A note on boundedness of the Hardy-Littlewood maximal operator on Morrey spaces

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by  A. Gogatishvili

A NOTE ON BOUNDEDNESS OF THE HARDY-LITTLEWOOD
MAXIMAL OPERATOR ON MORREY SPACES
1 2
A. GOGATISHVILI AND R.CH. MUSTAFAYEV
5 Abstract. In this paper we prove that the Hardy-Littlewood maximal oper-
1 ator is bounded on Morrey spaces M1,λ(Rn), 0≤λ<n for radial, decreasing
0 functions on Rn.
2

n
a
J
1. Introduction
9
Morrey spaces M ≡ M (Rn), were introduced by C. Morrey in [8] in order
] p,λ p,λ
A to study regularity questions which appear in the Calculus of Variations, and
C defined as follows: for 0 ≤ λ ≤ n and 1 ≤ p < ∞,
.
math Mp,λ := (cid:26)f ∈ Llpoc(Rn) : kfkMp,λ := x∈Rsnu,pr>0rλ−pnkfkLp(B(x,r)) < ∞(cid:27),
where B(x,r) is the open ball centered at x of radius r.
[
Note that M (Rn) = L (Rn) and M (Rn) = L (Rn), when 1 ≤ p < ∞.
1 p,0 ∞ p,n p
v These spaces describe local regularity more precisely than Lebesgue spaces
4 and appeared to be quite useful in the study of the local behavior of solutions to
8
partial differential equations, a priori estimates and other topics in PDE (cf. [4]).
4
6 Given a locally integrable function f on Rn and 0 ≤ α < n, the fractional
0 maximal function M f of f is defined by
. α
2
50 Mαf(x) := sup|Q|α−nn |f(y)|dy, (x ∈ Rn),
1 Q∋x ZQ
: where the supremum is taken over all cubes Q containing x. The operator M :
v α
Xi f → Mαf is called the fractional maximal operator. M := M0 is the classical
Hardy-Littlewood maximal operator.
r
a The study of maximal operators is one of the most important topics in har-
monic analysis. These significant non-linear operators, whose behavior are very
informativeinparticularindifferentiationtheory, providedtheunderstandingand
the inspiration for the development of the general class of singular and potential
operators (see, for instance, [9], [7], [3], [10], [5], [6]).
2010 Mathematics Subject Classification. Primary 42B25; Secondary 42B35.
Key words and phrases. Morrey spaces, maximal operator.
The research of A. Gogatishvili was partly supported by the grants P201-13-14743Sof the
GrantAgency ofthe CzechRepublic andRVO:67985840,by Shota RustaveliNationalScience
Foundation grants no. 31/48 (Operators in some function spaces and their applications in
Fourier Analysis) and no. DI/9/5-100/13 (Function spaces, weighted inequalities for integral
operators and problems of summability of Fourier series). The research of both authors was
partlysupportedby the jointprojectbetweenAcademy ofSciences ofCzechRepublic andThe
Scientific and Technological Research Council of Turkey.
1
2 A. GOGATISHVILIAND R.CH.MUSTAFAYEV
The boundedness of the Hardy-Littlewood maximal operator M in Morrey
spaces M was proved by F. Chiarenza and M. Frasca in [2]: It was shown that
p,λ
Mf is a.e. finite if f ∈ M and an estimate
p,λ
kMfk ≤ ckfk (1.1)
Mp,λ Mp,λ
holds if 1 < p < ∞ and 0 < λ < n, and a weak type estimate (1.1) replaces for
p = 1, that is, the inequality
t|{Mf > t}∩B(x,r)| ≤ crn−λkfk (1.2)
M1,λ
holds with constant c independent of x, r, t and f.
Inthispaperweshowthat(1.1)isnottrueforp = 1. According toourexample
the right result is (1.2). If restricted to the cone of radial, decreasing functions
on Rn, inequality (1.1) holds true for p = 1.
The paper is organized as follows. We start with notation and preliminary
results in Section 2. In Section 3, we prove that the Hardy-Littlewood maximal
operator M is bounded on M , 0 < λ < n, for radial, decreasing functions, and
1,λ
we give an example which shows that M is not bounded on M , 0 < λ < n.
1,λ
2. Notations and Preliminaries
Now we make some conventions. Throughout the paper, we always denote by
c a positive constant, which is independent of main parameters, but it may vary
from line to line. By a . b we mean that a ≤ cb with some positive constant
c independent of appropriate quantities. If a . b and b . a, we write a ≈ b
and say that a and b are equivalent. For a measurable set E, χ denotes the
E
characteristic function of E.
Let Ω be any measurable subset of Rn, n ≥ 1. Let M(Ω) denote the set
of all measurable functions on Ω and M (Ω) the class of functions in M(Ω)
0
that are finite a.e., while M↓(0,∞) (M+,↓(0,∞)) is used to denote the subset
of those functions which are non-increasing (non-increasing and non-negative)
on (0,∞). Denote by Mrad,↓ = Mrad,↓(Rn) the set of all measurable, radial,
decreasing functions on Rn, that is,
Mrad,↓ := {f ∈ M(Rn) : f(x) = ϕ(|x|), x ∈ Rnwithϕ ∈ M↓(0,∞)}.
Recall that Mf ≈ Hf, f ∈ Mrad,↓, where
1
Hf(x) := |f(y)|dy
|B(0,|x|)
ZB(0,|x|)
is n-dimensional Hardy operator. Obviously, Hf ∈ Mrad,↓, when f ∈ Mrad,↓.
For p ∈ (0,∞], we define the functional k·k on M(Ω) by
p,Ω
( |f(x)|pdx)1/p if p < ∞,
kfk := Ω
p,Ω
esssup |f(x)| if p = ∞.
( R Ω
The Lebesgue space L (Ω) is given by
p
L (Ω) := {f ∈ M(Ω) : kfk < ∞}
p p,Ω
A NOTE ON BOUNDEDNESS OF THE MAXIMAL FUNCTION ON MORREY SPACES 3
and it is equipped with the quasi-norm k·k .
p,Ω
The decreasing rearrangement (see, e.g., [1, p. 39]) of a function f ∈ M (Rn)
0
is defined by
f∗(t) := inf{λ > 0 : |{x ∈ Rn : |f(x)| > λ}| ≤ t} (0 < t < ∞).
3. Boundedness of M on M for radial, decreasing functions
1,λ
Reacall that
α−n
Mαf(x) ≈ sup|B| n |f(y)|dy
B∋x ZB
≈ sup|B(x,r)|α−nn |f(y)|dy, (x ∈ Rn),
r>0 ZB(x,r)
where the supremum is taken over all balls B containing x.
In order to prove our main result we need the following auxiliary lemmas.
Lemma 3.1. Assume that 0 < λ < n. Let f ∈ Mrad,↓(Rn) with f(x) = ϕ(|x|).
The equivalency
x
kfk ≈ supxλ−n |ϕ(ρ)|ρn−1dρ
M1,λ
x>0 Z0
holds with positive constants independent of f.
Proof. Recall that
kfkM1,λ ≈ sup|B|λ−nn f = kMλfk∞, f ∈ M(Rn).
B ZB
Switching to polar coordinates, we have that
λ−n
Mλ(f)(y) & |B(0,|y|)| n |f(z)|dz
ZB(0,|y|)
|y|
≈ |y|λ−n |ϕ(ρ)|ρn−1dρ.
Z0
Consequently,
|y|
kfk & esssup|y|λ−n |ϕ(ρ)|ρn−1dρ
M1,λ
y∈Rn Z0
x
= supxλ−n |ϕ(ρ)|ρn−1dρ,
x>0 Z0
where f(·) = ϕ(|·|).
4 A. GOGATISHVILIAND R.CH.MUSTAFAYEV
On the other hand,
|B|
kfkM1,λ . sup|B|λ−nn f∗(t)dt
B Z0
|B|
λ−n 1
= sup|B| n |ϕ(tn)|dt
B Z0
1
|B|n
≈ sup|B|λ−nn |ϕ(ρ)|ρn−1dρ
B Z0
x
= supxλ−n |ϕ(ρ)|ρn−1dρ,
x>0 Z0
where f(·) = ϕ(|·|). (cid:3)
Corollary 3.2. Assume that 0 < λ < n. Let f ∈ Mrad,↓(Rn) with f(x) = ϕ(|x|).
The equivalency
x x
kMfk ≈ supxλ−n ϕ(ρ)ρn−1ln dρ
M1,λ ρ
x>0 Z0 (cid:18) (cid:19)
holds with positive constants independent of f.
Proof. Let f ∈ Mrad,↓ with f(x) = ϕ(|x|). Since Mf ≈ Hf and Hf ∈ Mrad,↓,
by Lemma 3.1, switching to polar coordinates, using Fubini’s Theorem, we have
that
x 1
kMfk ≈ supxλ−n |f(y)|dy tn−1dt
M1,λ |B(0,t|
x>0 Z0 (cid:18) ZB(0,t) (cid:19)
x 1 t
≈ supxλ−n ϕ(ρ)ρn−1dρdt
t
x>0 Z0 Z0
x x
= supxλ−n ϕ(ρ)ρn−1ln dρ.
ρ
x>0 Z0 (cid:18) (cid:19)
(cid:3)
Lemma 3.3. Assume that 0 < λ < n. Let f ∈ Mrad,↓ with f(x) = ϕ(|x|). The
inequality
kMfk . kfk , f ∈ Mrad,↓
M1,λ M1,λ
holds if and only if the inequality
x x x
supxλ−n ϕ(ρ)ρn−1ln dρ . supxλ−n ϕ(ρ)ρn−1dρ, ϕ ∈ M+,↓(0,∞)
ρ
x>0 Z0 (cid:18) (cid:19) x>0 Z0
holds.
Proof. Thestatement immediatelyfollowsfromLemma3.1andCorollary3.2. (cid:3)
Lemma 3.4. Let 0 < λ < n. Then inequality
x x x
supxλ−n ϕ(ρ)ρn−1ln dρ . supxλ−n ϕ(ρ)ρn−1dρ (3.1)
ρ
x>0 Z0 (cid:18) (cid:19) x>0 Z0
holds for all ϕ ∈ M+,↓(0,∞).
A NOTE ON BOUNDEDNESS OF THE MAXIMAL FUNCTION ON MORREY SPACES 5
Proof. Indeed:
x x
supxλ−n ϕ(ρ)ρn−1ln dρ
ρ
x>0 Z0 (cid:18) (cid:19)
x 1 t
= supxλ−n ϕ(ρ)ρn−1dρdt
t
x>0 Z0 Z0
x t
= supxλ−n tn−λ−1tλ−n ϕ(ρ)ρn−1dρdt
x>0 Z0 Z0
t x
≤ suptλ−n ϕ(ρ)ρn−1dρ· supxλ−n tn−λ−1dt
t>0 Z0 (cid:18)x>0 Z0 (cid:19)
t
≈ suptλ−n ϕ(ρ)ρn−1dρ.
t>0 Z0
(cid:3)
Now we are in position to prove our main result.
Theorem 3.5. Assume that 0 < λ < n. The inequality
kMfk . kfk (3.2)
M1,λ M1,λ
holds for all f ∈ Mrad,↓ with constant independent of f.
Proof. The statement follows by Lemmas 3.3 and 3.4. (cid:3)
Remark 3.6. Note that inequality (3.2) holds true when λ = 0, for M (Rn) =
1,0
L (Rn) and M is bounded on L (Rn).
∞ ∞
Remark 3.7. It is obvious that the statement of Theorem 3.5 does not hold when
λ = n, for in this case M (Rn) = L (Rn) and the inequality
1,n 1
kMfk . kfk
L1(Rn) L1(Rn)
is true only for f = 0 a.e., which follows from the fact that Mf(x) ≈ |x|−n for
|x| large when f ∈ Lloc(Rn).
1
Example 3.8. We show that M is not bounded on M (Rn), 0 < λ < n. For
1,λ
simplicity let n = 1 and λ = 1/2. Consider the function
∞
f(x) = χ[k2,k2+1](x).
k=0
X
Then
kfkM1,1/2(R) = sup|I|−1/2 f ≤ sup |I|−1/2 f + sup |I|−1/2 f,
I ZI I:|I|≤1 ZI I:|I|>1 ZI
where the supremum is taken over all open intervals I ⊂ R. It is easy to see that
sup |I|−1/2 f ≤ sup |I|1/2 ≤ 1.
I:|I|≤1 ZI I:|I|≤1
6 A. GOGATISHVILIAND R.CH. MUSTAFAYEV
Note that
sup |I|−1/2 f = sup sup |I|−1/2 f
I:|I|>1 ZI m∈NI:m<|I|≤m+1 ZI
∞
= sup sup |I|−1/2 χ[k2,k2+1](x) dx
m∈NI:m<|I|≤m+1 ZI k=0 !
X
∞
= sup sup |I|−1/2 I ∩ [k2,k2 +1] .
m∈NI:m<|I|≤m+1 (cid:12) (cid:12)
(cid:12) k[=0 (cid:12)
(cid:12) (cid:12)
Since
(cid:12) (cid:12)
∞ (cid:12) ∞ (cid:12)
I ∩ [k2,k2 +1] ≤ [0,m+1]∩ [k2,k2 +1]
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) k[=0 (cid:12) (cid:12) k[=0 (cid:12)
for any interval(cid:12)I such that m < |(cid:12)I| ≤(cid:12)m+1, we obtain that (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
∞
sup |I|−1/2 f . supm−1/2 [0,m+1]∩ [k2,k2 +1]
I:|I|>1 ZI m∈N (cid:12)(cid:12) k[=0 (cid:12)(cid:12)
. supm−1/2m(cid:12)(cid:12) 1/2 = 1. (cid:12)(cid:12)
m∈N (cid:12) (cid:12)
Consequently, we arrive at
kfk . 2.
M1,1/2(R)
On the other hand, since
∞
1 1
Mf ≥ χ[k2,k2+1] + x−k2χ[k2+1,k2+k+1] + (k +1)2 +1−xχ[k2+k+1,(k+1)2] ,
k=0(cid:18) (cid:19)
X
we have that
k2 k−1 (i+1)2
kMfk ≥ supk−1 Mf ≥ supk−1 Mf
M1,1/2(R) k∈N Z0 k∈N i=1 Zi2
X
k−1
≥ supk−1 lnj & suplnk = ∞.
k∈N k∈N
j=1
X
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1 InstituteofMathematics oftheAcademyofSciences oftheCzechRepublic,
Z´itna 25, 115 67 Prague 1, Czech Republic.
E-mail address: [email protected]
2 Department of Mathematics, Faculty of Science and Arts, Kirikkale Uni-
versity, 71450 Yahsihan, Kirikkale, Turkey.
E-mail address: [email protected]