vjl C1 X kvjlkpp C2kuj kpp:
l p
l
Let us estimate
Cj0k0 (x)
X
j
(6:1)
qk
Ij0 k0 (x) = j0k0 (x)2j0 jM
i (t2 0 )uj
0
1
X
X
qk
qk
2(j0 qj ) @ M
vjl + M
vjl A
i (t2 0 )
i (t2 0 )
lk0
l<k0
= CIj0k0 (x) + CIj0k0 (x):
(1)
(2)
Using (5.7) and (5.12), we obtain
M X X
X
kIj(1)
k C 2j0 qj j0 k0 ( y t2qk ei )vjl 0 k0 p
p
=0 lk0 j
X X X j0qj X X qjj0qjj X
C1
2 vjl C2
2
j0k vjl :
j0k
lk0 j
l
jkqk0 j n0
jkqk0j n0 j
p
p
Combining assertion (5 ) of Lemma 5.1 and (6.1), we obtain
0 0
1 p1
p1 p1
X
X
X
X
@ kIj(1)0 k0 kpp A C2 2qjj0qjj @ j0k vjl A
j
k0
k
l
p
0
11
X
X qjj0qjj @X pA p
C3 2
vjl C4 2qjj0 qjjkuj kp
j
l
p
(6.2)
j
from the latter expression.
P
Now, we derive a similar estimate for k0 kIj(2)
kp . We have
0 k0 p
Ij(2)
(x) Cj0k0 (x)
0 k0
X
j j0 +n0
qk X vjl (x) :
2j0 qj M
i (t2 0 )
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Here, n0 is a certain number from N 0 independent of l, k0 (l < k0 ), j0 , and j the existence of this
number follows from (5.7) and (5.3). Using (5.12), we obtain the estimate
Ij(2)
(x) C1
0 k0
X
j
qk X 2(k0 ql)m(1+") vjl (x)
2qjj0 qj j" M
i (t2 0 )
l<k0
where " > 0 is chosen from the condition m(1 + ") = M y ".
Hence, by (5.7) and assertion (5 ) of Lemma 5.1, we have
X
qk
kIj(2)
k C2 2qjj0 qj j" 2(k0 ql)(M q") jM
i (t2 0 )vjl j
0 k0 p
l<k0
p
j
0
11
X qjj0qjj" @ X (k0 ql)(M q") M qk0 pA p
C3 2
2
ki (t2 )vjl kp :
X
j
(6.3)
l<k0
Setting 'jl (x ) = jl (x)'j (x ) and taking l from Lemma 5.5, we obtain the following
equation by virtue of (1.2):
l (D)'jl (x D)]D! = 'jl (x ) + rN (x )
(6:4)
so that vjl (x) = jl (x)uj (x) = c '1j 2j 'jl (x D)u satises the equation
f
g
1 j
qk
M qk
M qk
M
i (t2 0 )vjl (x) = i (t2 0 )l (D)vjl (x) y i (t2 0 )rlN (x D) cf'j g 2 u(x):
(6.5)
Let us estimate the Lp -norm of the rst summand on the right-hand side of (6.5). Note that
qk
M
i (t2 )l (D) = l k (D), where
l k () = (eit2
Let us show that
k i
k
y 1)M l ( ):
qk
q(kql)M kvjl kp :
kM
i (t2 )l (D)vjl kp C 2
(6:6)
For this purpose, we apply Theorem 1.2 and obtain
X
j (l k) ( )j C
0
2qk0 jeit2 k i y 1jM q0 jl() ( )j
k
0 M
Cd 2q(kql)M h iqjj (2ql h i)jjqd
(6.7)
with arbitrary d 2 (y1 1).
Taking d = jj, we obtain jl k j(N 1) C 2q(kql)M hence, by (1.4), we have (6.6).
Now, we estimate the Lp -norm of the second summand on the right-hand side of (6.5). First,
we note that, if the left-hand side of (6.5) is dierent from zero, then, by virtue of (5.7), (5.3),
and (2.2), the following estimate is valid for j in (6.5):
j lm + n0
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where n0 2 N 0 is independent of l. By (1.3), we have
(m2 )
1 +m2 qN (1q ))
1)
jrlN j((m
C jl j((m
L K)
L K ) j'jl j(L K ) :
0
(6:9)
0
However, for any 2 N n0 and d 2 (y1 1),
jl() ( )j c d 2qljj (2ql h i)d
so that
jl() ( )j c 2qlm1 h im1 qjj :
By (5.4), assertion (iii) of Lemma 2.1, and (5.3),
j'(jl()) (x )j C
X
jjl( ) (x)'(j() ) (x )j
0
00
j + j=jj
C1 2lj j h iqjj+j j C2 h iqjj+jj :
0
00
0
Now, from (6.9) we obtain
00
1 qN (1q ))
jrlN j((m
C 2qlm1 :
L K)
Henceforth, we assume that m1 > 0 and N is so large that N (1 y ) m1 + M . Then,
()
qlm1 h iqM qjj+jj
jrlN
( ) (x )j C 2
for jj L j j K:
Let r lN (x ) = iM q Dxi rlN (x ), = 0 1 : : : M . Then, it is obvious that
jr (lN)() (x )j C 2qlm1 h iqjj+jj
jj L
j j K y M qlm1 . Hence, by (1.4), we have
i.e., jr lN j(0)
(L K qM ) C 2
kr lN (x D)ukp C 2qlm1 kukp :
P c (r (x D)v)(x) therefore,
However, @x@ MMi (rlN (x D)u)(x) = M
=0 lN
M
@ (rlN ( D)u()) C 2qlm1 kukp @xMi
p
hence,
qk
M qkM 2qlm1 kuk :
kM
p
i (t2 )(rlN ( D)u())kp C jtj 2
Formulas (6.3), (6.5), (6.6), and (6.10) imply
kIj(2)
k C
0 k0 p
X
j
(6.10)
0
1 p1
X
X
2qjj0 qj j" @ 2q(k0 ql)"pkvjl kpp + 2jp2qk0 Mpqlm1 p kukpp A :
l<k0
l<k0
Combining this inequality with (6.8) and (5.13), we obtain
kIj(2)
k C1
0 k0 p
X
j
0
1 p1
X
2qjj0 qj j" @ 2q(k0 ql)"pkvjl kpp + 2qj0 "p2qk0 "pkukpp A :
l<k0
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This, combined with assertion (5 ) of Lemma 5.1 and (6.1), yields
0
p1 p1
B@X I (2) CA C2 X 2qjj0qjj"kuj kp + C2 2qj0"kukp :
j0k0 j
k0
p
(6.11)
From (6.2), (6.11), and the discrete analogue of the Young inequality for the convolution, we
obtain
0
11
2
3q q
0
11
B@X 4X kIj0 k0 kpp 5 p CA C @X kuj kqpA q + C kukp:
j0
(6:12)
j
k0
7. AN ESTIMATE FOR THE NORM kf jBpa q k IN TERMS OF DIFFERENCES
Let fk ( )gk2N0 and fek ( )gk2N0 be the systems of functions from Lemma 5.5.
Introduce a system of functions f#i gni=1 with the following properties:
(i) #i 2 C01 supp #i f : 21 < j j < 32 g
(ii) #i ( ) = 0 if j j = 1, i = ( ei ) < 2p1 n P
(iii) ni=1 #i ( ) = 1 for j j = 1.
For 0 < " 2qn0 q1 , we set
cjki (x ) = 'j (x )k ( )#i j j (ei"2 k i y 1)qM
jki(x ) = ek ()jk (x)(ei"2 k i y 1)M :
Let the numbers "0 > 0 and "1 > 0 be xed. Take N 2 N so large that
k
k
m(1 + "0 ) + "1 y N (1 y ) 0:
) (x D)
Let 2 N n0 , j j < N . Applying (1.2) to the product c(jki
jki( ) (x D)]D! , we obtain
) (x )
) (x D)
c(jki
jki( )(x ) = c(jki
jki( ) (x D)
X
D!
1 c( +) (x )
y
jki( +) (x ) + rjki N qj j(x ):
! jki
1 jj<N qj j
(7.1)
In this case, for any pair (L K ) 2 N N , there exists a pair (L0 K 0 ) 2 N N such that
) j(m1 ) j
(m2 )
1 +m2 q(N qj j)(1q ))
jrjki N qj jj((m
C (N )jc(jki
L K)
(L K ) jki( ) j(L K ) :
0
(7.2)
0
From the equations (7.1), we choose
that corresponding to = 0. The left-hand side of this
p
equation is equal to 'j (x )k ( )#i jj jk (x) from the right-hand side, we eliminate the terms
involving
) (x )
c(jki
jki()(x )
(7:3)
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with j j = 1 by substituting these expressions from (7.1) with j j = 1. Then, analogously, we
eliminate the terms involving (7.3) with j j = 2 from the right-hand side of the equation obtained.
Proceeding further in this way, we arrive at the equation
X
'j (x )k ()#i j j jk(x) =
{jkic(jki) (x D)
jki()(x D)D! + rjkiN (x )
j j<N
where
rjkiN =
X
j j<N
{~ jkirjki N qjj
(7:4)
(7:5)
here, sup j k i(j{jkij + j{~ jki j) C1 < 1.
Noting that on supp cjki
we readily obtain
c2j ja(x )j c0 him c00 2km
(ei"2 k i y 1) {0 > 0
h i 2k
k
) j(m1 ) C ()2qkm1 2qkj j(2km 2qj )
jc(jki
(L K )
0:
0
(7:6)
Using estimate (6.7) and (5.3), we obtain
jki() ((LjjK) ) C:
(7:7)
0
Taking = 1+ "0 , m1 = yj j + m + "1 , and m2 = j j, we obtain m1 + m2 y (N yj j)(1 y ) =
m + "1 y N (1 y ) 0, so that
qk"1 2qj qj"0
jrjki N qj jj(0)
(L K ) C 2
(7:8)
by virtue of (1.3), (7.6), and (7.7).
The multiplier k ( ) on the left-hand side of (7.4) can be rejected due to (5.3) and assertion (iii)
of Lemma 5.5. Then, summing (7.4) over i and k, we obtain
'j (x D)u(x) =
XX
k i j j<N
{jkic(jki) (x D)
jki()(x D)u(x) +
X
ki
rjkN (x D)u(x):
(7:9)
Taking into account property (5 ) of the system fjk (x)g from Lemma 5.1, we obtain the
following estimate from (7.9):
k'j ukpp C
XX
j j<N k i
p
X
) ( D)
kc(jki
jki( )( D)ukpp + rjkiN ( D)u :
ki
p
(7:10)
Taking into account (7.6) for m1 = 0 and = 0, (7.8), (1.4), and assertion (iii) of Lemma 5.5,
we rewrite this inequality as
2jp k'j ukpp C1
XX
j j<N k i
qk p
qj" p p
2jp2qkj jpkjk( ) M
i ("2 )ukp + C2 2 0 kukp :
(7:11)
By the function from (4.1) (which generates the family fjk g, see (4.2)) we construct a
function that satises (4.1) and certain additional conditions. Namely, taking the segment
t1 t2 ] ( 21 1) such that
0 (t) < 0 for t 2 t1 t2 ]
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we require that
(t) = 1 for t t1
(t) = 0 for t t2 :
Denote by jl the functions jl constructed by (4.2) on the basis of = . Then, there exists
{0 > 0 such that
jl {0 on supp jl 8j l 2 N 0 :
By virtue of assertion (3 ) of Lemma 5.1, we have
j jl() (x)j c~ 2ljj jl (x)
for 2 N n0 j l 2 N 0 :
(7:12)
From (7.11) with jk = jk and (7.12), we obtain
2jpk'j ukpp C3
XX
j j<N k i
C5
X
ki
qk p
p
2qk(1q)j jp 2jp kjk M
i ("2 )ukp + C4 kukp
p
qk p
kjk 2j M
i ("2 )ukp + C4 kukp
(7.13)
which yields the following estimate for 1 q 1 (with appropriate modication for q = 1):
2 0
2
3 q1
1 pq 3 q1
4X 2j k'j ukp q 5 C 64X @X kjk 2j Mi ("2qk )ukpp A 75 + C kukp:
j
j
ki
(7.14)
8. INTERPOLATION AND EMBEDDINGS OF THE SPACES Bpa p
The general problems of the interpolation of Banach spaces were stated in 10, 11]. Interpolation
theorems for the family of spaces Bps qa (1 < p < 1, 1 q 1, y1 < s < 1) were obtained
by Leopold in 3] by the construction of the retraction and coretraction operators, these theorems
were reduced to the interpolation theorems for the lq -spaces of Lp -valued sequences. In this case,
the form of the retraction and coretraction operators essentially depends on the PDO a(x D).
Here, we establish a retraction theorem for q = p and s > 0 in which we construct the retraction
and coretraction operators that are independent of a(x D). Thus, this theorem allows one to
characterize the interpolation spaces for a pair of spaces from fBpa pg that essentially dier by the
form of a(x D). To prove this theorem, we use the approach developed in 12{14]. The properties of
the retraction and coretraction operators constructed are also used for the proof of the embedding
theorem for Bpa p Brb r .
Let us introduce the spaces B~pa q , that coincide with Bpa p for 1 < q = p < 1 by Theorem 4.2.
De nition 8.1. Let a 2 S (m m0 #"), 1 p q 1, and M > m. For k0 2 N 0 and
" 2 (0 1], we set
9
8
21
3 q1
>
>
n X
=
<
X
a
0
n
a
k
1
M
q
k
q
~
~
4
5
Bp q = >u : u 2 S (R ) kujBp q k =
sup ka( 2 e )i ("2 )ukp + kukp < 1> :
:
i=1 k=k0 j"j "
(8:1)
When p = 1 or q = 1, the Lp - and lq -norms are understood as the L1- and l1 -norms,
respectively.
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Remark 8.1. The norms (8.1) for dierent k0 are equivalent since the inequality
sup ja(x 2k e1 )j C (k)
x
implies the estimate
M
ka( 2k e1 )M
i (t)ukp C (k)2 kukp :
Remark 8.2. It will be clear from the proofs that the requirement a 2 S (m m0 #") can be
essentially weakened in this section. However, we will not do it since we focus on the spaces Bpa p
introduced above.
Introduce the representation of a function as a sum of a series of convolutions of special type
that was given in 12{14].
R
Let 2 C01(R), supp (y1 1), (u)du = 1, (u) = u(u), and
M (y1)M qj M ! u X
1
^(u) = A (1 + j)2 j 1 + j
j =0
where
A = A(M
) = (y1)M
Z1
0
M
j
X
(1 y u )M du = (y1)M 1(y+1)j
j =0
!
M 6= 0
j
R
so that ^(u)duQ= 1.
Set ~ (y) = n1 ^(yi ), ~ k (y) = 2kn ~ (y2k y), and k = ~ k ~ k .
For f 2 L(Rn loc), consider the averaging
f(2 k )(x) = k k f (x):
Since f(2 k ) ! f (k ! 1) almost everywhere on Rn also in the sense of L(Rn loc), then the
following representation is valid almost everywhere on Rn :
1 p
X
yp
f
f = f +
k
k
0
0
or, in a dierent form,
f0 =
where
f0 =
+
0
k
k=1
f0 +
f
+
0 = 0
0
1p
X
k=1
kq1
k
k + kq1 fk =
fk =
p y
k
kq1
1
X
k=0
kq1 f
(8:2)
k fk
+
k1
k 1:
(8.3)
(8.4)
= k + kq1
Using 14, x7, formula (75)], we can represent fk for k 1 as
fk (x) =
Z X
n Z Z1
2kk+1
2 k
k
i=1 Rn q1
tq2qnKi
+
k
y v M
i
t i t i (v)f (x + y + ve ) dv dy dt
where Ki 2 C01 (Rn ) and i 2 C01(R1 ).
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BESOV
De nition 8.2. Let 1 p q 1, k (x) 2 C (Rn ), and k (x) > 0. We set
"
#
Lp k = u : u 2 L(Rn loc) kujLp k k = kk ujLp k < 1
n
o
lq (Lp k ) = h(x) = fhk (x)gk2N0 : fkk hk kp g 2 lq khjlq (Lp k )k = k kk hk kpjlq k < 1 :
De nition 8.3 10, 11]. The Banach space Y is called a retract of the Banach space X if
there exist linear continuous mappings R : X ! Y (retraction) and S : Y ! X (coretraction)
such that IY = RS : Y ! Y is the identity operator.
Theorem 8.1. Let 1 p q 1 and a 2 S (m m0 #"). Then, the space B~pa q is a retract of
the space lq (Lp k ) where k (x) = a(x 2k ) is taken from (3:6).
Here, the operators S (coretraction ) and R (retraction ) are given by
(S f )k (x) = fk (x)
k 2 N 0 x 2 Rn
(8.6)
Rh =
1
X
k hk
(8.7)
+
k=0
where +k and fk are the same as in (8.2){(8.5).
Proof. According to the denition of the retract, it is necessary to show that the operators S
and R possess the following properties:
(1) S : Bpa q ! lq (Lp k )
(2) R : lq (Lp k ) ! B~pa q (3) RS f = f 8f 2 B~pa q .
Property (3) is fullled by virtue of (8.2) and (8.3).
Let us establish (1). By the Minkowski inequality for integrals, k(S f )0 kp C kf kp. Taking into
account (5.11) and applying the Minkowski inequality for integrals and then the H!older inequality,
we obtain the following estimate from (8.6) and (8.5) (when i are concentrated in a suciently
small neighborhood of zero) for k 1:
kk (S f )k kp = kk fk kp
C1
n
X
i=1
22k
Z Z
2kk+1
2kk
v kKi k1 kk M
i (v)f kp i t
Hence, for 0 < " 1 and 1 q 1, we have
kf(S f )k gjlq (Lp k )k =
n "Z2
X
dv dt C2
"X
1
k=0
i=1
k(k S fk )kqp
k
k
k
kk M
i (v)f kp 2 dv:
0
# q1
2 0 "2 k
1q 3 1q
Z
n
1
X X k
C7
C3 kf kp + C3 64 B
kk M
@2
i (v)f kp dvA 5
i=1 k=1
0
8
"
# q1 9
n X
1
<
=
X
qk q
sup kk M
C4 :kf kp +
i ("2 )f kp i=1 k=1 0 " "
k
thus, we established property (1).
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The proof of property (2) is contained in the particular case r = p of following lemma.
Lemma
8.1. Let a(x n) n2 S (m m0 #"n) n1 p r 1 1 q 1 b(x ) =
n+n
q
h i p r a(x ) l (x) = 2ql( p q r ) a(x 2l ) = 2ql( p q r ) l (x) np y nr < m0 m and M > m y np + nr .
Then, for the operator R given by (8:7) we have
R : lq (Lp k ) ! B~rb q :
Proof. By the Young inequality, we have
X
1
1 + hk C1 X
k( np q nr ) kh k
2
k p
k
k=0
r
k=0
C2 2k( p q r qm ) kk hk kp C3 khjlq (Lp k )k:
n n
For 0 < j"j 1, we have the estimate
"X
l
ql
q
kl M
i ("2 )Rhkr
(8.8)
0
# q1 "X X
1
l
Let us show that
k=0
ql kl M
i ("2 )hk kr
!q # q1
:
ql qjlqkj{ kk hk kp
kl M
i ("2 )hk kr C 2
where { = minfM y m + np y nr m0 y np + nr g > 0.
Consider two cases: l k and l k.
Case l k 0.
ql M ql
l (x)jM
i ("2 )hk (x)j = l (x) i ("2 )
n n
C 2ql( p q r ) 2(kql)M
ZM Z a(x 2l ) a(x 2k )
+
k(Mei )
a(x 2k ) a(x y y 2k )
0
Z
(8:9)
k (x y y) hk (y) dy
+
(y + 2ql vei ) a(x y y 2k )jhk (x y y)j dy dv:
By (3.9), (5.11), and the Young inequality, this yields
q(lqk)(M qm+ p q r ) k h k :
ql kl M
k k p
i ("2 )hk kr C1 2
n n
(8:10)
Case l k. By (5.11), (3.9), and the Young inequality, we obtain
ql ( q )
ql M ql kl M
i ("2 )hk kr C1 2 p r kl i ("2 )hk kr
n n
C3 2ql( p q r ) 2(lqk)m kk hk kr
n n
C4
n n
2ql( p q r )+(lqk)m
0
0
Z
k (y)k ( y y)hk ( y y)dy
r
+
C5 2(lqk)(m q p + r ) kk hk kp :
0
n n
This expression, combined with (8.10), yields (8.9). Applying the discrete version of the Young
inequality, we obtain the following estimate from (8.9):
"X
l
ql
q
kl M
i ("2 )Rhkp
# q1
C khjlq (Lp k )k:
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BESOV
Combined with (8.8), this estimate proves the lemma.
Using Theorem 8.1 and the theorems on the interpolation of Banach-valued sequences and the
Lebesgue weight spaces (see 10, 11]), we obtain the following theorem.
Theorem 8.2. Let a0 a1 2 S (m m0 #") 1 p0 p1 1 1 q0 q1 1 0 < # < 1
1
1q#
# 1 1q# #
p = p0 + p1 q = q0 + q1 and p = q. Then,
~ a0
Bp0 q0 B~pa11 q1
h ~ a0 ~ a1 i ~ a#
=
Bp0 q0 Bp1 q1 p = Bp q
#p
where a# (x ) = a0 (x j j)1q# a1 (x j j)# .
Theorem 8.3. Let a 2 S (m m0 #") 1 p < r 1 1 q 1 b(x ) = hiq np + nr a(x ) np y nr < m0 m and M > m y np + nr . Then, B~pa q B~rb q .
Proof. Let us rewrite representation (8.2) of a function as
f = RS f
f 2 B~pa q :
The boundedness of
S : B~pa q ! lq (Lp k ) k (x) = a(x 2k )
established in Theorem 8.1 and the boundedness of
R : lq (Lp k ) ! B~rb q
established in Lemma 8.1 imply the boundedness of
RS : B~pa q ! B~rb q
and, hence, the assertion of the theorem.
ACKNOWLEDGMENTS
I am grateful to professor H.-G. Leopold for valuable remarks and recommendations made
during the rst version of the manuscript. These remarks and recommendations have been taken
into account in the nal version.
This work was supported by the Russian Foundation for Basic Research (project no. 99-0100868), the Program \Leading Scientic Schools" (project no. 96-15-96102), and the Visiting
Fellowship Grant of EPSRC (GR/M 02057).
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Translated by I. Nikitin
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