Littlewood - Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces
We point out that when the Hardy - Littlewood maximal operator is bounded on the space $L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R}$, the well-known characterization of spaces $L^{p(t)}(\mathbb{R}^n),\quad 1 < p < \infty$, by the L...
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| author | Kopaliani, T. S. Копаліані, Т. С. |
| author_facet | Kopaliani, T. S. Копаліані, Т. С. |
| author_sort | Kopaliani, T. S. |
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| datestamp_date | 2020-03-18T19:49:49Z |
| description | We point out that when the Hardy - Littlewood maximal operator is bounded on the space
$L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R}$,
the well-known characterization of spaces $L^{p(t)}(\mathbb{R}^n),\quad 1 < p < \infty$, by the
Littlewood - Paley theory extends to the space $L^{p(t)}(\mathbb{R}^n).$ We show that if $n > 1,$ the Littlewood -Paley operator is bounded on $L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R},$
if and only if $p(t) =$ const. |
| first_indexed | 2026-03-24T02:39:36Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDC 517.5
T. S. Kopaliani (Tbilisi Univ., Georgia)
LITTLEWOOD – PALEY THEOREM ON Lp(t) n( )R SPACES∗∗∗∗
TEOREMA LYTTLVUDA – PELI
PRO PROSTORY Lp(t) n( )R
We point out that when the Hardy – Littlewood maximal operator is bounded on the space Lp t( ) ( )R ,
1 < a ≤ p t( ) ≤ b < ∞ , t ∈R , the well-known characterization of spaces Lp( )R , 1 < p < ∞ , by the
Littlewood – Paley theory extends to the space Lp t( ) ( )R . We show that if n > 1, the Littlewood –
Paley operator is bounded on Lp t n( ) ( )R , 1 < a ≤ p t( ) ≤ b < ∞ , t n∈R , if and only if p t( ) = const .
Vstanovleno, wo koly maksymal\nyj operator Xardi – Littlvuda obmeΩenyj na prostori
Lp t( ) ( )R , 1 < a ≤ p t( ) ≤ b < ∞ , t ∈R , dobre vidoma xarakteryzaciq prostoriv Lp( )R , 1 < p <
< ∞, teori[g Littlvuda – Peli poßyrg[t\sq na prostir Lp t( ) ( )R . Pokazano, wo u vypadku
n > 1 operator Littlvuda – Peli obmeΩenyj na Lp t n( ) ( )R , 1 < a ≤ p t( ) ≤ b < ∞ , t n∈R , todi i
til\ky todi, koly p t( ) = const .
1. Introduction. Let m be a bounded function on Rn. The operator T defined by
the Fourier transform equation ( )̂ ( )Tf x = m x f x( ) ˆ( ), x ∈ R
n, is called a multiplier
operator with multiplier m . Let ρ be an ( n -dimensional) rectangle and χρ the cha-
racteristic function of ρ . The operator Sρ having multiplier m = ρ and defined by
the equation
( )̂ ( )S f xρ = χρ( ) ˆ( )x f x , x ∈ R
n,
is called a partial sum operator.
Let a collection of disjoint rectangles ∆ = { }ρ be a decomposition of R
n ( i.e.,
∪ρ∈∆ = Rn) . Given a function f in the Schwartz class S ( )R
n , define
G f x( )( ) =
ρ
ρ
∈
∑
∆
S f x( )
/
2
1 2
, x ∈ R
n.
Let { }nk k = −∞
+∞ , nk > 0, k ∈ Z , be a lacunary sequence ( i.e., there is an a > 1 such
that n nk k+1/ ≥ a for all k ) . Let ∆ be the collection of all intervals of the form
[ ],n nk k+1 and [ ],− +n nk k1 , k ∈ Z . Then ∆ is called a lacunary decomposition of R .
When nk = 2
k, k ∈ Z, the resulting ∆ is called the dyadic decomposition of R .
∗
The author was supported by grant GNSF / STO 7 / 3-171.
© T. S. KOPALIANI, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12 1709
1710 T. S. KOPALIANI
Let ∆i , i = 1, 2, … , n , be n lacunary decomposition of R . Let ∆ be the
collection of the intervals of the form ρ = ρ ρ ρ1 2× × … × n where ρi i∈∆ . Then ∆
is called a lacunary decomposition of Rn.
The important feature of the classical Littlewood – Paley theory is that a characteri-
zation of the spaces Lp n( )R , 1 < p < ∞ . It is well known (see [1, 2]) that if ∆ is a
lacunary decomposition of Rn then G f p( ) is equivalent to f p for 1 < p <
< ∞ ; i.e., there are constants A and B such that
A f p ≤ G f p( ) ≤ f p .
The purpose of this paper is to obtain analogously characterizations of variable ex-
ponent Lebesgue spaces Lp t n( )( )R .
Given a measurable functions p n( ) : [ , )⋅ → ∞R 1 , Lp t n( )( )R denotes the set of
measurable functions f on Rn such that for some λ > 0
R
n
f x
dx
p x
∫
( ) ( )
λ
< ∞ .
This set becomes a Banach function space when equipped with the norm
f p t( ) = inf : ( ) ( )
λ
λ
>
≤
∫0 1f x
dx
p x
.
Given a locally integrable function f, we define the Hardy – Littlewood maximal
function M f by
M f ( x ) = sup ( )
x Q Q
Q
f y dy
∈
∫1 ,
where the supremum is taken over all cubes containing x with sides parallel to the co-
ordinate axes. For conciseness, define P ( )R
n to be the set of measurable function
p n( ) : [ , )⋅ → ∞R 1 such that
1 < a ≤ p ( t ) ≤ b < ∞ : t ∈ R
n.
Let B ( )R
n be the set of p
n( ) ( )⋅ ∈P R such that M is bounded on Lp t n( )( )R . Con-
ditions for the boundedness of the Hardy – Littlewood maximal operator on spaces
Lp t n( )( )R have been studied in [3 – 8]. Diening [8] studied the necessary and suffici-
ent conditions in terms of the conjugate exponent ′ ⋅p ( ), ( 1 1/ /( ) ( )p t p t+ ′ = 1, t(∈
∈ R
n
) . He has proved that p
n( ) ( )⋅ ∈B R is equivalent to ′ ⋅ ∈p n( ) ( )B R , he also
proved that if p n( ) ( )⋅ ∈B R then p q n( ) ( )/⋅ ∈B R for some q > 1.
In harmonic analysis a fundamental operator is the Hardy – Littlewood maximal
operator. In many applications a crutial step has been to show that operator M is bo-
unded on a variable Lp space. Cruz-Uribe, Fiorenza, Martell and Perez [4] have show-
ed that many classical operators in harmonic analysis such as singular integrals, com-
mutators and fractional integrals are bounded on the variable Lebesgue space Lp t n( )( )R
whenever the Hardy – Littlewood maximal operator is bounded on Lp t n( )( )R .
If we consider, instead, the strong maximal operator
MR defined by
M f xR ( )( ) = sup ( )
x R R
R
f x dx
∈
∫1
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
LITTLEWOOD – PALEY THEOREM … 1711
where R is any rectangle in Rn , n > 1, with sides parallel to the coordinate axes
then the situation is different. For the strong Hardy – Littlewood maximal operator
MR we prove following theorem.
Theorem 1. Let 1 ≤ p ( t ) ≤ b < ∞ , t ∈ R
n. The strong Hardy – Littlewood
maximal operator
MR is bounded on Lp n( )( )⋅
R space if and only if p ( t ) =
= const = p and p > 1.
For function f L n∈ ( )R , the expression
H f ( x ) =
R
n i
n
k kx y
f y dy∫ ∏
= −1
1 ( )
is said to be n-dimensional ( )n > 1 Hilbert operator.
Analogously we may prove following theorem.
Theorem 2. Let 1 ≤ p ( t ) ≤ b < ∞ , t ∈ R
n. Then n-dimensional Hilbert ope-
rator ( )n > 1 is bounded on Lp n( )( )⋅
R space if and only if p ( t ) = const = p
and p > 1.
We prove following Littlewood – Paley type characterization of Lp t n( )( )R space.
Theorem 3. 1. Let ∆ be a lacunary decomposition of R and p( ) ( )⋅ ∈B R .
Then there are constants c, C > 0 such that for all f Lp t∈ ( )( )R
c f p t( ) ≤ G f p t( ) ( ) ≤ C f p t( ). (1)
2. Let ∆ be the dyadic decomposition of Rn, n > 1. If p( )⋅ ≠ const then
operator G is not bounded on Lp t n( )( )R .
2. Proof of theorems. Proof of Theorem 1. According to Jessen, Marcinkiewicz
and Zygmund [9] MR is bounded on all the Lp, p > 1, spaces and first part of
Theorem 1 is trivial.
Let MR is bounded on Lp n( )( )⋅
R . Virtue of interpolation theorem (see [10]), we
have
MR is bounded on Lp( )/⋅ θ = [ ]( )( ), ( )L Lp n n⋅ ∞
R R θ, 0 < θ < 1, and without
restriction of generality we may assume that 1 < inf ( )
R
n p t . Let 1 1/ /( ) ( )p t p t+ ′ =
= 1, t(∈ R
n
. Note that
sup ( ) ( )
R
R p t R p tR
1 χ χ ′ < ∞ (2)
condition is necessary for boundedness of MR on Lp t n( )( )R (see proof below).
We will give the proof of second part of Theorem 1 for the case n = 2 for simpli-
city, since the same argument holds when n > 2.
Let inf ( )
R
2 p t < sup ( )
R
2 p t . By Luzin’s theorem we can construct pairwise dis-
joint family of set Fi with the following condition: 1) R
2 \ ∪ Fi = 0, 2) functions
p Fi: → R are continuous, 3) for every fixed i all points of Fi are points of densi-
ty with respect to basis R .
Note that, we can find pair of points (( , ), ( , ))x y x y0 1 0 2 - or (( , ), ( , ))x y x y1 0 2 0 -type
from ∪ Fi such that p x y( , )0 1 ≠ p x y( , )0 2 or p x y( , )1 0 ≠ p x y( , )2 0 . Without loss of
generality, we may suppose that this pair is (( , ), ( , ))x y x y0 1 0 2 ; ( , )x y F0 1 1∈ ,
( , )x y F0 2 2∈ and y1 < y2
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
1712 T. S. KOPALIANI
Let 0 < ε < 1 be fixed. We may find δ > 0 such that for any rectangles
Q x y1 0 1' ( , ) ,
Q x y2 0 1' ( , ) with diameters loss than δ the following inequalities are
valid:
Q F1 1∩ > ( )1 1− ε Q , Q F2 2∩ > ( )1 2− ε Q , (3)
pQ1
= sup ( , )
Q F
p x y
1 1∩
< c1 < c2 <
inf ( , )
Q F
p x y
2 2∩
= pQ2
(4)
for some constant c1, c2.
Let Q1,τ , Q2,τ are rectangles with properties (3), (4) of the form ( , )x x0 0− +τ τ ×
× ( , )a b , ( , ) ( , )x x c d0 0− + ×τ τ , where a < b < c < d.
We have continuously embedding
L Q L Qp t pQ( )
, ,( ) ( )2 2
2
τ τO and L Qp t′( )
,( )1 τ O
O L Q
pQ′ 1
1( ),τ , where 1 1
1 1/ /′ +p pQ Q = 1 (see for example [11]). For rectangle Qτ =
= ( , ) ( , )x x a d0 0− + ×τ τ we have
Aτ = 1
Q Q p t Q p tτ
χ χ
τ τ( ) ( )′
≥
1
2 2 2 1 1τ
χ χ
τ τ( ) , ,( ) ( )d a Q F p t Q F p t− ′∩ ∩ ≥
≥ C
d a
d c b a
p pQ Q
2
2 2
1 1 1
2 1
τ
τ τ
( )
( ( )) ( ( ))
/ /
−
− − −
.
Note that if τ → 0 ( a, b, c, d is fixed ) Aτ → ∞ and consequently (2) is not va-
lid. This completes the proof.
Proof of Theorem 3. The inequalities (1) are consequence of the extrapolation
theorem given by Cruz-Uribe, Fiorenza, Martell and Perez [4] and the weighted norm
inequalities for G ( f ) function given by Kurtz [12]. We describe this results.
Let p− = ess inf{ ( ) : }p x x ∈R . By a weight we mean a nonnegative, locally in-
tegrable function ω . When 1 < p < ∞ , we say ω ∈Ap if for every interval Q
1 1 1
1
Q
x dx
Q
x dx
Q Q
p
p
∫ ∫ − ′
−
ω ω( ) ( ) ≤ C < ∞ .
The infimum over the constants on the right-hand side of the last inequality we de-
note by Ap,ω . By F will denote a family of ordered pairs of nonnegative, measurable
functions ( f , g ) . We say that an inequality
R
∫ f x x dxp( ) ( )0 ω ≤ C g x x dxp
R
∫ ( ) ( )0 ω , 0 < p0 < ∞ , (5)
holds for any ( f , g ) ∈ F and ω ∈ Aq (for some q, 1 < q < ∞ ) if it holds for any
pair in F such that the left-hand side is finite, and the constant C depends only on p0
and the Aq,ω constant of ω .
Theorem 4. Given a family F , assume that (5) holds for some 1 < p0 < ∞ ,
for every weight ω ∈ Ap0
and for all ( f , g ) ∈ F . Let p( ) ( )⋅ ∈P R be such that
there exists 1 < p1 < p– , with ( / )( ) ( )p p⋅ ′ ∈1 B R . Then
f p t( ) ≤ C g p t( )
for all ( f , g ) ∈ F such that f Lp t∈ ( )( )R .
Theorem 5 [12]. Let ∆ be a a lacunary decomposition of R , 1 < p < ∞ , and
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
LITTLEWOOD – PALEY THEOREM … 1713
ω(∈ Ap
. Then there exist constant c, C depending only on p , Ap,ω , and ∆ ,
such that
c f x x dxp
R
∫ ( ) ( )ω ≤
R
∫ ( ( )( )) ( )G f x x dxpω ≤ C f x x dxp
R
∫ ( ) ( )ω .
From assumption of Theorem 3 we get that there exists 1 < p1 < p– with
( / )( ) ( )p p⋅ ′ ∈1 B R (see [8]). Let Lcomp
∞ ( )R be the set of all bounded functions with
compact support. From Theorems 4, 5 with the pairs ( ),W f f we get right side in-
equality of (1) if f L∈ ∞
comp( )R . Note that Lcomp
∞ ( )R is dense in Lp t( )( )R (see [11])
and consequently this inequality is also valid for all f Lp t∈ ( )( )R . Analogously we ob-
tain left side inequality of (1).
Let n > 1. Fix a rectangle R = I I In1 2× × … × and let f be positive on R and
0 elsewhere function. Let kj be the greatest integer such that 2
kj ≤ ( )4 1n Ij
− and
ρ be the dyadic rectangle [ ],2 21 1 1k k + × … × [ ],2 2 1k kn n + . Note that (see [12, p. 246])
for all x ∈ R
S f xρ ( ) ≥ C
R
f x dx
R
∫ ( ) .
Let the operator G is bounded on Lp t n( )( )R . Then for some constant C we have
1
Q
f x dx
R
R p t∫ ( ) ( )χ ≤ C f p t( ). (6)
Note that ( ( ))( )Lp t n
R
∗ is isomorphic to the space Lp t n′( )( )R , where 1/ ( )p t +
+ 1/ ( )′p t = 1, t(∈ R
n (see [11]). Therefore, for all rectangle R , from (6) we get
condition (2). We use Theorem 1 to obtain the desired result.
3. Applications. We now consider applications of Theorem 3. In [7] is proved
following theorem.
Theorem 6. Let p( ) ( )⋅ ∈P R and exponent p( )⋅ is constant outside some
large ball. Then operator M is bounded on Lp t( )( )R if and only if (2) fulfilled
for intervals.
The estimate (2) is necessary for boundedness of operator M in Lp t( )( )R . Com-
bining the Littlewood – Paley type characterization of Lp t( )( )R space (Theorem 3)
with the previous theorem we can obtain the following corollary.
Corollary. Let p( ) ( )⋅ ∈P R and exponent p( )⋅ is constant outside some lar-
ge ball. Let ∆ be the dyadic decomposition of R . The following are equivalent:
1) p( ) ( )⋅ ∈B R ;
2) there are constants c, C > 0 such that for all f Lp t∈ ( )( )R
c f p t( ) ≤ G f p t( ) ( ) ≤ C f p t( ).
Let { }fk be a sequence of functions defined on R . By f Lkk
p t∑ ∈ ( )( )R we me-
an the partial sums fk
N
1∑ converge in Lp t( )( )R . We now will generalize Theorem 6
of Stein [13].
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
1714 T. S. KOPALIANI
Theorem 7. Let p( ) ( )⋅ ∈B R , and Sk be any collection of lacunary partial
sum operators. Then f Lp t∈ ( )( )R if and only if εk kk
S f∑ converges in
Lp t( )( )R for any sequence { }εk l∈ ∞ . Moreover , f p t( ) is equivalent to
sup { } ( )ε ε
k l
k kk p t
S f∞ = ∑1 .
Proof. Let p( ) ( )⋅ ∈B R and Sk be any collection of lacunary partial sum. For
f Lp t∈ ( )( )R we have S f Lkk
p t2 1 2∑( ) ∈
/ ( )( )R . Note that if { }εk l∈ ∞ then
εk kk
p tS f L2 1 2∑( ) ∈
/ ( )( )R and
εk k
k
S f 2
1 2
∑
/
≤ { }
/
εk l k
k
S f∞ ∑
2
1 2
.
If N > M using Theorem 3,
εk k
M
N
p t
S f
+
∑
1 ( )
≤ C S fk l k
M
N
p t
{ }
/
( )
ε ∞
+
∑
2
1
1 2
which implies εk k
N
S f
1 1
∑{ }∞
is Cauchy in Lp t( )( )R . From this follows
εk k
k p t
S f∑
( )
≤ c S fk l k
k p t
{ }
/
( )
ε ∞ ∑
2
1 2
.
Assume that Sk be any collection of lacunary partial sum operators and
εk kk
S f∑ ∈ Lp t( )( )R for all { }εk l∈ ∞ . We will prove that S f Lkk
p t2 1 2∑( ) ∈
/ ( )( )R
and there exists a constant c > 0 independent of f such that
S fk
k p t
2
1 2
∑
/
( )
≤ c S f
k l
k k
k p t
sup
{ } ( )ε
ε
∞ =
∑
1
. (7)
First we will prove that M = sup { } ( )ε ε
k l
k kk p t
S f∞ = ∑1 is finite. Indeed, consi-
der the collection of maps G l LN
p t: ( )( )∞ →{ }R defined by GN k( ){ }ε = εk kk
N
S f=∑ 1
.
Let G = G∞
. Each GN is continuous and by assumption GN k( ){ }ε converges to
G k( ){ }ε in Lp t( )( )R for each { }εk l∈ ∞ . Therefore GN k p t N
( ){ } ( )ε{ } =
∞
1
is bounded
for each { }εk l∈ ∞ . By the principle of uniform boundedness, there exists a constant
c > 0 such that GN ≤ c for all N. It follows that G ≤ c .
To proof of (7) will use Khinchine’s inequality for Rademacher series. Let r tk( ) =
= sgn(sin )2m tπ , m = 0, 1, 2, … , be the Rademacher functions, and set f =
= a rm m0
∞∑ . Then there are constants Bp and Cp such that for 0 < p < ∞
B f t dtp
p
p
0
1 1
∫
( )
/
≤
m
ma
=
∞
∑
0
2
1 2/
≤ C f t dtp
p
p
0
1 1
∫
( )
/
(8)
(see [2]). Let εk = r tk( ) for 0 ≤ t < 1. Then { }εk l∞ = 1 and
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
LITTLEWOOD – PALEY THEOREM … 1715
M ≥ r x S fk k
k p t
( )
( )
∑ .
Using (8) for p = 1 and Fubini’s theorem we have
S fk
k p t
2
1 2
∑
/
( )
≤ C r x S f dx
k
k k
p t
1
0
1
∫ ∑ ( )
( )
≤
≤ cC r x S f z dx g z dz
g
k k
kp t
1
1 0
1
sup ( ) ( ) ( )
( )′ ≤
∫ ∫ ∑
R
=
= cC r x S f z g z dz dx
g
k k
kp t
1
1 0
1
sup ( ) ( ) ( )
( )′ ≤
∫ ∫ ∑
R
≤
≤ cC r x S f z g z dz dx
g
k k
kq t
1
0
1
1
∫ ∫ ∑
≤
sup ( ) ( ) ( )
( ) R
≤
≤ cC r x S f dxk k
k p t
1
0
1
∫ ∑ ( )
( )
≤ cC M1 ,
which proves (7). This completes the proof of Theorem 7.
1. Stein E. M. Singular integrals and differentiability properties of functions. – Princeton: Princeton
Univ. Press, 1970.
2. Zygmund A. Trigonometric series. – 2nd ed. – London; New York: Cambridge Univ. Press, 1959.
3. Diening L. Maximal function generalized Lebesgue spaces Lp( )⋅ // Math. Inequal. Appl. – 2004.
– 7. – P. 245 – 254.
4. Cruz-Uribe D., Fiorenza A., Martell J. M., Perez C. The boundedness of classical operators on
variable Lp spaces // Ann. Acad. sci. fenn. math. – 2006. – 31. – P. 239 – 264.
5. Cruz-Uribe D., Fiorenza A., Neugebauer C. J. The maximal function on variable Lp spaces //
Ibid. – 2003. – 28. – P. 223 – 238; 2004. – 29. – P. 247 – 249.
6. Nekvinda A. Hardy – Littlewood maximal operator on L Rp x n( ) ( ) // Math. Inequal. Appl. – 2004.
– 7. – P. 255 – 266.
7. Kopaliani T. S. Infinitesimal convolution and Muckenhoupt Ap( )⋅ condition in variable Lp
spaces // Arch. Math. – 2007. – 89, # 2. – P. 185 – 192.
8. Diening L. Maximal function on Orlicz – Musielak spaces and generalized Lebesgue spaces //
Bull. sci. math. – 2005. – 129. – P. 657 – 700.
9. Jessen B., Marcinkiewicz J., Zygmund A. Note on the differentiability of multiple integrals //
Fund. Math. – 1935. – 25. – P. 217.
10. Diening L., Hästö P., Nekvinda A. Open problems in variable exponent Lebesgue and Sobolev
spaces // FSDONA 04 Proc. (Milovy, Czech. Rep., 2004) / Eds Drabek and Rakosnik. – P. 38 – 58.
11. Kováčik O., Rákosnik J. On spaces Lp t( ) and W k p x, ( ) // Czech. Math. J. – 1991. – 41, # 4. –
P. 592 – 618.
12. Kurtz D. S. Littlewood – Paley and multiplier theorems on weighted Lp spaces // Trans. Amer.
Math. Soc. – 1980. – 259. – P. 235 – 254.
13. Stein E. M. Classes H p , multiplicateurs et fonctions de Littlewood – Paley // C. r. Acad. sci. –
1966. – 263. – P. 716 – 719, 780 – 781.
Received 15.10.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 12
|
| id | umjimathkievua-article-3283 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:39:36Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1b/b008d17efe89cc9e87534e69a9a2531b.pdf |
| spelling | umjimathkievua-article-32832020-03-18T19:49:49Z Littlewood - Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces Tеорема Литтлвуда - Пелі про простори $L^{p(t)}(\mathbb{R}^n)$ Kopaliani, T. S. Копаліані, Т. С. We point out that when the Hardy - Littlewood maximal operator is bounded on the space $L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R}$, the well-known characterization of spaces $L^{p(t)}(\mathbb{R}^n),\quad 1 < p < \infty$, by the Littlewood - Paley theory extends to the space $L^{p(t)}(\mathbb{R}^n).$ We show that if $n > 1,$ the Littlewood -Paley operator is bounded on $L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R},$ if and only if $p(t) =$ const. Встановлено, що коли максимальний оператор Харді - Літтлвуда обмежений на просторі $L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R}$, добре відома характеризація просторів $L^{p(t)}(\mathbb{R}^n),\quad 1 < p < \infty$ теорією Літтлвуда - Пелі поширюється на простір $L^{p(t)}(\mathbb{R}^n).$ Показано, що у випадку $n > 1,$ оператор Літтлвуда - Пелі обмежений на $L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R}$, тоді і тільки тоді, коли $p(t) =$ const. Institute of Mathematics, NAS of Ukraine 2008-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3283 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 12 (2008); 1709 – 1715 Український математичний журнал; Том 60 № 12 (2008); 1709 – 1715 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3283/3312 https://umj.imath.kiev.ua/index.php/umj/article/view/3283/3313 Copyright (c) 2008 Kopaliani T. S. |
| spellingShingle | Kopaliani, T. S. Копаліані, Т. С. Littlewood - Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces |
| title | Littlewood - Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces |
| title_alt | Tеорема Литтлвуда - Пелі про простори
$L^{p(t)}(\mathbb{R}^n)$ |
| title_full | Littlewood - Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces |
| title_fullStr | Littlewood - Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces |
| title_full_unstemmed | Littlewood - Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces |
| title_short | Littlewood - Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces |
| title_sort | littlewood - paley theorem on $l^{p(t)}(\mathbb{r}^n)$ spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3283 |
| work_keys_str_mv | AT kopalianits littlewoodpaleytheoremonlptmathbbrnspaces AT kopalíaníts littlewoodpaleytheoremonlptmathbbrnspaces AT kopalianits teoremalittlvudapelíproprostorilptmathbbrn AT kopalíaníts teoremalittlvudapelíproprostorilptmathbbrn |