456–458.In this paper, the behavior of the sequence of L p⺢nnorms of primitives of distributions general ized functions is studied depending on their spectrum on the support of their Fou
Trang 1Original Russian Text © H.H Bang, V.N Huy, 2011, published in Doklady Akademii Nauk, 2011, Vol 440, No 4, pp 456–458.
In this paper, the behavior of the sequence of
L p(⺢n)norms of primitives of distributions (general
ized functions) is studied depending on their spectrum
(on the support of their Fourier transform)
The following result was proved in [1]: let 1 ≤ p ≤ ∞
and f (m) ∈ L p(⺢), m = 0, 1, 2, … Then the limit
always exists and
where is the Fourier transform of f
This result shows that the behavior of the sequence
is completely characterized by the spectrum
of f, which is the support of the generalized function
This issue has been investigated by numerous authors
(see, e.g., [1–12]) A natural question arises as to what
happens if the derivatives are replaced by primitives
For p = 2, Tuan answered this question in [13]: Let f ∈
L2(⺢) and σ := inf{|ξ|: ξ ∈ supp } > 0 Then there
exists primitives I m f, I m f ∈ L2(⺢) for all m, and
where supp is understood as the smallest closed set
outside of which = 0 a.e
In the general case of 1 ≤ p ≤ ∞ and σ ≥ 0, this prob
lem was solved in [14] Below, the problem is studied in
the case of many variables and 1 ≤ p ≤ ∞ For this pur
f( )m p
1/m
mlim→ ∞
f( )m p
1/m
mlim→ ∞ = σf = sup{ξ : ξ suppfˆ∈ },
f
ˆ
f( )m p
1/m
f
ˆ
f
ˆ
I m f 2
1/m
mlim→ ∞ σ– 1
,
=
f
ˆ
f
ˆ
pose, we need the concept of the primitive of general ized functions, which was introduced by Vladimirov
for generalized functions in D'(a, b), a, b∈⺢ [15] Developing his idea, we define this concept for tem pered generalized functions of many variables Let
S(⺢n ) and S'(⺢n) be the Schwartz spaces of test func
tions and generalized functions, respectively For h ∈
S'(⺢n) and ∀ϕ ∈ S(⺢ n ), we define h( ϕ) := 〈h, ϕ〉 Let f ∈ S'(⺢ n ) and e j = (0, …, 0, 1, 0, …, 0) ∈ ⺞n, where 1 is at the jth place f ∈ S'(⺢ n) is called the
e j th primitive of f if ( f ) = f; i.e.,
For ϕ ∈ S(⺢ n), we define
where
If f ∈ S'(⺢ n ) is some e jth primitive, then
(1)
where g j ∈ S'(⺢ n) is defined as
I e j
D e j
I e j
I e j
f D e j
ϕ ,
〈 〉 –〈f,ϕ〉 ∀ϕ S ⺢n
( )
∈
=
ψ x( ) ϕ x ( ) θ x ( ) ϕ x j ( 1, , ,x2 … x j– 1, ,t x j+1,
∞
–
+ ∞
∫
–
=
x j+2, … x, n )dt,
Φ x( ) ψ x( 1, , ,x2 … x j– 1, ,t x j+1, x j+2, … x, n )dt,
∞
–
x j
∫
=
θ C0
∞
1 – ,1
( ), θ t ( ) t d
⺢
∫
I e j
I e j
f,ϕ
〈 〉 = – 〈f,Φ〉+ 〈g j,ϕ〉,
g j,ϕ
〈 〉 = f j(x1, , ,x2 … x j– 1,x j+1,x j+2,…, x n),∫
× ϕ x( 1, , ,x2 … x j– 1, ,t x j+1,x j+2,…, x n ) t d
⺢
Behavior of Sequences of Norms of Primitives
of Functions Depending on Their Spectrum
H H Banga and V N Huyb
Presented by Academician V.S Vladimirov January 20, 2011
Received May 11, 2011
DOI: 10.1134/S1064562411060263
a Institute of Mathematics, Vietnamese Academy of Science
and Technology, Hoang Quoc Viet Road 18, Cau Giay, Hanoi,
10307 Vietnam
email: hhbang@math.ac.vn
b Department of Mathematics, College of Science,
Hanoi National University, Hanoi, Vietnam
email: nhat_huy85@yahoo.com
MATHEMATICS
Trang 2BEHAVIOR OF SEQUENCES OF NORMS OF PRIMITIVES OF FUNCTIONS 673
Here, f j ∈ S'(⺢ n – 1) and
for every h ∈ S(⺢ n – 1 ) Then g j is the jth constant tem
pered generalized function Note that, if n = 1, then
g j is a constant Conversely, for any jth constant tem
pered generalized function g j, the functional f over
S(⺢n ) defined by formula (1) is the e j th primitive of f.
Thus, we have proved the following result: each func
tion f ∈ S'(⺢ n) has its own primitive f in S'(⺢n) and
every e j th primitive of f is given by (1), where g j is any
jth constant tempered generalized function.
Let I0f = f In what follows, for j = 1, 2, …, n and a
function f ∈ S'(⺢ n), the symbol f stands for some of
its e jth primitives; i.e., f ∈ ( f ) For any multi
index α ∈ ⺞n with |α| ≥ 1, we define Iαf = ( f),
where jα := max{j: αj ≥ 1}; i.e., Iαf ∈ ( f ).
Therefore, Dα(Iαf ) = f for all α ∈ ⺞n Let 1 ≤ p ≤ ∞ and
f ∈ L p(⺢n ) We write (Iαf ⊂ L p(⺢n) if for any α ∈
⺞n there exists an th primitive of f (denoted by
f) that belongs to L p(⺢n)
For δ > 0, let
It was shown above that ∀f ∈ S'(⺢ n) and ∀α ∈ ⺞n
there always exists a primitive Iαf ∈ S'(⺢ n) The ques
tion arises as to what happens if S'(⺢n) is replaced
byL p(⺢n)
A function f ∈ S'(⺢ n) is said to satisfy condition (O)
if there exists δ > 0 such that supp ⊂ (⺢n, δ) It turns
out that condition (O) ensures the existence of some
primitive of any order of f ∈ L p(⺢n)
Theorem 1 Let 1 ≤ p ≤ ∞, f ∈ L p(⺢n ), and condition
(O) be satisfied Then, for any j = 1, 2, …, n, there exists
precisely one e j th primitive of f, which is denoted by f,
such that f belongs to L p(⺢n ) and satisfies condition (O).
Moreover,
Assume that f ∈ L p(⺢n) but condition (O) does not
hold Then it is possible that there is no primitive f∈
L p(⺢n ), j = 1, 2, …, n This occurs, for example, if
p = ∞ and f ≡ 1 (then supp = {0}); if p = ∞ and f(x) =
f j,h
= I e j
f x ( ) θ x, ( )h x j ( 1, , ,x2 … x j– 1,x j+1,x j+2,…, x n)
I e j
I e j
I e j
I e j
ᏼe j
I
e
jα
I α e jα
–
ᏼe jα
I α e jα
–
)
α ⺞n
∈
e j
–
I α e jα
–
⺢n
δ
,
( ) ξ (ξ1, , ,ξ2 … ξn) ⺢n
:
∈
= {
= min{ξ1, ξ2, ,… ξn } δ }.>
f
ˆ
I e j
I e j
suppI e j
f = suppfˆ
I e j
f
ˆ
(then supp = {ξ ∈ ⺢n: ξj ∈ {0, –2, 2}, j = 1,
2, …, n}); and if 1 ≤ p < ∞ and f(x) = (then supp = [–2, 2]n)
Note also that, if p < ∞, f ∈ L p(⺢n ), and j = 1, 2, …, n,
then there exists at most one f ∈ L p(⺢n) Moreover,
if p = ∞ and f, f ∈ L∞(⺢n ) for some j ∈ {1, 2, …, n},
then f(x) + c = g(x) ∈ L∞(⺢n ) for all c ∈ ⺓, c ≠ 0, while there is no g ∈ L∞(⺢n)
Let p = ∞ and f(x) = Then supp =
{x∈⺢n : x j ∈ {–1, 1}, j = 1, 2, …, n} and there exists any αth primitive of f in L∞(⺢n)
Theorem 2 Let 1 ≤ p ≤ ∞ and f ∈ L p(⺢n ) Then there
exists at most one sequence (Iαf ⊂ L p(⺢n)
Theorem 3 Let 1 ≤ p ≤ ∞, f ∈ L p(⺢n ), and condition (O) be satisfied Then there exists precisely one sequence
of primitives (Iαf ⊂ L p(⺢n ) Moreover, supp = supp ∀α ∈ ⺞n
The following results hold for the sequence of norms of generalized primitives
Theorem 4 Let 1 ≤ p ≤ ∞, (Iαf ⊂ L p(⺢n ), f 0,
and condition (O) be satisfied Then there always exists the limit
Note that, if condition (O) is not satisfied, then Theorem 4 does not hold, since = 0 for all
α ∈ ⺞n: |αj | ≥ 1, j = 1, 2, …, n
Theorem 5 Let 1 ≤ p ≤ ∞, (Iαf ⊂ L p(⺢n),
andσ = (σ1, σ2, …, σn) ∈ Then supp ⊂ –∞, –σk] ∪ [σk, +∞) if and only if
(2)
Theorem 6 Let 1 ≤ p < ∞, (Iαf ⊂ L p(⺢n ), and
σ = (σ1, σ2, …,σn) ∈ Assume that supp ⊂ –∞, –σk] ∪ [σk, +∞) Then
x j
sin2
j= 1
n
x j
sin2
x j2
j= 1
n
∏
f
ˆ
I e j
I e j
I e j
x j
cos
j= 1
n
)
α ⺞n
∈
)
α ⺞n
α
f f
ˆ
)
α ⺞n
ξα
ξ suppfˆ∈
inf
ξα
ξ suppfˆ∈
inf
)
α ⺞n
∈
⺢+
n
f
k= 1
n
∏
σα Iαf p
)
α ⺞n
∈
⺢+
n
f
k= 1
n
∏
Trang 3674 BANG, HUY
(3)
Remark 1 Theorem 6 does not hold if p = ∞
Indeed, let σ ∈ and
Then supp ⊂ –∞, –σj] ∪ [σj, +∞) and
σα||Iαf||∞ = 1, α ∈ ⺞n
Let f ∈ S'(⺢ n ) and P(x) be a polynomial in n vari
ables Define
where the differential operator P(D) is obtained from
P(x) by making the substitution x j→ – Each ele
ment h ∈ ⺠( f ) is called a Pth primitive of f Let ᏼ0f = f
and ᏼf be some Pth primitive of f; i.e., ᏼf ∈ ⺠( f ), and
ᏼm + 1 f ∈ ⺠(ᏼm f ), m = 0, 1, 2, … Then P k (D)ᏼm + k f =
ᏼm f for k, m = 0, 1, … Let 1 ≤ p ≤ ∞ and f ∈ L p(⺢n) If
for any m = 0, 1, … there is a Pth primitive of ᏼm f
(denoted by ᏼm + 1 f ) that belongs to L p(⺢n), then we
write (ᏼm f ⊂ L p(⺢n)
Theorem 7 Let 1 ≤ p ≤ ∞, P(x) be a polynomial in n
variables, f 0, and (ᏼm f ⊂ L p(⺢n ) Then, for all
m ∈ ⺞,
Theorem 8 Let 1 ≤ p ≤ ∞, P(x) be a polynomial in n
variables, f 0, and (ᏼm f ⊂ L p(⺢n ) Then
Theorem 9 Let 1 ≤ p ≤ ∞, P(x) be a polynomial in n
variables, f 0, and (ᏼm f ⊂ L p(⺢n ) Assume that
has a compact support Then there always exist the limits
and
ACKNOWLEDGMENTS This work was supported by the Vietnamese National Foundation for Science and Technology Development, project no 101.01.50.09
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σα Iαf p
αlim→ ∞ = 0.
⺢+
n
f x( ) sinσj x j
j= 1
n
∏
=
f
j= 1
n
∏
⺠ f( ) h S' ⺢n
( ): P D ( )h f=
∈
=
i ∂
∂x j
)m∞=0
suppfˆ suppᏼm
f
ᏼm
f p
1/m
ξ suppfˆ∈
inf
≥
ᏼm
f p
1/m
,
mlim→ ∞
ᏼm
f p
1/m
mlim→ ∞ = P( )ξ
ξ suppfˆ∈
inf