Continuity of certain pseudodifferential operators in spaces of generalized smoothness
We investigate the continuity of a pseudodifferential operator in some spaces of generalized smoothness. Some properties of spaces of generalized smoothness and generalized Lipschitz spaces are established. Досліджено неперервність псевдодиференціального оператора у деяких просторах узагальненої гла...
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Knopova, V.P. 2020-02-11T20:51:34Z 2020-02-11T20:51:34Z 2006 Continuity of certain pseudodifferential operators in spaces of generalized smoothness / V.P. Knopova // Український математичний журнал. — 2006. — Т. 58, № 5. — С. 638–652. — Бібліогр.: 14 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165120 519.21 We investigate the continuity of a pseudodifferential operator in some spaces of generalized smoothness. Some properties of spaces of generalized smoothness and generalized Lipschitz spaces are established. Досліджено неперервність псевдодиференціального оператора у деяких просторах узагальненої гладкості. Доведено деякі властивості просторів узагальненої гладкості та узагальнених просторів Ліпшиця. en Інститут математики НАН України Український математичний журнал Статті Continuity of certain pseudodifferential operators in spaces of generalized smoothness Неперервність деяких псевдодиференціальних операторів у просторах узагальненої гладкості Article published earlier |
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Continuity of certain pseudodifferential operators in spaces of generalized smoothness |
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Continuity of certain pseudodifferential operators in spaces of generalized smoothness Knopova, V.P. Статті |
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Continuity of certain pseudodifferential operators in spaces of generalized smoothness |
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Continuity of certain pseudodifferential operators in spaces of generalized smoothness |
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Continuity of certain pseudodifferential operators in spaces of generalized smoothness |
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Continuity of certain pseudodifferential operators in spaces of generalized smoothness |
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continuity of certain pseudodifferential operators in spaces of generalized smoothness |
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Knopova, V.P. |
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Knopova, V.P. |
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Неперервність деяких псевдодиференціальних операторів у просторах узагальненої гладкості |
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We investigate the continuity of a pseudodifferential operator in some spaces of generalized smoothness. Some properties of spaces of generalized smoothness and generalized Lipschitz spaces are established.
Досліджено неперервність псевдодиференціального оператора у деяких просторах узагальненої гладкості. Доведено деякі властивості просторів узагальненої гладкості та узагальнених просторів Ліпшиця.
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1027-3190 |
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https://nasplib.isofts.kiev.ua/handle/123456789/165120 |
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Continuity of certain pseudodifferential operators in spaces of generalized smoothness / V.P. Knopova // Український математичний журнал. — 2006. — Т. 58, № 5. — С. 638–652. — Бібліогр.: 14 назв. — англ. |
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AT knopovavp continuityofcertainpseudodifferentialoperatorsinspacesofgeneralizedsmoothness AT knopovavp neperervnístʹdeâkihpsevdodiferencíalʹnihoperatorívuprostorahuzagalʹnenoígladkostí |
| first_indexed |
2025-11-27T01:42:50Z |
| last_indexed |
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1850791620187783168 |
| fulltext |
UDC 519.21
V. P. Knopova (V. M. Glushkov Inst. Cybern. Nat. Acad. Sci. Ukraine, Kyiv)
CONTINUITY OF CERTAIN PSEUDODIFFERENTIAL
OPERATORS IN SPACES OF GENERALIZED SMOOTHNESS
NEPERERVNIST| DEQKYX PSEVDODYFERENCIAL|NYX
OPERATORIV U PROSTORAX UZAHAL|NENO} HLADKOSTI
We investigate the continuity of a pseudodifferential operator in some spaces of generalized smoothness.
Some properties of the spaces of generalized smoothness and the generalized Lipschitz spaces are
proved.
DoslidΩeno neperervnist\ psevdodyferencial\noho operatora u deqkyx prostorax uzahal\neno]
hladkosti. Dovedeno deqki vlastyvosti prostoriv uzahal\neno] hladkosti ta uzahal\nenyx
prostoriv Lipßycq.
1. Introduction. In this paper we investigate the continuity of a pseudodifferential
operator p ( x, D ), the symbol p ( x, ξ ) of which for fixed x is a continuous negative
definite function, in the spaces of generalized smoothness. Such a continuity result
enables us to add p ( x, D ) as a perturbation to certain generator of an Lp-sub-
Markovian semigroup, and still obtain the generator of an Lp-sub-Markovian
semigroup.
For the survey on pseudodifferential operators with continuous negative definite
symbol we refer to [1, 2], and the literature given therein, in particular, we mention [3,
4], where the symbolic calculus in L 2 for such operators was developed. In [5]1 the
symbolic calculus for pseudodifferential operators in Lp was developed, and it was
proved that under certain conditions on the symbol p ( x, ξ ) of a pseudodifferential
operator p ( x, D ), this operator is continuous between some (related) ψ-Bessel
potential spaces. Such conditions are similar to those in L 2-case, but much stronger,
since to guarantee the continuity of p ( x, D ) between subspaces of L p ( R
n
) the
Lizorkin’s Fourier multiplier theorem is applied.
Our approach is a bit different. We will pose the conditions on the Lévy measure of
p ( x, ξ ), and prove that under such conditions p ( x, D ) is continuous between related
generalized Lipschitz spaces. Assuming also that p ( x, D ) is continuous between
certain ψ-Bessel potential spaces, we obtain our main result by interpolation
arguments. Thus, our conditions on the symbol are different from those given in [5].
In the second section we give the necessary definitions concerning the spaces of
generalized smoothness. The third section contains some technical statements on such
spaces as well as some properties of the generalized Lipschitz spaces. In the fourth
section under some conditions on the Lévy measure of continuous negative define
symbol p ( x, ξ ) we prove the continuity of p ( x, D ) in generalized Lipschitz spaces.
In the last section we prove that p ( x, D ) is continuous in the related spaces of
generalized smoothness. This result allows us to use p ( x, D ) as the perturbation of a
generator of an Lp-sub-Markovian semigroup.
If it is not specially indicated, all the function spaces, considered in the paper, are
over R
n.
2. Preliminaries. We start with some preliminary definitions and results.
Definition 1. A. A sequence ( )γ j j∈N0
of positive numbers is called strongly
increasing if there is a positive constant d and a natural number κ such that
equations
1 This manuscript can also be found on the web-page of the author, http:// www.math.etzh.ch//farkas/.
© V. P. KNOPOVA, 2006
638 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
CONTINUITY OF CERTAIN PSEUDODIFFERENTIAL OPERATORS IN THE SPACES … 639
d γj ≤ γk for all j, k, 0 ≤ j ≤ k,
and
2 γj ≤ γk for all j, k, with j + κ ≤ k
are satisfied.
B. A sequence ( )γ j j∈N0
of positive numbers is of bounded growth if there exists
a positive constant d and a number J ∈ N0 , such that
γj + 1 ≤ d γj for any j ≥ J.
C. A sequence ( )σ j j∈N0
is called admissible if
d0 σj ≤ σj + 1 ≤ d1 σj for all j ∈ N
holds for some positive d0 and d1
.
For a fixed strongly increasing sequence N = ( )Nj j∈N0
and fixed J ∈ N define
the covering
Ω j
N J, = ξ ξ κ∈ ≤{ }+R
n
j JN: if j = 0, 1, … , J κ,
and
Ω j
N J, = ξ ξκ κ∈ ≤ ≤{ }− +R
n
j J j JN N: if j = J κ, J κ + 1, …
for some κ.
Definition 2. Let Φ
N, J be a collection of functions ϕ j
N J
j
,( ) ∈N0
such that
B1
) ϕ j
N J, ∈ C n
0
∞( )R and ϕ ξj
N J, ( ) ≥ 0 if ξ ∈ R
n for J ∈ N0 ;
B2
) sup ,pϕ j
N J ⊂ Ω j
N J, ;
B3
) for any γ ∈ N0
n there exists a constant cγ > 0 such that for any J ∈ N0
D j
N Jγ ϕ ξ, ( ) ≤ | ξ |
γ
for any γ ∈ R
n
;
B4
) there exists a constant cϕ > 0 such that
0 <
j
j
N J
=
∞
∑
0
ϕ ξ, ( ) = cϕ < ∞ for any ξ ∈ R
n.
We will give a general definition of the spaces of generalized smoothness, Fpq
Nσ,
and Bpq
Nσ, , which are the generalizations of Triebel – Lizorkin and Besov spaces
respectively (see [6]).
Definition 3. Let ( )Nj j∈N0
be a strongly increasing sequence of bounded
growth, and let ( )σ j∈N0
be an admissible sequence.
i) Let 0 < p, q ≤ ∞. The Besov space of generalized smoothness is
Bpq
Nσ, = f S f D fB j j
N J
j l Lpq
N
q p
∈ ′ = ( ) < ∞
∈
: ( ),
,
( )
σ σ ϕ
N0
.
ii) The 0 < p < ∞, 0 < q ≤ ∞. The Triebel – Lizorkin space of generalized
smoothness is
Fpq
Nσ, = f S f D fF j j
N J
j L lpq
N
p q
∈ ′ = ( ) < ∞
∈
: ( ),
,
( )
σ σ ϕ
N0
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
640 V. P. KNOPOVA
Remark 1. For 0 < p , q < ∞ the spaces Bpq
Nσ, and Fpq
Nσ, can be defined
without the restriction that ( )Nj j∈N0
is of bounded growth.
Such spaces are the generalizations (due to the standardization theorem) of those
introduced by Kaljabin [7, 8]; they are equivalent to the spaces given in [7, 8] if σ is
strongly increasing and of bounded growth (see [6] for details).
Denote by σ
s the sequence ( )σ j
s
j∈N0
= ( )2
0
sj
j∈N
.
Remark 2. For Nj = 2
j and σ = σ
s the spaces Fpq
Nsσ , and Bpq
Nsσ , coincide with
Fpq
s and Bpq
s respectively (see [6]).
We are interested in the situation when the strongly increasing sequence N is
associated with a function which satisfies some additional conditions. For the
following definition we refer to [6] (see also [5]).
Definition 4. Let A be the class of all nonnegative functions a : R
n → R of
class C
∞ with the following properties:
A) lim ( )
ξ
ξ
→∞
a = ∞;
B) a ( ξ ) is almost increasing in | ξ |, i.e., there exist constants δ0 ≥ 1 and R >
> 0 such that a ( ξ ) ≤ δ0 a ( η ) if R ≤ | ξ | ≤ η;
C) there exists m > 0 such that a ( ξ ) | ξ |
– m is almost decreasing in | ξ |, i.e.,
there exists a constant δm
, 0 < δm ≤ 1, and R > 0 such that
a ( ξ ) | ξ |
– m ≥ δm
a ( η ) | η | – m if R ≤ | ξ | ≤ η;
D) for every multiindex α = ( α1 , … , αn ), α i ∈ N ∪ { 0 } , i = 1, … , n, there
exists some cα > 0 such that
| D α
a ( ξ ) | ≤ cα a ( ξ ) 1 2+( )−
ξ
α
, if | ξ | ≥ R.
The functions from A are called admissible functions.
It was proved that for an admissible symbol a ( ξ ) the sequence N a = Nj
a r
j
,( ) ∈N0
,
where
Nj
a r, = sup ξ ξ: ( )a rj≤{ }2 , j ∈ N0 , (1)
is strongly increasing, see Lemma 3.1.16 from [6]. Note that the definition of the
strongly increasing sequence by (1) does not require the radial symmetry of the
symbol.
Recall the definition of the ψ-Bessel potential space of order s:
Hp
s nψ, ( )R = u F Fus
Lp
: ( ) ( )− +( )( ) < ∞
/1 21 ψ ξ ξ ,
where ψ is a continuous negative definite function, Fu is the Fourier transform of
function u, and s > 0 (see [9] for details).
For the admissible continuous negative functions a ( ξ )
Fp
Ns
,
,
2
σ = Hp
a s, , s > 0
(see [6]).
For the following definition we refer to [2, p. 293, 294], or [10] (§ 1.9.1).
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, #5
CONTINUITY OF CERTAIN PSEUDODIFFERENTIAL OPERATORS IN THE SPACES … 641
Let G = { z ∈ C; 0 < Re z < 1 }. For two complex Banach spaces X X0 0
, ⋅( ) and
X X1 1
, ⋅( ) both embedded into some Hausdorff space χ, set X : = X0 + X1
, equipped
with the norm || ⋅ || X : = max ,⋅ ⋅( )X X0 1
which is equivalent to the norm ⋅ X0
+
+ ⋅ X1
, and which turns X into a Banach space. Denote by W ( G, X ) the space of
all continuous functions ω : G → X with the following properties:
1) ω G is analytic and sup ( )
z G
Xz
∈
ω < ∞;
2) ω ( i y ) ∈ X0 and ω ( 1 + i y ) ∈ X1 , for y ∈ R with continuous maps y �
� ω ( i y ) and y � ω ( 1 + i y );
3) ω W G X( , ) : = max sup ( ) , sup ( )ω ωi y i yX X0 1
1 +( ) < ∞.
By the maximum principle W G X W G X( , ), ( , )⋅( ) is the Banach space. We call
{ X0 , X 1 } an interpolation couple, and for any interpolation couple define its
complex interpolation space
[ X0 , X1 ] θ : = { u ∈ X; there exists ω ∈ W ( G, X ) such that ω ( θ ) = u}.
On [ X0 , X1 ] θ we introduce the norm
u X X0 1,[ ]θ
: = inf , ( , ) ( )( , )ω ω ω θW G X W G X u∈ =( )and . (2)
With this norm the space X X X X0 1 0 1
, , ,[ ] ⋅( )[ ]θ θ
is a Banach space.
3. Some properties of the spaces of generalized smoothness and generalized
Lipschitz spaces. In this section we give some auxiliary technical results, which will
be necessary to prove our main results in Section 5.
Lemma 1. Let 0 < p0
, q0
, p1 , q1 ≤ ∞, and 0 < θ < 1. Then for
1
p
=
1 − θ
p
+
+
θ
p1
,
1
q
=
1
0
− θ
q
+
θ
q1
and s = ( 1 – θ ) s0 + θ s1 we have
B Bp q
N
p q
Ns s
0 0
0
1 1
1
σ σ
θ
, ,,
= Bpq
Nsσ , .
Proof. Let ϕ j
N J
j
,{ } ≥0
∈ Φ N, J, and S f = f j
N J
j*
,ϕ{ } ≥0
. From the definition of
Bpq
Nsσ , we have
f ∈ Bpq
Nsσ , ⇔ S f ∈ l Lq
s
p( ),
where lq
s = ( ) : ( )a a lj j
s
j j q≥ ≥ ∈{ }0 02 . Therefore, by the definition of W ( G, X ) and
(2) we see that for 0 < θ < 1
B Bp q
N
p q
Ns s
0 0
0
1 1
1
σ σ
θ
, ,,
= f B Bp q
N
p q
Ns s
∈ +
0 0
0
0 0
1
σ σ, , : ∃ ω ∈ W G B Bp q
N
p q
Ns s
, , ,
0 0
0
1 1
1
σ σ+
,
such that ω ( θ ) = f } =
= { S f ∈ l Lq
s
p0
0
0
( ) + l Lq
s
p1
1
1( ) : ∃ ω ∈ W G l L l Lq p q
s
p
s
,
0
0
0 1
1
1
σ ( ) + ( )
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
642 V. P. KNOPOVA
such that g ( θ ) = S f } =
= S f S f l L l Lq
s
p q
s
p: ,∈ ( ) ( )[ ]{ }0
0
0 1
1
1 θ
= S f S f l Lq
s
p: ( )∈{ } = Bpq
Nsσ , .
The lemma is proved.
Remark 3. Similarly, for 0 < p0
, q0
, p1 , q1 < ∞ and 0 < θ < 1,
1
p
=
1 − θ
p
+
+
θ
p1
,
1
q
=
1
0
− θ
q
+
θ
q1
, s = ( 1 – θ ) s0 + θ s1 , we can prove that
F Fp q
N
p q
Ns s
0 0 1 1
0 1σ σ
θ
, ,,[ ] = Fpq
Nsσ , .
The embedding properties of the spaces Fpq
Nsσ , and Bpq
Nsσ , are similar to those of
Fpq
s and Bpq
s , since the construction of the spaces of generalized smoothness is
different to those of Fpq
s and Bpq
s only in the choice of the decomposition of unity
and the sequence σ
s. We will give a chain of embeddings, which will be useful for us
when we prove the mapping properties of certain pseudodifferential operators.
Lemma 2. Let N = Nj
a
j
,2
0
( ) ∈N
, 0 < q ≤ p < ∞, and 0 < ε < s < ∞. Then the
following embedding take place:
Fp
Ns
,
,
2
σ ⊂ Fp q
Ns
,
,σ ⊂ Bp p
Ns
,
,σ ⊂ Fp q
Ns
,
,σ ε−
.
Proof. Two left-hand embeddings follow from l q ⊂ l p for q ≤ p and the
observation that Fp p
Ns
,
,σ = Bp p
Ns
,
,σ , 2 ≤ p < ∞. Indeed, for f ∈ Fp q
Ns
,
,σ
f Bp p
s N
,
,σ ≤ c
f Fp p
s N
,
,σ = c D f
j
js
j
N J p
p
Lp
=
∞
∑
/
0
1
2 ψ ( ) ≤
≤ c D f
j
js
j
N J q
q
Lp
=
∞
∑
/
0
1
2 ψ ( ) = c
f Fp q
Ns
,
,σ ≤
≤ c D f
j
js
j
N J
Lp
=
∞
∑
/
0
2
1 2
2 ψ ( ) = c
f Fp
Ns
,
,
2
σ ,
i.e., we obtain two left-hand embeddings.
To obtain the right-hand embedding, we will follow Proposition 2.3.2/2 [7]. Note,
that for a sequence aj j( ) ≥0
j
j s
j
q
q
a
=
∞
−∑
/
0
1
2 ( )ε ≤ sup
j
j s
j
j
j q
q
a
≥ =
∞
−∑
/
0 0
1
2 2 ε ≤
≤ c a
j
js
jsup
≥0
2 .
Then for f ∈ Bpp
Nsσ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, #5
CONTINUITY OF CERTAIN PSEUDODIFFERENTIAL OPERATORS IN THE SPACES … 643
f Fpq
Nsσ ε− , = c D f
j
j s
j
N J q
q
Lp
=
∞
−∑
/
0
1
2 ( ) ( )ε ψ ≤
≤ c D f
j
js
j
N J
Lp
sup ( )
≥0
2 ψ ≤ c D f
j
js
j
N J p
p
Lp
=
∞
∑
/
0
1
2 ψ ( ) = c f Bpp
Nsσ , ,
which proves the right-hand embedding.
The lemma is proved.
Remark 4. Since S ( R
n
) is dense in Hp
a s, = Fp
Ns
,
,
2
σ , ( )Nj j∈N0
= Nj
a
j
,2
0
( ) ∈N
,
then all the embeddings in Lemma 2 are dense.
Denote by a ( D ) the operator with symbol a ( ξ ). For the following theorem we
again refer to [6] (see also Theorem 2.3.8 from [12]). Let a ( ξ ) and ( )Nj j∈N0
be as
before. For any real s we have
1 +( ) /a D r( ) µ
: Bpq
Ntσ µ+ , → Bp q
Nt
,
,σ , 0 < p, q ≤ ∞,
isomorphically.
Similarly to the Lipschitz spaces of order α one can define the spaces Λλµ ( )t
(see
[11]), where the function | t | α is substituted by some function λ
µ
(t). In what follows,
we denote by Λ1 the space of Lipschitz functions on R
n.
Definition 5. Let λ (t) : R
n → R+ be a continuous nonnegative function and µ >
> 0 be a real number such that
1) λ (t) → 0 as | t | → 0, and λ (t) → ∞ as | t | → ∞;
2) lim
( )
t
t
t→
+
0
1λµ
= ∞;
3) λ (t) has no other zeros except at t = 0.
Define
Λλµ ( )t
= f C f x t f x A tC∈ − − ≤{ }∞ ∞
: ( ) ( ) ( )λµ ,
and let
f
t
Λ
λµ ( )
= f ∞ + sup
( ) ( )
( )t
Cf x t f x
t>
− −
∞
0 λµ
be the norm in Λλµ ( )t
.
We need to make an additional assumption about the relation between the operator
a (D) and the function λµ( )t .
Assumption 1. Assume, that λµ( )t is such that 1 +( )− /a D r( ) µ maps C ∞
continuously into Λλµ ( )t
.
Of course the choice of such a function λµ( )t may be not unique.
Lemma 3. Let σ s = ( )2 0
j s
j≥ , N = Nj
a r
j
,( ) ≥0
be defined as in (1), and the
function ( λ
µ
(t) ) satisfy Assumption 1.
Then
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
644 V. P. KNOPOVA
B N
∞,
,
1
0σ ⊂ C∞ ⊂ B N
∞ ∞,
,σ0
(3)
and
B N
∞,
,
1
σµ
⊂ Λλ µ( )t
⊂ B N
∞ ∞,
,σµ
. (4)
Proof. The proof of (3) is similar to the proof of Proposition 2.5.7 [12]. Let
ϕ j
N J
j
,( ) ∈N0
∈ Φ
N,
J
, and f ∈ B N
∞,
,
1
0σ . Then
f L∞
=
j
j
N J
L
D f x
=
∞
∑
∞
1
ϕ , ( ) ( ) ≤
j
j
N J
L
D f x
=
∞
∑
∞1
ϕ , ( ) ( ) = f B N
∞,
,
1
0σ .
Since ϕ j
N J D f x, ( ) ( ) is bounded and continuous (by Paley – Wiener theorem), and
hence uniformly continuous in R
n, then f ∈ C∞ , and thus we have the left-hand
embedding in (3).
Now let f ∈ C∞ . Since by B3 the function ϕ j
N J x, ( ) is a Fourier multiplier, then
f B N
∞ ∞,
,σ 0 = sup ( ),
j
j
N J
L
D f
∈ ∞N0
ϕ ≤ sup ( ),
j
j
N J
L LD f
∈ ∞ ∞
N0
ϕ ≤ c f C∞
,
which proves the right-hand embedding. (By ϕ j
N J
L
D, ( )
∞
we understand the norm
of the operator which corresponds to the symbol ϕ ξj
N J, ( ) .)
By the lifting property of 1 +( )− /a D r( ) µ (see [6]) we get (4) from (3).
The lemma is proved.
Remark 5. Similarly to Proposition 2.3.2/2 [12], one can prove that B N
∞ ∞
+
,
,σµ ε
⊂
⊂ B N
∞,
,
1
σµ
for some ε > 0, which leads to
B∞ ∞
+
,
σµ ε
⊂ Λλ µ( )t
⊂ B N
∞ ∞,
,σµ
. (5)
Proof. We only need to show that B N
∞ ∞
+
,
,σµ ε
⊂ B N
∞,
,
1
σµ
holds for ε > 0, then (5)
will follow from (4).
Since for a sequence ( )aj j≥0 with aj -finite
j
j s
ja
=
∞
∑
0
2 ≤ sup ( )
j
j s
j
j
ja
≥
+
=
∞
−∑
0 0
2 2ε ε ≤ c a
j
j s
jsup ( )
≥
+
0
2 ε
holds, then for f ∈ B N
∞ ∞
+
,
,σµ ε
we obtain
f B N
∞,
,
1
σµ =
j
j s
j
N J
L
D f
=
∞
∑
∞1
2 ϕ , ( ) ≤ sup ( )( ) ,
j
j s
j
N J
L
j
jD f
≥
+
=
∞
−
∞
( ) ∑
0 0
2 2ε εϕ ≤
≤ c D f
j
j s
j
N J
L
sup ( )( ) ,
≥
+
∞
( )
0
2 ε ϕ = c f B N
∞ ∞
+
,
,σµ ε .
Let 0 < µ < 1. Define now by Λλ µ( )t
0 the closure of C0
∞ with respect to
⋅ Λ
λ µ( )t
-norm. Similarly to Theorem III.3.3 from [13], we have the following lemma.
Lemma 4. The space Λλ µ( )t
0 , 0 < µ < 1, coincides with the space of functions
from Λλ µ( )t
for which
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lim
( ) ( )
( )t
Cf t f
t→
⋅ − − ⋅
∞
0 λ µ = 0. (6)
Proof. Since C0
∞ ⊂ Λ1 ⊂ Λλ µ( )t
and C0
∞ is dense in Λ1
, we see, that the
space Λ
Λ
1
⋅
λµ ( )t is equivalent to C t
0
∞ ⋅ Λ
λµ ( ) .
We will follow the proof given in [13]. Clearly, for all functions from Λ 1 (6)
holds. Let f m ∈ Λ 1 and f m → f in Λλµ ( )t
. Fix ε > 0. There exists N0
, such that
fm Λ1
< c, f fm
t
− Λ
λµ ( )
<
ε
2
for all m ≥ N0
and let δ be such that for t < δ we have
t
tλµ ( )
<
ε
2c
.
Then
f x t f x
t
( ) ( )
( )
− −
λµ ≤
f x t f x t f x f x
t
m m( ) ( ) ( ) ( )
( )
− − − − −( )
λµ +
+
f x t f x
t
m m( ) ( )
( )
− −
λµ ≤ f fm
t
− Λ
λµ ( )
+ f
t
t
m Λ1 λµ ( )
<
ε
2
+
ε
2
= ε,
and f satisfies (6).
Let now f ∈ Λλµ ( )t
and satisfy (6). We will show that f belongs to Λλµ ( )t
0 , i.e.,
we will show that there exists a sequence ( )fm m≥0 from Λ 1 , that converges to f in
⋅ Λ
λµ ( )t
-norm.
Consider
f m
(
x
) = m f d
x
x m+ /
∫
1
( )τ τ – m f d
m
0
1/
∫ ( )τ τ.
Here and further we denote by
x
x m
d
+ /
∫ …
1
τ =
x
x m
x
x m
m
m
n
d d
1
1 1 1
1
+ +/ /
∫ ∫… … …τ τ ,
and analogously for
0
1/
∫ …
m
dτ .
The functions f m , m ≥ 0, are once continuously differentiable (by each x i ), and
therefore belong to Λ 1 . Changing the variables, we obtain
f m ( x ) = m f x f d
m
0
1/
∫ + −[ ]( ) ( )θ θ θ ,
which gives
f m ( x ) – f ( x ) = m f x f f x d
m
0
1/
∫ + − −[ ]( ) ( ) ( )θ θ θ .
Define
f x t f x tm( ) ( )+ − −[ ] – f x f xm( ) ( )−[ ] = Ψ ( f m – f, t );
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
646 V. P. KNOPOVA
in this notation
f fm
t
− Λ
λµ ( )
= f fm C−
∞
+ sup
( , )
( )t
mf f t
t>
−
0
Ψ
λµ .
Since f ∈ Λλµ ( )t
and (6) holds, then for chosen ε > 0 there exists δ > 0 such that for
| t | < δ f satisfies
f x t f x
t
( ) ( )
( )
+ −
λµ <
ε
2
∀x ∈ R
n.
Then
Ψ( , )
( )
f f t
t
m −
λµ =
m
t
f x t f x t f x f x d
m
λ
θ θ θµ ( )
( ) ( ) ( ) ( )
0
1/
∫ + + − + − + +[ ] ≤
≤ m
f x t f x
t
f x t f x
t
d
m
0
1/
∫ + + − + + + −
( ) ( )
( )
( ) ( )
( )
θ θ
λ λ
θµ µ < m
m
1
2 2
ε ε+
= ε.
Let | t | ≥ δ. Then
Ψ( , )
( )
f f t
t
m −
λµ = m
t
f x t f x t f x f x
d
m
0
1/
∫ + + − + + + −
λ θ
λ
θ
λ θ
θ
λ θ
θ
µ
µ µ µ
( )
( )
( ) ( )
( )
( ) ( )
( )
≤
≤ 2
0
1
m f
t
d
t
m
Λ
λµ
λ θ
λ
θ
µ
µ
( )
( )
( )
/
∫ ≤ 2
0
1
C f m d
t
m
Λ
λµ
λ θ θµ
( )
( )
/
∫ ,
where C is such that λ µ− ( )t ≤ C for | t | ≥ δ, and we again may chose large N0
such that for all m ≥ N0
m d
m
0
1/
∫ λ θ θµ ( ) <
ε
λµ
2C f
t
Λ
( )
.
Therefore f fm
t
− Λ
λµ ( )
< ε for m ≥ N0
, and in such a way f ∈ Λλµ ( )t
0 .
Lemma 4 is proved.
Let λ :
(
0, 1
) → R
+ be a nondecreasing, continuous function, lim ( )t t→0 λ = 0
and let for 1 ≤ p, q ≤ ∞ and M ∈ N
Bpq
nλ ( )R = f L
f t
t
d t
tp
p
M q q
∈
< ∞
∫
/
:
( , )
( )
( )
( )
0
1
1
ω
λ
λ
λ
,
where
ω p
M f t( , ) = sup ( )
h t
p
M
L
u
p<
⋅∆ ,
and ∆ p
M is the finite difference of order M in h.
In addition, let t �
λ( )t
tM be increasing, and t �
λ
δ
( )t
t
be almost decreasing, then
Bpq
nλ ( )R = Bpq
Nσ1, ,
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CONTINUITY OF CERTAIN PSEUDODIFFERENTIAL OPERATORS IN THE SPACES … 647
where (
σ
1
) = ( )2 0
j
j≥ , Nj =
1
hj
, where hj is such that λ
(
hj
) = 2 1− j λ( ) (see [14]).
In the following we suppose that λ
(
1
) = 1.
Clearly, if the function λµ ( )t satisfies the conditions above, we obtain from the
definition of Λλµ that
Λλµ ( )R
n = B n
∞∞
λµ
( )R . (7)
Moreover, if an operator 1 +( )a D( ) is an isomorphism between Bpq
Nσ µ1+ , and
Bpq
Nσµ , , it is an isomorphism between B∞∞
+σ µ1
and B∞∞
σµ
, where the function λ is
uniquely determined by a
(
ξ
).
4. The continuity of a pseudodifferential operator in generalized Lipschitz
spaces. Let the pseudodifferential operator be of the form
p ( x, D ) f ( x ) =
R
n
f x y f x x dy
\
( ) ( ) ( , )
0{ }
∫ − −( )ν , (8)
where ν( , )x dy is a Lévy measure, which depends on x.
Theorem 1. Let p ( x, D ) be as in (8), ν( , )x dy = g ( x, y ) dy, where the
function g ( x, y ) is differentiable in x and satisfies
sup ( ) ( , ) ( , )
x y
h
y
h
n
y g x y dy g x y dy
∈ <
+
≥
∫ ∫′ + ′
R 1
1
1
λµ < ∞ (9)
for any direction h, and
sup ( ) ( ) ( , ) ( ) ( , )
x y h y h
h
n
h y g x y dy h g x y dy
∈
−
<
+
>
∫ ∫+
R
λ λ λµ µ 1 → 0 as | h | → 0,
(10)
where ′g x yh( , ) is the derivative of g ( x, y ) with respect to x in direction h.
Then
p ( x, D ) : Λλµ+1
0
( )t
→ Λλµ ( )t
0 .
Proof. Let f ∈ Λλµ+1
0
( )t
. For such f we have
sup
( ) ( )
( )y
Cf x y f x
y>
+
− −
∞
0
1λµ < ∞.
Since
p x D f C( , )
∞
≤ c f
tΛλ
µ
( )
+1 , (11)
we will check weather the following inequality is satisfied:
sup
( )
( ) ( ) ( , )
h h
f x h y f x h x h dy
n>
∫ − − − −( ) −
0
1
λ
νµ
R
–
–
R
n
f x y f x x h dy
C
∫ − −( ) −
∞
( ) ( ) ( , )ν ≤ c f
tΛλ
µ
( )
+1 .
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648 V. P. KNOPOVA
Again, for | h | ≥ 1 we obtain
1
λ
νµ ( )
( ) ( ) ( , )
h
f x h y f x h x h dy
n
R
∫ − − − −( ) − –
R
n
f x y f x x dy∫ − −( )( ) ( ) ( , )ν ≤
≤ C f x h y f x h x h dy
n
R
∫ − − − −( ) −( ) ( ) ( , )ν +
+ C f x y f x x dy
n
R
∫ − −( )( ) ( ) ( , )ν ≤ 2C p x D f C( , )
∞
,
and then for f ∈ Λλµ+1
0
( )t
sup
( , ) ( ) ( , ) ( )
( )h
Cp x h D f x h p x D f x
h≥
− − −
∞
1 λµ ≤ c f
tΛλ
µ
( )
+1 . (12)
Next consider the case | h | < 1. It is convenient to decompose
p x h D f x h( , ) ( )− − – p x D f x( , ) ( ) =
=
R
n
f x h y f x h f x y f x g x y dy∫ − − − −( ) − − −( ){ }( ) ( ) ( ) ( ) ( , ) +
+
R
n
f x h y f x h g x h y g x y dy∫ − − − −( ) − −( )( ) ( ) ( , ) ( , ) = I1 + I2
,
and consider I1 and I2 separately.
From the mean-value theorem we have
g x h y g x y( , ) ( , )− − = h g x yh′ ( , )0
for some x0
, where ′g x yh( , ) is the derivative of g with respect to x in direction h,
and consequently in view of (9) we obtain
I C2 ∞
≤ h
f x h y f x h
y
y g x y dy
y
C
y
hsup
( ) ( )
( )
( ) ( , )
>
+
<
+− − − −
′∞ ∫
0
1
1
1
0λ
λµ
µ +
+ 2
1
0h f g x y dyC
y
h∞
≥
∫ ′ ( , ) ≤ h f
t
Λ
λµ +1 ( )
.
Therefore for I2 we obtain
I
h
C2 ∞
λµ ( )
≤
h
h
f
tλµ
λµ( ) ( )
Λ +1
= o h f
t
( )
+Λ
λµ 1 ( )
as | h | → 0.
For
I
h
C1 ∞
λµ ( )
we have using (9) and (10)
I
h
C1 ∞
λµ ( )
=
=
1
λ
µµ ( )
( ) ( ) ( ) ( ) ( , ) ( )
h
f x h y f x h f x y f x g x y dy
y h<
∫ − − − −( ) − − −( ){ }
+
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CONTINUITY OF CERTAIN PSEUDODIFFERENTIAL OPERATORS IN THE SPACES … 649
+
y h C
f x h y f x h f x y f x g x y dy
≥
∫ − − − −( ) − − −( ){ }
∞
( ) ( ) ( ) ( ) ( , ) ( )µ ≤
≤ sup
( ) ( ) ( ) ( )
( )y
Cf x h y f x h f x y f x
y>
+
− − − −( ) − − −( )
∞
0
1λµ ×
×
1
0
1
λ
λµ
µ
( )
( ) ( , )
h
y g x y dy
h
∫ + +
+ sup
( ) ( ) ( ) ( )
( )0 1
1
< <
+
− − − −( ) − − −( )
∞
h
Cf x h y f x y f x h f x
hλµ λ( ) ( , )h g x y dy
h
∞
∫ ≤
≤ C f o h
t
Λ
λµ+
( )
1 ( )
as h → 0,
and thus
sup
( , ) ( ) ( , ) ( )
( )h
Cp x h D f x h p x D f x
h≤
− − −
∞
1 λµ ≤ C f
t
Λ
λµ+1 ( )
. (13)
Combining (13) with (12) and (11), we arrive at
p x D f
t
( , )
( )
Λ
λµ
≤ C f
t
Λ
λµ+1 ( )
.
Moreover, we can see from (13) that
lim
( , ) ( ) ( , ) ( )
( )h
Cp x h D f x h p x D f x
h→
− − −
∞
0 λµ = 0,
which completes the proof.
5. Continuity of a pseudodifferential operator in some spaces of generalized
smoothness. In this section we give the theorem on the continuity of some
pseudodifferential operator in the Besov spaces of generalized smoothness.
We start with an auxiliary theorem, see [13] or [1] for the reference.
Theorem 2. Let X X0 0
, ⋅( ) and X X1 1
, ⋅( ) be two Banach spaces as above,
and let Y Y0 0
, ⋅( ) and Y Y1 1
, ⋅( ) be two Banach spaces satisfying the same
conditions as X0 and X 1
. Suppose that T :
X0 → X 1 is a bounded linear
operator such that A f ∈ Yk for f ∈ Xk
, and
A f Yk
≤ M fk Xk
, k = 0, 1.
Then A maps continuously Xθ = X X0 1,[ ]θ into Yθ = Y Y0 1,[ ]θ, and we have the
estimate:
A f Yθ
≤ M M f X0
1
1
−θ θ
θ
, θ ∈ [
0, 1
].
Next we need a theorem which gives the continuity of pseudodifferential operator
between the generalized Bessel potential spaces in L2
. For our convenience we quote
the necessary conditions.
Let us split the symbol p ( x, D ) into two parts:
p ( x, ξ ) = p1 ( ξ ) + p2 ( x, ξ ), (14)
where p1 :
R
n → C is a continuous negative definite function, and p2 :
R
n × R
n → C
is continuous.
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650 V. P. KNOPOVA
Assumption 2. We assume that the function p ( x, ξ ) admits the decomposition
(14), where p1 :
R
n → C is some continuous negative definite function, and p2 :
R
n ×
× R
n → C is continuous, and suppose that the following conditions are satisfies:
C1
. The function p1 satisfies for some γ0 > 0 and γ1
, γ2 ≥ 0:
γ0 a ( ξ ) ≤ Re p1 ( ξ ) ≤ γ1 a ( ξ ) for all | ξ | ≥ 1
and
| Im p1 ( ξ ) | ≤ γ2 Re p1 ( ξ ) for all ξ ∈ R
n.
C2
. For m ∈ N0 the function x � p2 ( x, ξ ) belongs to C
m, and the estimate
∂ ξα
x q x2( , ) ≤ ϕ ξα ( ) ( )x a1 +( )
holds for all α ∈ N0
n , | α | ≤ m, with ϕα ∈ L1 .
(See Assumption 2.3.5 from [2] for the reference.)
For the following theorem we refer to Proposition 2.3.6 and Theorem 2.3.11
from [2].
Theorem 3. Let the conditions C1 and C2 with m ≥ n + [ t ] + 1 of Assumption
2 hold for the symbol p ( x, ξ ) of the pseudodifferential operator p ( x, D ). Then
p ( x, D ) is continuous from H
a,
t
+
2 to H
a,
t
for any t ≥ 0.
Theorem 4. Let p ( x, D ) be as in Theorem 1, and in addition assume that it
satisfies the conditions of Theorem 3, and for large | ξ | it holds
a ( ξ ) ≥ | ξ |
α, 0 < α < 2 .
Then for s > p
n
p− + + −
1 2
2
α
µ( ) , p ≥ 2,
p ( x, D ) :
Bp p
Ns
,
,σ +1
→ Bp p
Ns
,
,σ
continuously, where N = ( )Nj j≥0 , Nj = ( ) ( )a j−1 22 .
Proof. From Theorem 3 we have
p ( x, D ) :
Ha t
2
2, + → Ha t
2
,
(15)
continuously, in particular, (15) holds for all t >
2
2
+ n
α
. For such t the space H
a,
2
is
continuously embedded into C∞ .
We know that B N
∞∞
+σ µ1 , = B∞∞
+λ µ1
= Λλµ+1( )t
, µ ≥ 0. By Theorem 1 the operator
p ( x, D ) is continuous from Λ
�
λµ+1( )t = B N�
∞∞
+σ α1 , to Λ
�
λµ ( )t = B N�
∞∞
σα , , i.e.,
p x D f B N( , ) ,
�
∞ ∞σα ≤
c f B N
�
∞ ∞
+σ α1 , .
Further, for t >
2
2
+ n
α
, we have
B
t N
22
1σ + , ⊂ Λ1 ⊂ B
N�
∞∞
+σ α1 , ,
B
t N
22
σ , ⊂ Λ1 ⊂ B
N�
∞∞
σα , ,
and these embeddings are dense.
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Since the norms in the interpolation spaces B B
t sN N
22
σ σ
θ
, ,, ∞∞[ ] and
B B
t sN N
22
σ σ
θ
, ,,
�
∞∞
coincide for t, s > 0, 0 < θ < 1, we obtain by Theorem 2 that p ( x, D ) is continuous
from Bp p
Ns
,
,σ +1
to Bp p
Ns
,
,σ for p =
2
1 − θ
and s = ( 1 – θ ) t + θ µ =
2 2t p
p
+ −( )µ
. Since
t >
2
2
+ n
α
then s > p
n
p− + + −
1 2
2
α
µ( ) = s̃ , and since s depends on t linearly
then for any s0 > s̃ there exists t0 >
2
2
+ n
α
such that s0 =
2 20t p
p
+ −( )µ
. Therefore
p ( x, D ) is continuous between Bp p
Ns
,
,σ +1
and Bp p
Ns
,
,σ for all s > p
n− +
1 2
α
+
+ ( )p − )2 µ and p ≥ 2.
Theorem 4 is proved.
Theorem 4 together with Lemma 2 give us that under the conditions of Theorem 4
p ( x, D ) :
Fp
Ns
,
,
2
1σ +
→ Bp p
Ns
,
,σ for s > p
n
p− + + −
1 2
2
α
µ( ) and p ≥ 2
continuously, or to make our notation easier we will write
p ( x, D ) :
Hp
a s, +1 → Bp p
Ns
,
,σ ,
where Hp
a s, = Fp
Ns
,
,
2
σ is an a-Bessel potential space.
Theorem 4 allows us to use such operators p ( x, D ) as perturbations of some
generators of Lp-sub-Markovian semigroups. To do this, we will quote Theorem 2.8.1
from [2], from where our result easily follows.
Theorem 5. Let ( – A, D ( A ) ) be a pseudodifferential operator which generates
a sub-Markovian semigroup in Lp , 1 < p < ∞. If an operator p ( x, D ) is L p-
dissipative, A-bounded, i.e., D ( p ( x, D ) ) ⊂ D ( A ), and for some ε ∈ [ 0, 1 ) and
δ > 0
p x D u Lp
( , ) ≤ ε Au Lp
+ δ u Lp
, u ∈ D ( A ),
and in addition ( – A – p ( x, D ), D ( A ) ) is an L p-Dirichlet operator, then ( – A –
– p ( x, D ), D ( A ) ) is a generator of an Lp-sub-Markovian semigroup.
We arrive at the following theorem:
Theorem 6. Let −( )ψ ψ( ), ,D Hp
2 be the generator of an Lp-sub-Markovian
semigroup, and let p ( x, D ) satisfy conditions of Theorem 4. Assume that for ψ ,
such that
˜ ( )( )ψ λ( )
− −1 1 1
x
x
= 1, (16)
the operator ˜ ( ), ˜ ,ψ ψD H 2( ) is ψ ( D )-bounded. Then the operator ( – ψ ( D ) – p ( x,
D ), Hp
ψ, 2
) is the generator of an Lp-sub-Markovian semigroup.
Proof. From (7) we see, that if ψ̃ satisfies (16) and p ( x, D ) satisfies the
conditions of Theorem 4, then p ( x, D ) is continuous from Hp
s˜ ,ψ +1 to Bp p
Ns
,
, ˜σ , where
Ñ = Ñ j j( ) ≥0
, Ñ j = sup : ˜ ( )ξ ψ ξ ≤{ }22 j .
Since our operator is a Dirichlet operator (as an operator with continuous negative
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
652 V. P. KNOPOVA
define symbol), and therefore it is dissipative, the statement of the theorem follows
from Theorem 5.
Acknowledgement. The author would like to thank Prof. Walter Farkas for fruitful
discussions and remarks.
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Received 26.10.2004,
after revision — 18.03.2005
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