THEMATICS MA
Boundedness of the maximal operator and singular integral operator in generalized Morrey spaces Ali Akbulut, Vagif Guliyev and Rza Mustafayev Abstract. In this paper we provide the conditions on the pair (ω1 , ω2 ) which ensures the boundedness of the maximal operator and Calder´on-Zygmund singular integral operators from one generalized Morrey space Mp,ω1 to another Mp,ω2 , 1 < p < ∞, and from the space M1,ω1 to the weak space W M1,ω2 . As applications, by these results we get some estimates for uniformly elliptic operators on generaized Morrey spaces.
1. Introduction The theory of boundedness of classical operators of real analysis, such as maximal operator and singular integral operators etc, from one weighted Lebesgue space to another one is well studied by now. These results have good applications in the theory of partial differential equations. However, in the theory of partial differential equations, along with weighted Lebesgue spaces, general Morrey-type spaces also play an important role. n Let f ∈ Lloc 1 (R ). The maximal operator M is defined by Z 1 M f (x) = sup |f (y)|dy, t>0 |B(x, t)| B(x,t) where |B(x, t)| is the Lebesgue measure of the ball B(x, t). Definition 1.1. Let k(x) : Rn \{0} → R. We call k(x) a Calder´ on-Zygmund kernel (C-Z kernel) if 0
2000 Mathematics Subject Classification. 42B20, 42B25, 42B35. Key words and phrases. Generalized Morrey spaces, maximal operator, Hardy operator, singular integral operator. The research of V. Guliyev was partially supported by the grant of the Azerbaijan-U. S. Bilateral Grants Program II (project ANSF Award / AZM1-3110-BA-08) The research of R. Mustafayev was partially supported by the Institutional Research Plan no. AV0Z10190503 of AS CR, by the grant of BGP II (project ANSF Award / AZM1-3110-BA-08) and by a Post Doctoral Fellowship of INTAS (Grant 06-1000015-6385). 1
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Aca
Preprint, Institute of Mathematics, AS CR, Prague. 2010-1-26
INSTITU
TE
of
2
(i) k ∈ C ∞ (Rn \{0}); (ii) k(x) is homogeneous of degree − n; Z (iii) k(x)dσξ = 0, where Σ = {x ∈ Rn : |x| = 1}
is the unit sphere in Rn .
Σ
Theorem 1.2 ([9]). Let k be a real measurable function in Rn × (Rn \{0}) such that (i) k(x, z) is a C-Z kernel for a.a. x ∈ Rn ; (ii)
max k(∂ j /∂z j )k(x, z)kL∞ (Rn ×Σ) = M < ∞.
|j|≤2n
For ε > 0 set Z k(x, x − y)f (y)dy.
Tε f (x) := |x−y|>ε
Then there exists T f ∈ Lp (Rn ) such that lim kTε f − T f kLp (Rn ) = 0
ε→0+
and, moreover, there exists a positive constant C such that kT f kLp (Rn ) ≤ Ckf kLp (Rn ) . Morrey spaces Mp,λ were introduced by C. Morrey in 1938 [14] and defined as n follows. For 0 ≤ λ ≤ n, 1 ≤ p ≤ ∞, f ∈ Mp,λ if f ∈ Lloc p (R ) and kf kMp,λ ≡ kf kMp,λ (Rn ) =
sup x∈Rn , r>0
λ
r− p kf kLp (B(x,r)) < ∞,
where B(x, r) is the open ball centered at x of radius r. Note that Mp,0 = Lp (Rn ) and Mp,n = L∞ (Rn ). If λ < 0 or λ > n, then Mp,λ = Θ, where Θ is the set of all functions equivalent to 0 on Rn . These spaces appeared to be quite useful in the study of the local behaviour of solutions to partial differential equations, apriori estimates and other topics in the theory of partial differential equations. n We also denote by W Mp,λ the weak Morrey space of all functions f ∈ W Lloc p (R ) for which kf kW Mp,λ ≡ kf kW Mp,λ (Rn ) =
sup x∈Rn , r>0
λ
r− p kf kW Lp (B(x,r)) < ∞,
where W Lp denotes the weak Lp -space. F. Chiarenza and M. Frasca [8] studied the boundedness of the maximal operator M in these spaces. Their results can be summarized as follows: Theorem 1.3. Let 1 ≤ p < ∞ and 0 < λ < n. Then for 1 < p < ∞ M is bounded from Mp,λ to Mp,λ and for p = 1 M is bounded from M1,λ to W M1,λ .
3
G.D.Fazio and M.A.Ragusa [9] studied the boundedness of the Calder´on-Zygmund singular integral operators in Morrey spaces, and their results imply the following statement for Calder´on-Zygmund operators T . Theorem 1.4. Let 1 ≤ p < ∞, 0 < λ < n. Then for 1 < p < ∞ Calder´ onZygmund singular integral operator T is bounded from Mp,λ to Mp,λ and for p = 1 T is bounded from M1,λ to W M1,λ . Note that in the case of the classical Calder´on-Zygmund singular integral operators Theorem 1.4 was proved by J. Peetre [18]. If λ = 0, the statement of Theorem 1.4 reduces to Theorem 1.2 for Lp (Rn ) (see also [6], [21]). In the present work, we study the boundedness of maximal operator M and Calder´on-Zygmund singular integral operators T from one generalized Morrey space Mp,ω1 to another Mp,ω2 , 1 < p < ∞, and from the space M1,ω1 to the weak space W M1,ω2 . As applications, by these results we get some estimates for uniformly elliptic operators on generaized Morrey spaces. 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.
2. Generalized Morrey spaces For the sake of completeness we recall the definition of the spaces and some properties of the spaces we are going to use. If in place of the power function rλ in the definition of Mp,λ we consider any positive measurable weight function ω(x, r), then it becomes generalized Morrey space Mp,ω . Definition 2.1. Let ω(x, r) be a positive measurable weight function on Rn × (0, ∞) and 1 ≤ p < ∞. We denote by Mp,ω the generalized Morrey space, the n space of all functions f ∈ Lloc p (R ) with finite quasinorm kf kMp,ω (Rn ) =
sup x∈Rn ,r>0
1
ω(x, r)− p kf kLp (B(x,t)) .
Definition 2.2. We say that (ω1 , ω2 ) belong to the class Zp,n , p ∈ [0, ∞), if there is a constant C such that, for any x ∈ Rn and for any t > 0 1 !p Z ∞ ess inf r 1 M is bounded from Mp,ω1 to Mp,ω2 and for p = 1 M is bounded from M1,ω1 to W M1,ω2 . Proof. By Lemma 3.3 and Theorem 3.1 we get kM f kMp,ω2 (Rn ) . .
sup
− p1
ω2 (x, r)
x∈Rn ,r>0
sup x∈Rn ,r>0
r
n p
−n p
sup t t>r
1
kf kLp (B(x,t))
ω1 (x, r)− p kf kLp (B(x,t)) = kf kMp,ω1 (Rn ) ,
8
if p ∈ (1, ∞) and kM f kW M1,ω2 (Rn ) . .
sup
−1 n
ω2 (x, r) r
x∈Rn ,r>0
sup x∈Rn ,r>0
−n
sup t t>r
kf kL1 (B(x,t))
ω1 (x, r)−1 kf kL1 (B(x,t)) = kf kM1,ω1 (Rn ) ,
if p = 1.
Corollary 3.5. Let p ∈ [1, ∞] and ω : (0, ∞) → (0, ∞) is an increasing function. Assume that the mapping t 7→ ω(t) is almost decreasing (there exists a constant tn ω(s) c such that for s < t, we have sn ≥ c ω(t) ). Then there exists a constant C > 0 tn such that kM f kMp,ω (Rn ) ≤ Ckf kMp,ω (Rn ) , if 1 < p ≤ ∞, and kM f kW M1,ω (Rn ) ≤ Ckf kM1,ω (Rn ) .
4. Singular integrals and Hardy operator In this section we are going to use the following statement on the boundedness of the Hardy operator Z 1 t (Hg)(t) := g(r)dr, 0 < t < ∞. t 0 Theorem 4.1. ([7]) The inequality ess sup w(t)Hg(t) ≤ c ess sup v(t)g(t) t>0
(4.1)
t>0
holds for all non-negative and non-increasing g on (0, ∞) if and only if Z w(t) t ds A := sup < ∞, t t>0 0 ess sup0 0 does not depend on x0 , r and f . Moreover, for any x0 ∈ Rn and r > 0 Z ∞ n t−n+1 kf kL1 (B(x0 ,t)) dt, kT f kW L1 (B(x0 ,r)) ≤ C r
(4.4)
2r
where constant C > 0 does not depend on x0 , r and f . Proof. Let p ∈ (1, ∞). For arbitrary x0 ∈ Rn , set B = B(x0 , r) for the ball centered at x0 and of radius r. Write f = f1 + f2 with f1 = f χ2B and f2 = f χRn \(2B) . Since f1 ∈ Lp (Rn ), T f1 (x) exists for a.a. x ∈ Rn and from the boundedness of T in Lp (Rn ) ([9]) it follows that: kT f1 kLp (B) ≤ kT f1 kLp (Rn ) ≤ Ckf1 kLp (Rn ) = Ckf kLp (2B) , where constant C > 0 is independent of f. Let us prove that the non-singular integral T f2 (x) exists for all x ∈ B. It’s clear that x ∈ B, y ∈ Rn \(2B) implies 12 |x0 − y| ≤ |x − y| ≤ 32 |x0 − y|. We get Z |f (y)| n |T f2 (x)| ≤ 2 dy. n Rn \(2B) |x0 − y| By Fubini’s theorem we have Z ∞ Z Z dt |f (y)| |f (y)| dy ≈ dy n n+1 |x0 −y| t Rn \(2B) Rn \(2B) |x0 − y| Z ∞Z dt |f (y)|dy n+1 ≈ t 2r 2r≤|x0 −y|0 B(x0 , r), we get existence of T f (x) for a.a. x0 ∈ Rn . Moreover, for all p ∈ [1, ∞) the inequality Z ∞ n dt kT f2 kLp (B) . r p kf kLp (B(x0 ,t)) n +1 . (4.5) tp 2r
10
is valid. Thus kT f kLp (B) . kf kLp (2B) + r
n p
Z
∞
kf kLp (B(x0 ,t)) 2r
dt t
n +1 p
.
On the other hand, ∞ dt kf kLp (2B) ≈ r kf kLp (2B) n +1 2r t p Z ∞ n dt kf kLp (B(x0 ,t)) n +1 . . rp tp 2r
Z
n p
Thus kT f kLp (B) . r
n p
Z
∞
kf kLp (B(x0 ,t)) 2r
dt t
n +1 p
.
Let p = 1. From the weak (1,1) boundedness of T ([9]) it follows that: kT f1 kW L1 (B) ≤ kT f1 kW L1 (Rn ) ≤ Ckf1 kL1 (Rn ) = Ckf kL1 (2B) , where the constant C > 0 is independent of f. Then by (4.5) we get the inequality (4.4).
Theorem 4.3. Let p ∈ [1, ∞) and (ω1 , ω2 ) ∈ Zp,n . Then Calder´ on-Zygmund n singular integral T f (x) exists for a.a. x ∈ R and for p > 1 the operator T is bounded from Mp,ω1 (Rn ) to Mp,ω2 (Rn ) and for p = 1 the operator T is bounded from M1,ω1 (Rn ) to W M1,ω2 (Rn ). Moreover, for p > 1 kT f kMp,ω2 . kf kMp,ω1 , and for p = 1 kT f kW M1,ω2 . kf kM1,ω1 . Proof. By Lemma 4.2 and Theorem 4.1 we have for p > 1 Z ∞ dt − p1 n p kf kLp (B(x,t)) n +1 kT f kMp,ω2 (Rn ) . sup ω2 (x, r) r x∈Rn , r>0 tp r n Z r− p − p1 n p kf kLp (B(x,t− np )) dt ≈ sup ω2 (x, r) r x∈Rn , r>0 0 Z r p − p1 1 −n = sup ω2 (x, r ) kf kLp (B(x,t− np )) dt r 0 x∈Rn , r>0 .
sup x∈Rn ,r>0
p
1
ω1 (x, r− n )− p kf kLp (B(x,r− np )) = kf kMp,ω1 (Rn )
11
and for p = 1 kT f kW M1,ω2 (Rn ) . ≈ =
−1 n
sup
∞
kf kL1 (B(x,t))
ω2 (x, r) r
x∈Rn , r>0
r −1 n
sup
Z
sup
kf kL1 (B(x,t−n )) dt 0
ω2 (x, r
1 −1 1 −n
)
r
Z
r
kf kL
1 (B(x,t
0
1
sup x∈Rn ,r>0
dt tn+1
r −n
ω2 (x, r) r
x∈Rn , r>0
x∈Rn , r>0
.
Z
ω1 (x, r− n )−1 kf kL
1 (B(x,r
1 −n
))
1 −n
))
dt
= kf kM1,ω1 (Rn ) .
Corollary 4.4. Let p ∈ [1, ∞) and (ω1 , ω2 ) ∈ Zep,n (Rn ). Then for p > 1 T is bounded from Mp,ω1 (Rn ) to Mp,ω2 (Rn ) and for p = 1 T is bounded from M1,ω1 to W M1,ω2 . Note that Theorem 2.7 and Corollary 4.4 coincide.
5. Estimates for uniformly elliptic operators on generaized Morrey spaces In this section we consider uniformly elliptic operators L=−
n X
∂i (aij (x)∂j ) + V (x)
i,j=1
with non-negative potentials V on Rn (n ≥ 3) which belong to certain reverse 1 H¨older class. We show several estimates for V L−1 , V 2 ∇L−1 and ∇2 L−1 on generalized Morrey spaces under certain assumptions on aij (x), V and p. Our results generalize some results of K. Kurata and S. Sugano [12]. For the Schr¨odinger operators −∆ + V (x) with nonnegative polynomials V , several authors ([20], [23], [24]) studied Lp boundedness for 1 < p < ∞ of 1 1 1 ∇(−∆ + V )− 2 , (−∆ + V )− 2 ∇, and ∇(−∆ + V )−1 ∇, V 2 ∇(−∆ + V )−1 , and ∇2 (−∆ + V )−1 . In particular, J. Zhong [24] proved that if V is a non-negative 1 polynomial, ∇2 (−∆ + V )−1 , ∇(−∆ + V )− 2 , and ∇(−∆ + V )−1 ∇ are Calder´onZygmund operators. Recently, Z. Shen [19] generalized these results. He proved 1 1 that ∇(−∆ + V )− 2 , (−∆ + V )− 2 ∇, and ∇(−∆ + V )−1 ∇ are Calder´on-Zygmund operators, if V belongs to the reverse H¨older class Bn (see Definition 6.1), which includes non-negative polynomials and allows some non-smooth potentials. Moreover, Z. Shen also showed Lp boundedness for V (−∆ + V )−1 , and ∇2 (−∆ + V )−1 1 when V ∈ Bn/2 and V 2 ∇(−∆ + V )−1 when V ∈ Bn .
12
In this section we consider uniformly elliptic operators n X
L = L0 + V (x) = −
∂i (aij (x)∂j ) + V (x)
i,j=1
with certain non-negative potentials V on Rn (n ≥ 3), where aij (x) is a measurable function satisfying the conditions: (A1 ) There exists a constant λ ∈ (0, 1] such that 2
aij (x) = aji (x), λ|ξ| ≤
n X
aij (x)ξi ξj ≤ λ−1 |ξ|2 , x, ξ ∈ Rn ;
i,j=1
(A2 ) There exist constants α ∈ (0, 1] and K > 0 such that kaij kC α (Rn ) ≤ K. Throughout this section we use the following notation: n
∂j = ∇j = ∇xj
X ∂ = , |∇u(x)|2 = |∇j u(x)|2 ∂xj j=1
The purpose of this section is to show boundedness of the operators T1 = V L−1 , 1 T2 = V 2 ∇L−1 and T3 = ∇2 L−1 from one generalized Morrey space Mp,ω1 to another Mp,ω2 . Although it is known T1 and T3 are Calder´on-Zygmund operators for the case L = −∆ + V with non-negative polynomials V , it is not known that whether Tj (j = 1, 2, 3) are Calder´on-Zygmund operators or not under the general condition V ∈ B∞ . We show, under the same conditions as in [19] 1 for V , boundedness of T1 = V L−1 and T2 = V 2 ∇L−1 on generalized Morrey spaces Mp,ω (Rn ). Actually, we used pointwise estimates of Tk f (x), k = 1, 2, by the Hardy-Littlewood maximal function (see [12], Theorem 1.3). We also show boundedness of T3 = ∇2 L−1 on generalized Morrey spaces under the additional assumption (A3 ) There exist a constant α ∈ (0, 1] such that aij ∈ C 1+α (Rn ), aij (x + z) = aij (x), for all x ∈ Rn , for all z ∈ Zn , and n X
∂i (aij (x)) = 0, j = 1, . . . , n.
i −1
Here L is the integral operator with the fundamental solution (or the minimal Green function (see e. g. [15])) of L as its integral kernel. We can also define L−1 f for f ∈ C0∞ (Rn ) as the unique solution of Lu = f on certain generalized Morrey space M2,ω (Rn ), and can see it is a bounded operator on certain generalized Morrey spaces M2,ω (Rn ) (see e. g. [20]).
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Definition 5.1. Let V (x) ≥ 0. (1) A nonnegative locally Lq integrable function V on Rn is said to belong to the reverse H¨older class Bq (1 < q < ∞) if there exists C > 0 such that the reverse H¨older inequality 1q Z Z 1 C q V (x) dx ≤ V (x)dx |B| B |B| B holds for every ball B in Rn . (2) We say V ∈ B∞ , if there exists a constant C > 0 such that Z C kV kL∞ (B) ≤ V (x)dx |B| B holds for every ball B in Rn . Clearly, B∞ ⊂ Bq for 1 < q < ∞. But it is important that the Bq class has a property of ”self-improvement”; that is, if V ∈ Bq , then V ∈ Bq+ε for some ε > 0 (see [17]). K. Kurata and S. Sugano [12] proved the following pointwise estimate for T1 and T2 which generalize the results in [24], Lemma 3.2 to uniformly elliptic operators with general potentials V ∈ B∞ . Theorem A. Suppose that A(x) satisfies (A1 ) for T1 , (A1 ) − (A2 ) for T2 , and V ∈ B∞ . Then there exist positive constants Ck , k = 1, 2 such that |Tk f (x)| ≤ M f (x), f ∈ C0∞ (Rn ), k = 1, 2. Hence Theorem A and Theorem 3.4 in Section 2 imply Corollary 5.2. Let A(x) and V (x) satisfy the same assumptions as in Theorem A. 1 (1) Suppose 1 < p < ∞, and (ω1 , ω2 ) ∈ Z0,n . Then V L−1 and V 2 ∇L−1 are bounded from Mp,ω1 to Mp,ω2 . (2) Suppose 1 < p < ∞, (ω1 , ω2 ) ∈ Z0,n and (A3 ) for A(x). Then ∇2 L−1 is bounded from Mp,ω1 to Mp,ω2 . Theorem B. (1) Suppose A(x) satisfies (A1 ) and V ∈ Bq , q > n/2. Then there exist a positive constant C such that 0 1/q 0 |T1∗ f (x)| ≤ CM |f |q (x), f ∈ C0∞ (Rn ), where 1/q + 1/q 0 = 1. (2) Suppose A(x) satisfies (A1 ) − (A2 ). When V ∈ Bq with n > q > n/2 we have 1/p1 (x), f ∈ C0∞ (Rn ), |T2∗ f (x)| ≤ CM |f |p1 where 1/p1 = 1 + (1/n) − (3/2q). When V ∈ Bq with q > n we have 1/p1 (x), f ∈ C0∞ (Rn ), |T2∗ f (x)| ≤ CM |f |p1
14
where 1/p1 = 1 − (1/2q). Hence Theorem B and Theorems 3.4 and 4.3 imply Corollary 5.3. (1) Suppose A(x) satisfies (A1 ). Suppose V ∈ Bq with q > n/2, and (ω1 , ω2 ) ∈ Z0,n and q 0 < p < ∞. Then T1 is bounded from Mp,ω1 to Mp,ω2 . (2) Suppose A(x) satisfies (A1 ) − (A2 ). Suppose V ∈ Bq with n/2 < q < n, p1 < p < ∞, 1/p1 = 1 + (1/n) − (3/2q) and (ω1 , ω2 ) ∈ Z0,n . Then T2∗ is bounded from Mp,ω1 to Mp,ω2 . (3) Suppose A(x) satisfies (A1 ) − (A2 ). Suppose V ∈ Bq with q > n, p1 < p < ∞, 1/p1 = 1 − (1/2q) and (ω1 , ω2 ) ∈ Z0,n . Then T2∗ is bounded from Mp,ω1 to Mp,ω2 . (4) Suppose A(x) satisfies (A1 ) − (A3 ), 1 < p < ∞ and (ω1 , ω2 ) ∈ Zp,n . Then 2 −1 ∇ L is bounded from Mp,ω1 to Mp,ω2 . Acknowledgements. The authors thank Dr. A. Gogatishvili for valuable comments.
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Ali Akbulut Ahi Evran University, Department of Mathematics, Kirsehir, Turkey E-mail address:
[email protected] Vagif Guliyev Ahi Evran University, Department of Mathematics, Kirsehir, Turkey Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan E-mail address:
[email protected] Rza Mustafayev Institute of Mathematics of Academy of Sciences of Czech Republic Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan E-mail address:
[email protected]
