Rend. Sem. Mat. Univ. Padova 1xx (201x) Rendiconti del Seminario Matematico
della Università di Padova
c European Mathematical Society
⃝
Marcinkiewicz-type interpolation theorem
for Morrey-type spaces and its corollaries
Burenkov V.I.
∗
– Chigambayeva D.K.∗∗– Nursultanov E.D.‡
λ
Abstract – We introduce a class of Morrey-type spaces Mp,q
, which includes classical
Morrey spaces and discuss their properties. We prove a Marcinkiewicz-type interpolation theorem. This theorem is then applied to obtaining the boundedness in the
introduced Morrey-type spaces of the Riesz potential and singular integral operator.
Mathematics Subject Classification (2010). 42B35, 47B06, 47B38, 46B70, 47G10.
Keywords. Morrey spaces, Morrey-type spaces, interpolation theorem, Riesz potential,
singular integral operator.
∗ The author was supported by the Russian Foundation for Basic Research (project 14-0100684).
∗∗ The author is grateful to the Padova University (Padova) and L.N. Gumilyov Eurasian
National University (Astana) for hospitality during the writing of this paper.
‡ The author is grateful to Lomonosov Moscow State University (Kazakhstan branch, Astana)
and L.N. Gumilyov Eurasian National University (Astana) for hospitality during the writing of
this paper.
Burenkov V.I., Sreklov Institute of Mathematics, 8 Gubkin st., Moscow, Russia, and Cardiff
School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK
E-mail: burenkov@cardiff.ac.uk
Chigambayeva D.K., Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian
National University, 2 Satpayev st., Astana, Kazakhstan
E-mail: d.darbaeva@yandex.kz
Nursultanov E.D., Department of Mechanics and Mathematics, Lomonosov Moscow State
University (Kazakhstan branch), 11 Kazhimukan st., Astana, Kazakhstan
E-mail: er-nurs@yandex.ru
2
Burenkov V.I. – Chigambayeva D.K. – Nursultanov E.D.
1. Introduction
Let 0 < p ≤ ∞ and 0 ≤ λ ≤ np . The Morrey spaces Mpλ were defined as the spaces
n
of all functions f ∈ Lloc
p (R ) such that
||f ||Mpλ ≡ ||f ||Mpλ (Rn ) = sup sup t−λ ∥f ∥Lp (Bt (x)) < ∞,
x∈Rn t>0
where Bt (x) is the open ball of radius t > 0 with center at the point x ∈ Rn .
n/p
If λ = 0, then Mp0 (Rn ) = Lp (Rn ), while if λ = np , then Mp (Rn ) = L∞ (Rn ).
If λ < 0 or λ > np , then Mpλ = Θ, where Θ is the set of all functions that are
equivalent to zero on Rn . See [1].
Classical Morrey spaces were introduced by Morrey in 1938 and arose in connection with some problems of the theory of partial differential equations and
theory of variations [1]. There is a number of books and survey papers on Morrey
and Morrey-type spaces and classical operators of real analysis in Morrey-type
spaces, for example, [1]-[14].
This paper is devoted to the interpolation properties of Morrey-type spaces.
Some results for classical Morrey spaces were obtained in Stampacchia [15], Campanato and Murthy [16], Peetre [6]. In particular in [6] it is proved that
(Mpλ0 , Mpλ1 )θ,∞ ⊂ Mpλ ,
where 1 ≤ p < ∞, λ = (1 − θ)λ0 + θλ1 , 0 < θ < 1. In Ruiz and Vega [17], Blasco,
Ruiz and Vega [18] it is proved that this inclusion is strict.
In [5] a more detailed investigation of the interpolation problem for Morrey
spaces was carried out. In particular, it was proved that the inclusion
(Mpλ00 , Mpλ11 )θ,∞ ⊂ Mpλ ,
θ
where 1 ≤ p0 , p1 < ∞, 0 < λ0 < pn0 , 0 < λ1 < pn1 , 0 < θ < 1 and p1 = 1−θ
p0 + p1 ,
λ = (1 − θ)λ0 + θλ1 holds if and only if p0 = p1 .
λ
The case of local Morrey-type spaces LMp,q
was considered in Burenkov and
λ
n
Nursultanov [19]. Let 0 < p, q ≤ ∞, λ ≥ 0. A function f ∈ LMp,q
if f ∈ Lloc
p (R )
and
∥f ∥LMp,q
= ∥t−λ ∥f ∥Lp (B0 (t)) ∥Lq (0,∞) < ∞.
λ
λ
In [19] it was proved, in particular, that local Morrey-type spaces LMp,q
form an
interpolation scale when p is fixed, i.e.
λ0
λ1
λ
,
(LMp,q
, LMp,q
) = LMp,q
0
1 θ,q
where 0 < p, q0 , q1 , q ≤ ∞, 0 < θ < 1 and λ = (1 − θ)λ0 + θλ1 .
Further generalization of interpolation properties for general local Morrey-type
λ
spaces LMp,q
(G, µ) was discussed in [20], namely, it was proved that
λ0
λ1
λ
(LMp,q
(G, µ), LMp,q
(G, µ))θ,q = LMp,q
(G, µ) ,
0
1
Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries 3
under the same assumptions on the numerical parameters.
λ
In this paper we introduce a generalized class of Morrey-type spaces Mp,q
(Ω),
λ
which coincide with the classical Morrey space Mp in the case of q = ∞ and
Ω = Rn . According to the above results and also [21] it follows that the classical
interpolation theorems for this scale of the spaces do not take place. Nevertheless,
in contrast to the classical interpolation theorems we prove some analog of interpolation theorem. Compared with the classical interpolation theorems the condition
λ
of theorem is formulated in terms of local Morrey-type spaces LMp,q,x
and the
λ
statement in terms of generalized Morrey-type spaces Mp,q (Ω). At the same time
we are saying that this theorem is an interpolation theorem of Marcinkiewicz-type,
because it allows us to obtain from the weak estimates for linear and quasi-linear
operators in terms of local Morrey-type spaces the stronger estimates in terms of
generalized Morrey-type spaces.
Let f ∈ Lloc
1 . In this paper we also study estimates for the norm of Riesz
operator, which is defined as follows
∫
Iγ f (x) =
Rn
f (y)dy
, 0 < γ < n.
|x − y|n−γ
Let us consider the well-known Hardy-Littlewood-Sobolev inequality. Let
1 < p < q < ∞,
γ=
n n
− ,
p
q
then operator Iγ is bounded from Lp to Lq .
The boundedness of Iγ in classical Morrey spaces and their generalizations
was investigated by Adams [9], Chiarenza and Frasca [10], Peetre [6], Nakai [11],
Sawano and Tanaka [12], and others. Further results of the boundedness of Riesz
potential for local Morrey-type spaces were obtained in Burenkov, Gogatishvili,
Guliyev and Mustafayev [13]. In this paper we continue the study of the boundedness of Iγ in the case of Morrey-type spaces. In particular, by applying the
Marcinkiewicz-type interpolation theorem we obtain estimate for the norm of Riesz
operator in the Morrey-type spaces.
Also we study the boundedness of singular operators. The classical results for
Calderon-Zygmund operators of different classes state that if 1 < p < ∞, then
this operator is bounded from Lp to Lp (see, for example, [22], [23]). J. Peetre [7]
studied the boundedness of singular operator in Morrey spaces. Further results
of the boundedness of Calderon-Zygmund operator for local Morrey-type spaces
were obtained in Burenkov, Guliyev, Serbetci and Tararykova [14], and others.
Given functions F and G, in this paper F . G means that F ≤ CG, where C is
a positive number depending only on numerical parameters that may be different
on different occasions. Moreover, F ≍ G means that F . G and G . F .
4
Burenkov V.I. – Chigambayeva D.K. – Nursultanov E.D.
λ
2. Spaces Mp,q
(Ω)
Let Ω ⊂ Rn , 0 < p < ∞, 0 < q ≤ ∞ and 0 ≤ λ ≤ np . We consider generalλ
ized Morrey-type spaces Mp,q
(Ω) that are defined for q < ∞ as the spaces of all
loc
n
functions f ∈ Lp (R ) such that
( ∫∞ (
||f ||Mp,q
λ (Ω) =
t
)q
−λ
sup ∥f ∥Lp (Bt (x))
x∈Ω
dt
t
) q1
,
0
and for q = ∞,
−λ
||f ||Mp,∞
∥f ∥Lp (Bt (x)) .
λ
(Ω) = sup sup t
x∈Ω t>0
λ
Note that if p = ∞, then M∞,q
(Ω) = Θ. Also note that the introduced spaces
coincide with the classical Morrey spaces in the case of q = ∞ and Ω = Rn , i.e.
λ
Mp,∞
(Rn ) = Mpλ .
If Ω = {x} – one-point set, then
λ
λ
Mp,q
(Ω) = LMp,q,x
,
λ
– local Morrey-type spaces [24].
where LMp,q,x
λ
However, these spaces are distinct from the global Morrey-type spaces GMp,q
,
which were introduced by V.I. Burenkov, V.S. Guliyev and H.V. Guliyev [24] and
defined as the space of all functions f Lebesque measurable on Rn with finite
quasi-norm
( ∫∞ (
)q ) q1
dt
−λ
∥f ∥GMp,q
= sup
t ∥f ∥Lp (Bt (x))
λ
n
t
x∈R
0
if q < ∞ and usual modification if q = ∞.
Note also that the generalized Morrey-type spaces are closed to “net” spaces
Np,q (M ) [25]. We begin with the following lemma, which describes the properties
λ
of spaces Mp,q
(Ω).
Lemma 2.1. Let Ω ⊂ Rn .
(1) If 0 < p0 < p1 < ∞ and λ0 = λ1 + n
continuously embedded into Mpλ00,q (Ω).
(
1
p1
−
1
p0
)
, then the space Mpλ11,q (Ω) is
λ
(2) If 0 < q0 < q1 ≤ ∞, then the space Mp,q
(Ω) is continuously embedded into
0
λ
Mp,q1 (Ω).
Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries 5
Proof. Let f ∈ Mpλ11,q (Ω). Applying Hölder’s inequality and noting that |Bt (x)| ≍
tn , we get
( ∫∞ (
∥f ∥M λ0
p0 ,q (Ω)
t−λ0 sup ∥f ∥Lp0 (Bt (x))
=
)q
x∈Ω
dt
t
) q1
0
( ∫∞ (
t−λ0 sup ∥f ∥Lp1 (Bt (x)) |Bt (x)|
≤
1
p0
− p1
1
)q
x∈Ω
0
( ∫∞ (
.
(
t
− λ0 −n
(
1
p1
− p1
)q
))
sup ∥f ∥Lp1 (Bt (x))
0
x∈Ω
dt
t
dt
t
) q1
) q1
= ∥f ∥M λ1
p1 ,q (Ω)
,
0
which means inclusion Mpλ11,q (Ω) ,→ Mpλ00,q (Ω).
λ
λ
(ii). First let q1 = ∞ and f ∈ Mp,q
(Ω). Let us proof that Mp,q
(Ω) ,→
0
0
λ
Mp,∞ (Ω).
−λ
∥f ∥Mp,∞
sup ∥f ∥Lp (Bt (x))
λ
(Ω) = sup t
t>0
x∈Ω
= (λq0 )
1
q0
( ∫∞
sup
t>0
dτ
τ −λq0
τ
) q1
0
sup ∥f ∥Lp (Bt (x)) .
x∈Ω
t
Note that Bt (x) ⊂ Bτ (x) if t ≤ τ , then we have
∥f ∥Mp,∞
λ
(Ω) ≤ (λq0 )
1
q0
( ∫∞ (
sup
τ
−λ
t>0
)q0
sup ∥f ∥Lp (Bτ (x))
x∈Ω
dτ
τ
) q1
0
t
= (λq0 )
1
q0
∥f ∥Mp,q
λ
0
(Ω) .
If q1 < ∞, then
( ∫∞ (
∥f ∥Mp,q
λ
1
(Ω)
=
τ −λ sup ∥f ∥Lp (Bτ (x))
)q1 −q0 (
x∈Ω
τ −λ sup ∥f ∥Lp (Bτ (x))
)q0
x∈Ω
t
q
1− q0
1
≤ (∥f ∥Mp,∞
λ
(Ω) )
(∥f ∥Mp,q
λ
0
(Ω) )
q0
q1
. ∥f ∥Mp,q
λ
0
(Ω) .
This completes the proof of the second part of the lemma.
Lemma 2.2. Let 0 < p, q ≤ ∞, 0 < λ < ∞, then for any Ω ⊂ Rn
(
−λ
2
(ln2)
1
q
)q )1/q
∑(
−λm
≤ ∥f ∥Mp,q
2
sup ∥f ∥Lp (B2m (x))
λ (Ω)
m∈Z
x∈Ω
dτ
τ
) q1
1
6
Burenkov V.I. – Chigambayeva D.K. – Nursultanov E.D.
(
≤ 2 (ln2)
λ
∑(
1
q
−λm
2
)q )1/q
sup ∥f ∥Lp (B2m (x))
.
x∈Ω
m∈Z
Proof. Let t ∈ (0, ∞), then there exists m ∈ Z, such that 2m ≤ t < 2m+1 .
Therefore,
( (
)q )1/q
∞
∫
−λ
t sup ∥f ∥Lp (Bt (x)) dt
t
x∈Ω
0
(
∑
=
2m+1
∫
m∈Z 2m
(
Thus,
∑
2m+1
∫
m∈Z 2m
(
t
sup ∥f ∥Lp (Bt (x))
t
sup ∥f ∥Lp (Bt (x))
x∈Ω
dt
t
.
dt
t
x∈Ω
(
≤ 2λ (ln2)
)1/q
)q
−λ
)1/q
)q
−λ
(
1
q
∑
m∈Z
(
)q )1/q
2−λ(m+1) sup ∥f ∥Lp (B2m+1 (x))
.
x∈Ω
On the other hand,
(
)q )1/q
m+1 (
∑ 2∫
t−λ sup ∥f ∥Lp (Bt (x)) dt
t
m∈Z 2m
x∈Ω
(
−λ
≥2
(ln2)
1
q
∑
m∈Z
(
)q )1/q
−λm
2
sup ∥f ∥Lp (B2m (x))
,
x∈Ω
and we obtain the required equivalence.
3. Interpolation theorem
An operator T is called quasi-additive if for some A > 0 and for almost all y ∈ Rn
the following inequality
|T (f + g)(y)| ≤ A(|(T f )(y)| + |(T g)(y)|)
β0
β1
holds for all functions f, g ∈ LMq,σ,x
+ LMq,σ,x
.
Theorem 3.1. Let Ω ⊂ Rn , 0 < α0 , α1 , β0 , β1 < ∞, α0 ̸= α1 , β0 ̸= β1 ,
0 < p, q ≤ ∞, 0 < σ ≤ τ ≤ ∞, 0 < θ < 1 and
α = (1 − θ)α0 + θα1 , β = (1 − θ)β0 + θβ1 .
β0
β1
Let T be a quasi-additive operator given on LMq,σ,x
+ LMq,σ,x
, x ∈ Ω. Suppose
that for some M1 , M2 > 0 the following inequalities hold
(3.1)
βi
αi
∥T f ∥LMp,∞,x
≤ Mi ∥f ∥LM βi , x ∈ Ω, f ∈ LMq,σ,x
, i = 0, 1,
q,σ,x
Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries 7
then
1−θ
α (Ω) ≤ cAM
∥T f ∥Mp,τ
M1θ ∥f ∥Mq,τ
β
0
(Ω)
β
for all functions f ∈ Mq,τ
(Ω) and c > 0 depends only on α0 , α1 , β0 , β1 , p, q, σ, θ.
β
Proof. 1. Let f ∈ Mq,τ
(Ω). For every x ∈ Ω, s > 0, we define the functions
f0,s = f χBs (x) , f1,s = f − f0,s ,
where χBs (x) denotes the characteristic function of a ball Bs (x). Then f =
f0,s + f1,s and
∥T f ∥Lp (Bt (x)) = ∥T (f0,s + f1,s )∥Lp (Bt (x))
≤ A∥|T f0,s | + |T f1,s |∥Lp (Bt (x))
≤ 2( p −1)+ A(∥T f0,s ∥Lp (Bt (x)) + ∥T f1,s ∥Lp (Bt (x)) ).
1
By inequality (3.1) we have
∥T f0,s ∥Lp (Bt (x)) = tα0 t−α0 ∥T f0,s ∥Lp (Bt (x))
α0
≤ tα0 sup r−α0 ∥T f0,s ∥Lp (Br (x)) = tα0 ∥T f0,s ∥LMp,∞,x
r>0
( ∫∞
≤ M0 t ∥f0,s ∥LM β0
α0
q,σ,x
( ∫s
= M0 t
α0
≤2
( ∫∞
(
r
−β0
+
)σ dr
∥f0,s ∥Lq (Br (x))
+
r
(( ∫s
M0 t
α0
r
−β0
∫∞
) σ1
( −β0
)σ dr
r
∥f0,s ∥Lq (Br (x))
r
s
(
r
−β0
)σ dr
∥f0,s ∥Lq (Br (x))
r
0
(
( −β0
)σ dr
r
∥f0,s ∥Lq (Br (x))
r
0
0
1
(σ
−1)+
= M0 t
α0
)σ dr
∥f0,s ∥Lq (Br (x))
r
) σ1
) σ1
) σ1 )
.
s
For 0 < r ≤ s and y ∈ Br (x) we have that f0,s (y) = f (y)χBs (x) (y) = f (y),
therefore ∥f0,s ∥Lq (Br (x)) = ∥f ∥Lq (Br (x)) . For r > s and y ∈
/ Br (x) we get that
f0,s (y) = 0, therefore ∥f0,s ∥Lq (Br (x)) = ∥f ∥Lq (Bs (x)) . Hence,
( ∫s
0
(
r
−β0
)σ dr
∥f0,s ∥Lq (Br (x))
r
) σ1
( ∫s
=
0
(
r
−β0
)σ dr
∥f ∥Lq (Br (x))
r
) σ1
8
Burenkov V.I. – Chigambayeva D.K. – Nursultanov E.D.
and
( ∫∞
(
r
−β0
)σ dr
∥f0,s ∥Lq (Br (x))
r
) σ1
( ∫∞
= ∥f ∥Lq (Bs (x))
s
( −β0 σ dr
r
r
) σ1
s
= (β0 σ)
= (β0 σ)
−β0
1
−σ
∥f ∥Lq (Bs (x)) s
1
−σ
∥f ∥Lq (Bs (x)) s−β0 (β1 σ) sβ1
1
σ
( β1 ) σ1 β1 −β0
s
=
β0
( ∫∞
( ∫∞
(
s
( −β1
)σ dr
r
∥f ∥Lq (Br (x))
r
dr
r−β1 σ
r
) σ1
) σ1
.
s
Thus, for all σ > 0
∥T f0,s ∥Lp (Bt (x))
( s
) σ1
( ∫∞
) σ1
∫
(
)
(
)
dr
dr
σ
σ
.
≤ c1 M0 tα0
r−β0 ∥f ∥Lq (Br (x))
+ sβ1 −β0
r−β1 ∥f ∥Lq (Br (x))
r
r
s
0
Similarly, according to inequality (3.1), since f1,s (y) = 0 if y ∈ Bs (x) and
|f1,s (y)| ≤ |f (y)| if y ∈ Rn , we get that
∥T f1,s ∥Lp (Bt (x)) = tα1 t−α1 ∥T f1,s ∥Lp (Bt (x))
( ∫s
) σ1
∫∞
(
)
(
)
dr
dr
σ
σ
≤ M1 tα1
r−β1 ∥f1,s ∥Lq (Br (x))
+
r−β1 ∥f1,s ∥Lq (Br (x))
r
r
0
( ∫∞
≤ M1 tα1
(
r−β1 ∥f ∥Lq (Br (x))
)σ dr
r
s
) σ1
.
s
So, for all t > 0 and s > 0 we obtain
[( ∫s
(
∥T f ∥Lp (Bt (x)) ≤ c2 A M0 t
+ sβ1 −β0
α0
( ∫∞
(
r
−β0
)σ dr
∥f ∥Lq (Br (x))
r
0
(
)σ dr
r−β1 ∥f ∥Lq (Br (x))
r
s
( ∫∞
+ M1 tα1
(
)σ dr
r−β1 ∥f ∥Lq (Br (x))
r
s
where c2 > 0 depends only on p, β0 , β1 and σ.
) σ1 ]
) σ1 )
,
) σ1
Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries 9
2. Suppose that β0 < β1 , α0 < α1 and set s = ctγ , where γ =
will be taken further. Then
( ∫∞
α (Ω) =
∥T f ∥Mp,τ
(
)τ dt
t−α sup ∥T f ∥Lp (Bt (x))
t
x∈Ω
α1 −α0
β1 −β0 ,
and c > 0
) τ1
0
( τ1 +1)+
≤3
c2 A(M0 I1 + M0 I2 + M1 I3 ),
where
( ∫∞ (
I1 =
γ
)σ ) στ ) τ1
∫ct (
dr
dt
tα0 −α
r−β0 sup ∥f ∥Lq (Br (x))
,
r
t
x∈Ω
0
0
I2 = (ctγ )β1 −β0
( ∫∞ (
)σ ) στ ) τ1
∫∞ (
dt
dr
,
tα0 −α
r−β1 sup ∥f ∥Lq (Br (x))
r
t
x∈Ω
ctγ
0
and
( ∫∞ (
I3 =
t
α1 −α
0
∫∞ (
r
−β1
)σ
sup ∥f ∥Lq (Br (x))
x∈Ω
ctγ
dr
r
) στ
dt
t
) τ1
.
Making the change of variable ctγ = u, we obtain
I1 = γ − τ cθ(β1 −β0 ) J1 , I2 = γ − τ cθ(β1 −β0 ) J2 , I3 = γ − σ c−(1−θ)(β1 −β0 ) J3 ,
1
1
1
where
( ∫∞ (
J1 =
y
−θ(β1 −β0 )
( ∫y (
)σ ) σ1 )τ ) τ1
dt
dr
−β0
,
r
sup ∥f ∥Lq (Br (x))
r
t
x∈Ω
0
( ∫∞ (
J2 =
y
0
(1−θ)(β1 −β0 )
( ∫∞ (
)σ ) σ1 )τ ) τ1
dt
dr
−β0
,
r
sup ∥f ∥Lq (Br (x))
r
t
x∈Ω
y
0
and
( ∫∞ (
( ∫∞ (
)σ ) σ1 )τ ) τ1
dr
dt
y (1−θ)(β1 −β0 )
r−β1 sup ∥f ∥Lq (Br (x))
.
r
t
x∈Ω
J3 =
y
0
Note that γ − τ ≤ max{1, γ − σ }.
3. For the integral estimates J1 , J2 , J3 apply the following variants of Hardy
inequalities: if µ > 0, −∞ < ν < ∞ and 0 < σ ≤ τ ≤ ∞, then
( ∫∞ (
( ∫∞ (
)τ ) τ1
( ∫y (
)σ ) σ1 )τ ) τ1
1
dy
dy
dr
−µ
−ν
−σ
y −µ−ν |g(y)|
y
r |g(r)|
≤ (µσ)
r
y
y
1
0
0
1
0
10
Burenkov V.I. – Chigambayeva D.K. – Nursultanov E.D.
and
( ∫∞ (
( ∫∞ (
)σ ) σ1 )τ ) τ1
( ∫∞ (
)τ ) τ1
1
dr
dy
dy
µ
−ν
−σ
µ−ν
y
r |g(r)|
≤ (µσ)
y
|g(y)|
.
r
y
y
y
0
0
According to these inequalities we have
1
−σ
( ∫∞
J1 ≤ (θ(β1 − β0 )σ)
(
y −β sup ∥f ∥Lq (By (x))
x∈Ω
)τ dy
y
) τ1
0
= (θ(β1 − β0 )σ)− σ ∥f ∥LMq,τ
β
(Ω) ,
1
J2 ≤ (θ(β1 − β0 )σ)− σ ∥f ∥Mq,τ
β
(Ω) ,
1
and
J3 ≤ ((1 − θ)(β1 − β0 )σ)− σ ∥f ∥Mq,τ
β
(Ω) .
1
Hence,
−θ(β1 −β0 )
α (Ω) ≤ c3 (M0 c
∥T f ∥Mp,τ
+ M1 c−(1−θ)(β1 −β0 ) )∥f ∥Mq,τ
β
(Ω) ,
where c3 > 0 depends only on α0 , α1 , β0 , β1 , p, q, σ and θ.
1
( M1 ) β −β
1
0 , then
Let now c = M
0
1−θ
α (Ω) ≤ 2c3 M
∥T f ∥Mp,τ
M1θ ∥f ∥Mq,τ
β
0
(Ω) .
4. If β1 < β0 , α0 < α1 or β0 < β1 , α1 < α0 or β1 < β0 , α1 < α0 , then the
assumptions of Part 2 and 3 are similar, changing only the choice of parameters γ
1
( M1 ) β −β
−α0
0
1 , in the second γ = α0 −α1 , c =
and c. In the first case γ = αβ01 −β
, c= M
β1 −β0
1
0
1
1
( M1 ) β −β
( M1 ) β −β
0
1 , in the third γ = α0 −α1 , c =
0
1 .
M0
β0 −β1
M0
Corollary 3.1. Let 0 < α0 , α1 , β0 , β1 < ∞, α0 ̸= α1 , β0 ̸= β1 , 0 < p, q ≤ ∞,
0 < σ ≤ τ ≤ ∞, 0 < θ < 1 and
α = (1 − θ)α0 + θα1 , β = (1 − θ)β0 + θβ1 .
Then there exists c > 0, depending only on α0 , α1 , β0 , β1 , p, q, σ, θ, such that if
β0
β1
x ∈ Rn , T is a quasi-additive operator on LMq,σ,x
+ LMq,σ,x
, and for some
M1 , M2 > 0 the following inequalities hold
αi
∥T f ∥LMp,∞,x
≤ Mi ∥f ∥LM βi
q,σ,x
for all f ∈
βi
LMq,σ,x
,
i = 0, 1, then
α
∥T f ∥LMp,τ,x
≤ cAM01−θ M1θ ∥f ∥LMq,τ,x
β
β
for all f ∈ LMq,τ,x
.
Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries 11
Remark 3.1. If T is a linear operator, then the statement of Corollary 3.1
follows using the standard arguments of interpolation theory from the equality
λ0
λ1
λ
(LMp,r
, LMp,r
) = LMp,τ,x
,
0 ,x
1 ,x θ,τ
(3.2)
where 0 < p, r0 , r1 , τ ≤ ∞, 0 < λ0 , λ1 < ∞, λ0 ̸= λ1 , 0 < θ < 1, λ = (1−θ)λ0 +θλ1 ,
and respectively by equivalence of quasi-norm, which were proved in [19] under the
additional assumptions λ0 , λ1 ≤ np and in general case in [20]. In the case of T
is quasi-linear operator, then the statement does not follow from (3.2).
4. Riesz potentionals in Morrey-type spaces
Lemma 4.1. Let 1 < p, q, r < ∞, p1 + 1r = 1 + 1q , g ∈ Lr,∞ (D), f ∈ Lp (B − D),
where B, D ⊂ Rn are measurable sets, then
∫
f (· − y)g(y)dy ≤ c∥f ∥Lp (B−D) ||g||Lr,∞ (D) ,
Lq (B)
D
where c > 0 depends only on the parameters p, q and r.
Proof. Let f ∈ Lp (B − D), g ∈ Lr,∞ (D). Set
{
g(y), y ∈ D,
g̃(y) =
0, y ∈
/D
{
and
f˜(y) =
f (y), y ∈ B − D,
0, y ∈
/ B − D.
Using O’Neil’s inequality, we have
∫
∫
˜
f (· − y)g(y)dy = f (· − y)g(y)dy D
Lq (B)
D
∫
˜(· − y)g̃(y)dy ≤
f
Rn
Lq (B)
∫
˜
= f (· − y)g̃(y)dy Lq (B)
Rn
≤ c∥f˜∥Lp (Rn ) ||g̃||Lr,∞ (Rn )
Lq
(Rn )
= c∥f ∥Lp (B−D) ||g||Lr,∞ (D) ,
which completes the proof of the lemma.
1
p
Theorem 4.1. Let z ∈ Rn , 1 < p < q < ∞, 0 < ν ≤ λ <
1
1
ν
n
+ 1r + λ−ν
n and s = r − n . If k ∈ Lr,∞ (R ) and
(4.1)
sup 2νm ∥k∥Ls,∞ (Dm ) ≤ M,
m∈Z
n
q,
1+
1
q
=
12
Burenkov V.I. – Chigambayeva D.K. – Nursultanov E.D.
where Dm = B2m+1 (0) \ B2m (0). Then convolution operator
∫
(k ∗ f )(x) = k(x − y)f (y)dy
Rn
ν
λ
is bounded from LMp,1,z
to LMq,∞,z
. Moreover, the following estimate holds
ν
||k ∗ f ||LMq,∞,z
≤ c(||k||Lr,∞ (Rn ) + M )||f ||LMp,1,z
.
λ
Proof. Let z ∈ Rn , l ∈ Z and Dm = B2m+1 (0)\B2m (0), m ∈ Z. Applying
Minkowski’s inequality, we get
∥k ∗ f ∥Lq (B2l (z))
∫
=
k(y)f (x − y)dy +
B2l (0)
k(y)f (x − y)dy ∫
≤
Lq (B2l (z))
B2l (0)
k(y)f (x − y)dy ∞ ∫
∑
k=1D
l+k
∞ ∫
∑
k(y)f (x − y)dy +
m=l D
m
Lq (B2l (z))
Lq (B2l (z))
= I1 + I2 .
Let us estimate each part. Applying Lemma 4.1 with 1/r1 = 1 + 1/q − 1/p we
obtain
I1 ≤ ∥k∥Lr1 ,∞ (B2l (0)) ∥f ∥Lp (B2l+1 (z))
ν
.
. 2νl ∥k∥Lr1 ,∞ (B2l (0)) ∥f ∥LMp,∞,z
For the second estimate we use the Hölder’s inequality and Lemma 4.1, Lemma
2.2 and we get
I2 . 2
λl
≤ 2λl
≤ 2λl
∞ ∫
∑
k(y)f (x − y)dy m=l D
m
∞
∑
Lq1 (B2l (z))
∥k∥Ls,∞ (Dm ) ∥f ∥Lp (B2l (z)−Dm )
m=l
∞
∑
2ν(m+2) ∥k∥Ls,∞ (Dm ) (2−ν(m+2) ∥f ∥Lp (B2m+2 (z)) )
m=l
< 2 sup 2ν(m+2) ∥k∥Ls,∞ (Dm )
λl
m≥l
∑
(2−ν(m+2) ∥f ∥Lp (B2m+2 (z)) )
m∈Z
≍ 2λl sup 2ν(m+2) ∥k∥Ls,∞ (Dm ) ∥f ∥LM ν
m≥l
p,1,z
,
where 1/q1 = 1/q − λ/n, 1/s = 1 + 1/q − λ/n − 1/p = 1/r − ν/n.
Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries 13
According Lemma 2.1 we have
2−λl ∥k ∗ f ∥Lq (B2l (z))
(
)
(ν−λ)l
νm
. 2
∥k∥Lr1 ,∞ (B2l (0)) ∥f ∥LMp,∞,z
+ sup 2 ∥k∥Ls,∞ (Dm ) ∥f ∥LM ν
ν
p,1,z
m∈Z
(
)
. 2(ν−λ)l ∥k∥Lr1 ,∞ (B2l (0)) + sup 2νm ∥k∥Ls,∞ (Dm ) ∥f ∥LM ν .
p,1,z
m∈Z
Since ν ≤ λ, then
2(ν−λ)l ||k||Lr1 ,∞ (B2l (0)) ≍ 2(ν−λ)l
= 2(ν−λ)l
sup s
λ−ν
n
s r1 −
1
1
sup s r1 k ∗ (s)
0<s<2ln
k ∗ (s) . sup s r1 −
λ−ν
n
1
λ−ν
n
k ∗ (s).
s>0
0<s<2ln
Therefore,
2(ν−λ)l ||k||Lr1 ,∞ (B2l (0)) . ||k||Lr,∞ (Rn ) ,
where 1r = r11 − λ−ν
n .
Given an arbitrariness of l, by Lemma 2.2 we get
(
)
∥k ∗ f ∥LMp,∞,z
≍ sup 2−λl ∥k ∗ f ∥Lp (B2l (z)) . ∥k∥Lr,∞ (Rn ) + M ∥f ∥LM ν
λ
p,1,z
l∈Z
,
which completes the proof of the theorem.
Note that the condition (4.1) in Theorem 4.1 is essential, i.e. the following
statement does not hold
(4.2)
ν
∥T ∥LMp,1,z
≤ c∥k∥Lr,∞ = c sup t r k ∗ (t).
λ
→LMq,∞,z
1
t>0
Example 4.1. Let z = 0, 1 < p ≤ q < ∞, 0 < ν ≤ λ <
1
α > νp
. Define the functions
{
1,
f (x) =
0,
and
|x| ∈ [lα , lα + 1],
otherwise
{
1
(l + 1)− r ,
k(x) =
0,
n
q,
1+ 1q = p1 + 1r + λ−ν
n ,
l ∈ Z+
|x| ∈ [lα , lα + 1], l ∈ Z+
otherwise.
ν
We show that the function f belongs to the space LMp,1,0
. Indeed,
∫∞
ν
∥f ∥LMp,1,0
=
1
tν+1
0
(∫
)1/p
+t
−t
|f (y)|p dy
dt.
14
Burenkov V.I. – Chigambayeva D.K. – Nursultanov E.D.
Furthermore,
∫+t
∑
|f (y)|p dy = 2 min t,
1 ≤ 2 min(t, t1/α ),
lα ≤t
−t
hence,
ν
∥f ∥LMp,1,0
< ∞.
Since k ∗ (t) ≤ t−1/r , therefore k ∈ Lr,∞ (R), but also for x ∈ [−1/2, 1/2] we
have
+∞
∫
∞
1∑
(f ∗ k)(x) =
k(x − y)f (y)dy ≥
(l + 1)−1/r = ∞,
2
l=1
−∞
and the inequality of type (4.2) is impossible.
Theorem 4.2. Let Ω ⊂ Rn , 1 < p < q < ∞, 0 < τ ≤ ∞, 0 < ν ≤ λ <
γ = np − nq + λ − ν > 0,
∫
Iγ f (x) =
Rn
f (x − y)
n
q
and
dy
.
|y|n−γ
ν
λ
If f ∈ Mp,τ
(Ω), then convolution operator Iγ f ∈ Mq,τ
(Ω), and the following
inequality holds
∥Iγ f ∥Mq,τ
λ (Ω) ≤ c∥f ∥M ν (Ω) .
p,τ
(4.3)
Proof. Let k(x) = 1/|x|n−γ . Check the conditions of Theorem 4.1
∥k∥Lr,∞ (Rn ) = sup t1/r k ∗ (t) ≍ sup t1/r
t>0
t>0
sup 2νm ∥k∥Ls,∞ (Dm ) ≍ sup 2νm
m∈Z
m∈Z
tn−n
1
sup
0<t≤2mn
ν
1
γ
t1− n
= 1,
1
γ ≤ 1.
(2mn + t)1− n
Then applying Theorem 4.1 we obtain the weak inequality
ν
∥Iγ f ∥LMq,∞,z
≤ c∥f ∥LMp,1,z
, z ∈ Ω.
λ
Next, since 0 < ν <
n
p,
0<λ<
0 < ν1 < ν < ν0 <
n
q,
then there exist ν0 , ν1 and λ0 , λ1 , such that
n
,
p
0 < λ1 < λ < λ0 <
n
.
q
Let ν0 − λ0 = ν1 − λ1 = ν − λ. Then the weak inequalities hold
∥Iγ f ∥LM λi
q,∞,z
≤ c ∥f ∥LM νi
p,1,z
, i = 0, 1, z ∈ Ω.
Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries 15
Hence, by interpolation Theorem 3.1 the strong inequality holds
∥Iγ f ∥Mq,τ
λ (Ω) ≤ c∥f ∥M ν (Ω) ,
p,τ
where 0 < τ ≤ ∞, ν = (1 − θ)ν0 + θν1 , λ = (1 − θ)λ0 + θλ1 and 0 < θ < 1,
1+
1 1 (1 − θ)(λ0 − ν0 ) + θ(λ1 − ν1 )
1
= + +
q
p r
n
1 1 λ−ν
1
,
=1+ = + +
q
p r
n
which completes the proof of the theorem.
Note that a similar description of the boundedness of Iγ f in classical Morrey
spaces with restriction λq = νp was given by Adams [9], Chiarenza and Frasca
[10], Sawano and Tanaka [12] and Burenkov in survey paper [3], who used the
different methods of the proof. In our case instead of this condition we have condition ν ≤ λ, which is necessary in Theorem 4.2. Indeed, from (4.3), in particular,
it follows that
ν
∥Iγ f ∥LMq,q,0
≤ c∥f ∥LMp,p,0
,
λ
that is equivalent to inequality
∥Iγ f ∥Lq (|x|−qλ ) ≤ c∥f ∥Lp (|x|−pν ) ,
(4.4)
and in this case the condition ν ≤ λ is necessary. Indeed, let N ∈ N, zN =
(N, 0, ..., 0),
f (x) = χB2 (zN ) (x)|x|ν ,
∫
f (x − y)
Iγ f (y) =
dx.
|x|γ
Rn
Then by (4.4) we have
|B2 |1/p c ≥ ∥Iγ f ∥Lq (|x|−qλ )
1/q
q
∫
∫ ν
|x − y|
1
dx dy .
≥
λ
|y| B2 (zN )+y |x|γ
B1 (zN )
Since B2 (zN ) + y ⊃ B1 (0) for any y ∈ B1 (zN ) we get for N > 2
16
Burenkov V.I. – Chigambayeva D.K. – Nursultanov E.D.
∫
|B2 |1/p c ≥
B1 (zN )
∫
≥
B1 (0)
≥ c1
1/q
q
1 ∫
ν
|x − y|
dx dy
λ
|y| B1 (0) |x|γ
1
|x|γ
1/q
q
ν
(|y| − 1)
|y|λ dy dx
∫
B1 (zN )
∫
1
|y|(λ−ν)q
1/q
dy
,
B1 (zN )
and this is possible only if ν ≤ λ.
5. Singular operators in Morrey-type spaces
Definition 5.1. Let 1 ≤ a < b ≤ ∞. We call a linear operator T an (a, b) singular if
(1) the operator T is bounded from Lp (Rn ) to Lp (Rn ) for a < p < b;
(2) there exists a measurable function K on Rn such that for all x, y ∈ Rn ,
|K(x, y)| ≤
(5.1)
C
|x − y|n
and for any finite, locally integrable function f with compact support the operator
T is defined by
∫
(5.2)
T f (x) = K(x, y)f (y)dy a.e. on Rn \ suppf.
Rn
Theorem 5.1. Let Ω ⊂ Rn , 1 ≤ a < b ≤ ∞ and operator T is an (a, b)singular. Then for p ∈ (a, b), 0 < ν < np , 0 < τ ≤ ∞ operator T is bounded from
ν
ν
Mp,τ
(Ω) to Mp,τ
(Ω).
ν
Proof. Let z ∈ Ω, l ∈ Z and Dm (z) = B2m+1 (z) \ B2m (z), m ∈ Z, f ∈ LMp,1,z
.
Applying Minkowski’s and Hölder’s inequalities we get
)
( ∑
∞
(
)
∥T f ∥Lp (B2l (z)) ≤ T f χB l+1 (z) + T
f χDm (z) 2
Lp (B2l (z))
Lp (B2l (z))
m=l+1
( ∞
)
(
)
∑
νl . T f χB l+1 (z) +
2
T
f
χ
= I1 + I2 ,
D
(z)
m
2
Lp (B l (z))
2
m=l+1
Lq (B2l (z))
Marcinkiewicz-type interpolation theorem for Morrey-type spaces and its corollaries 17
where 1/p = 1/q + ν/n.
Let us estimate each part. By definition of the singular operator we have
(
)
I1 ≤ T f χB l+1 (z) ≤ C∥f ∥Lp (B2l+1 (z) ) .
n
Lp (R )
2
(
)
Since B2l (z) ∩ ∪∞
m=l+1 Dm (z) = ∅, for operator T and function f χDm (z) if
m ≥ l + 1 the representation (5.2) and estimate (5.1) hold for all x ∈ B2l (z), hence
∞
∑
(
)
T f χ (z) Dm
L
I2 ≤ 2νl
=2
νl
. 2νl
m=l+1
∫
∞
∑
q (B2l (z))
K(x, y)f (y)χDm (y) dy m=l+1 Rn
∫
∞
∑
m=l+1
Dm (z)
|f (y)|
dy |x − y|n Lq (B
Lq (B2l (z))
.
(z))
2l
Since m > l, B2l (z) − Dm (z) = B2l (0) + z − (Dm (0) + z) = B2l (0) − Dm (0) ⊂
Dm−1 (0) ∪ Dm (0) ∪ Dm+1 (0), application of Lemma 4.1 yields
∞
∑
I2 . 2νl
≤ 2νl
m=l+1
∞
∑
∥|x|−n ∥Lr,∞ (B2l (z)−Dm (z)) ∥f ∥Lp (Dm (z))
∥|x|−n ∥Lr,∞ (Dm−1 (0)∪Dm (0)∪Dm+1 (0)) ∥f ∥Lp (B2m+1 (z)) ,
m=l+1
where 1/r = 1 + 1/q − 1/p = 1 − ν/n.
Note that
∥|x|−n ∥Lr,∞ (Dm−1 (0)∪Dm (0)∪Dm+1 (0)) ≍
sup
0<t<2mn
t1−ν/n
≤ 2−νm .
2mn + t
Therefore,
2−νl ∥T f ∥Lp (B2l (z))
(
.
−νl
2
∥f ∥Lp (B2l+1 (z)) +
∞
∑
)
−ν(m+1)
2
∥f ∥Lp (B2m+1 (z))
m=l+1
ν
. ∥f ∥LMp,1,z
.
Given an arbitrariness of l for all z ∈ Ω we have the weak inequality
ν
ν
∥T f ∥LMp,∞,z
. ∥f ∥LMp,1,z
.
Interpolation Theorem 3.1 completes the proof of the theorem.
18
Burenkov V.I. – Chigambayeva D.K. – Nursultanov E.D.
Note that Theorem 5.1 answers the question what singular operators which are
bounded in the spaces Lp , will be bounded in the generalized Morrey-type spaces.
Let T be a Calderon-Zygmund operator, i.e. a linear operator taking C0∞ into
Lloc
1 , bounded on L2 and represented by
∫
T f (x) = K(x, y)f (y)dy a.e. on Rn \suppf
Rn
for every function f ∈ L∞ (Rn ) with compact support. Here K(x, y) is a continuous
function away from the diagonal and satisfies the standard estimates: for some
c1 > 0 and 0 < ε ≤ 1
|K(x, y)| ≤ c|x − y|−n ,
for all x, y ∈ Rn , x ̸= y and
|K(x, y) − K(x′ , y)| + |K(y, x) − K(y, x′ )|
(
)ε
|x − x′ |
≤ c1
|x − y|−n
|x − y|
whenever 2|x − x′ | ≤ |x − y| for some constants c > 0, ε ∈]0, 1]. This class of
operators was introduced by Coifman and Meyer [26].
Corollary 5.1. [3],[12] Let 1 < p < ∞, 0 ≤ λ <
operators T are bounded from Mpλ to Mpλ .
n
p.
Then Calderon-Zygmund
Proof. It is known that Calderon-Zygmund operator is bounded in the Lebesgue
spaces Lp (Rn ) if p ∈ (0, ∞). Hence, for the interval (a, b) = (0, ∞) the conditions
of Theorem 5.1 are satisfied, which completes the proof.
Acknowledgements. The paper was started when the authors were staying at
the Department of Mathematics of Padova University (Padova, Italy) in Summer
2014.
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