- Email: [email protected]
Fractional integral operators between Banach function lattices
Nonlinear Analysis 117 (2015) 148–158
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Fractional integral operators between Banach function lattices Vakhtang Kokilashvili a,b , Mieczysław Mastyło c,d , Alexander Meskhi a,e,∗ a
Department of Mathematical Analysis, A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, University Str., 0186 Tbilisi, Georgia b
International Black Sea University, Agmashenebeli Ave., Tbilisi 0131, Georgia
c
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
d
Institute of Mathematics, Polish Academy of Science (Poznań branch), Umultowska 87, 61-614 Poznań, Poland
e
Department of Mathematics, Faculty of Informatics and Control Systems, Georgian Technical University, 77, Kostava St., Tbilisi, Georgia
article
abstract
info
Article history: Received 21 October 2014 Accepted 22 January 2015 Communicated by Enzo Mitidieri
We study the generalized fractional integral transforms associated to a measure on a quasimetric space. We give a characterization of those measures for which these operators are bounded between Lp -spaces defined on nonhomogeneous spaces. The key in the proof of one of the main theorems is the boundedness of the modified sublinear Hardy–Littlewood maximal operator in the classical Lebesgue space with general measure. We also provide necessary and sufficient conditions for some classes of integral operators to be bounded from Lorentz to Marcinkiewicz spaces. © 2015 Elsevier Ltd. All rights reserved.
MSC: primary 42B20 26A33 secondary 47B38 47G10 Keywords: Fractional integral operators Non-doubling measure Lorentz spaces Marcinkiewicz spaces
1. Introduction The aim of this paper is to study the boundedness of generalized fractional integral operators between Banach function lattices. The paper is motivated by applications of classical fractional integrals in the theory of Sobolev embeddings. We recall that in the Euclidean space Rn , the fractional integral operator Iα of order α , 0 < α < n, is defined by Iα f (x) =
Rn
f (y) dy, |x − y|n−α
f ∈ L1loc (Rn ).
By the well-known Hardy–Littlewood–Sobolev theorem, Iα is a bounded operator from Lp (Rn ) to Lq (Rn ) if and only if p > 1 and 1/p − 1/q = α/n. In the last two decades potential operators over quasi-metric measure spaces (X , d, µ) are studied intensively; we refer e.g. to the book [3]. We recall that by a quasi-metric we mean a mapping that satisfies the following conditions d(x, y) = 0
∗ Corresponding author at: Department of Mathematical Analysis, A. Razmadze Mathematical Institute, Javakhishvili Tbilisi State University, University Str., 0186 Tbilisi, Georgia. E-mail addresses: [email protected] (V. Kokilashvili), [email protected] (M. Mastyło), [email protected] (A. Meskhi). http://dx.doi.org/10.1016/j.na.2015.01.014 0362-546X/© 2015 Elsevier Ltd. All rights reserved.
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
149
if and only if x = y; d(x, y) ≤ Ct (d(x, z ) + d(z , y)) for some Ct ≥ 1 and all x, y, z ∈ X ; there exists a constant Cs such that d(x, y) ≤ Cs d(y, x) for all x, y ∈ X . As usual, for a ball B = B(x, r ) := {y ∈ X ; d(x, y) < r } with x ∈ X , r > 0 and a > 0, the notation aB := B(x, ar ) stands for the concentric dilation of B. We assume that all balls are measurable with positive finite measure. This implies that our measures are σ -finite and that the set {x ∈ X ; µ({x}) > 0} of atoms is at most countable. We say that the measure µ is lower α -Ahlfors regular (resp., upper β -Ahlfors regular) with α > 0 and β > 0, if there exists C > 0 such that
µ(B(x, r )) ≥ Cr α (resp.,
µ(B(x, r )) ≤ Cr β ), for all x ∈ X and 0 < r < diam(X ). When α = β , the measure µ is called α -Ahlfors regular. We use standard definitions and notation from the theory of Banach lattices (see, e.g., [8]). Let (Ω , Σ , µ) be a complete σ -finite measure space and let L0 (µ) = L0 (Ω , µ) denote the space of all equivalence classes of µ-a.e. finite real-valued measurable functions on Ω with the topology of convergence in measure on µ-finite sets and let L0 (µ, Ω ) denote the space 0 of extended real valued measurable functions on Ω . A linear subspace E of L (µ) is said to be an ideal if f ∈ E and |g | ≤ |f | in L0 (µ), then g ∈ E. By E+ we denote the collection of nonnegative functions in E. Let E and F be two vector subspaces of L0 (S , Σ1 , µ) and L0 (T , Σ2 , ν), respectively. A linear operator T : E → F is said to be an integral operator if there exists a Σ1 × Σ2 measurable function K (called the kernel of the operator) such that we have Tf (x) =
K (x, y)f (y) dν(y) T
for every f ∈ E and for µ-almost all x ∈ S . A Banach (function) lattice (E , ∥·∥E ) on (Ω , Σ , µ) is an ideal of L0 (µ), which is complete with respect to a norm ∥·∥E . We also assume that the support of the space E is Ω (supp(E ) = Ω ), that is, there is an element u ∈ E with u > 0 µ-a.e. on Ω . It is said that E has the Fatou property (or E is maximal) if for any f ∈ L0 , fn ∈ E+ such that fn ↑ f a.e. and sup ∥fn ∥E < ∞, we have that f ∈ E and ∥fn ∥E ↑ ∥f ∥E . The Köthe dual space E ′ of a Banach lattice E on (Ω , Σ , µ) is defined as the space of all f ∈ L0 (µ) such that Ω |fg | dµ < ∞ for every g ∈ E. It is a Banach lattice on (Ω , Σ , µ) when equipped with the norm
∥f ∥E ′ = sup ∥g ∥E ≤1
Ω
|fg | dµ,
f ∈ E′.
Let us remark that E ′ of E is a maximal Banach lattice on (Ω , µ) as well as a number of classical spaces such as Lebesgue spaces Lp , 1 ≤ p ≤ ∞, Orlicz spaces or more general Musielak–Orlicz spaces. Notice that E is a maximal Banach lattice if and only if E = E ′′ := (E ′ )′ with equality of norms (see, e.g., [8, p. 30]). Throughout the paper by c and C we denote various absolute positive constants which may have different values even on the same line. 2. Generalized fractional integral operators Let (X , d, µ) be a quasi-metric space. We say that an integral operator TK is an operator of potential type if it is of the form TK f (x) =
K (x, y)f (y) dµ(y),
x ∈ X,
X
where the kernel K : X × X → [0, ∞] is a function which satisfies the following monotonicity conditions: For every k2 > 1 there exists k1 > 1 such that K (x, y) ≤ k1 K (x′ , y) whenever d(x′ , y) ≤ k2 d(x, y), and K (x, y) ≤ k1 K (x, y′ ) whenever d(x, y′ ) ≤ k2 d(x, y). Important examples of potential type operators are the fractional integrals. We mention that the fractional operators Iα f (x) =
f (y)
X
d(x, y)n−α
dµ(y),
x ∈ X,
where 0 < α < n, has been studied intensively for both the doubling and non-doubling setting. Another type of fractional integral is given by Tf (x) :=
f (y)
X
µ(B(x, d(x, y)))1−γ
dµ(y),
x ∈ X,
150
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
where γ ∈ (0, 1). We refer e.g., to [3] (Chapter 6) and also to the papers [5,7] and references therein where this type operator is studied. Recall that the ϕ : [0, a) → [0, ∞) defined on [0, a) with 0 < a ≤ ∞ is called quasi-concave if it is non-decreasing on [0, a) with ϕ(0) = 0 and t → ϕ(t )/t is non-increasing on [0, a). To simplify presentation we denote the set of quasi-concave . For any quasi-concave function ϕ ∈ P , we functions on [0, ∞) by P . In the case when ϕ ∈ P is concave we write ϕ ∈ P define ϕ∗ (t ) := t /ϕ(t ) for all t > 0. For any ϕ ∈ P , we define ϕ by
ϕ(t ) := sup s>0
ϕ(st ) , ϕ(s)
t > 0.
Since ϕ is sub-multiplicative, it follows that there exist the limits (see, e.g., [11, pp. 52–54]) ln ϕ(t )
αϕ := lim
ln t
t →0+
,
βϕ := lim
ln ϕ(t )
t →∞
ln t
,
which satisfy 0 ≤ αϕ ≤ βϕ ≤ 1. Throughout the paper we consider functions ϕ ∈ P which satisfy 0 < αϕ ≤ βϕ < 1. For general examples of such functions we refer to [6]. For the sake of completeness we include an example; if 0 < θ < 1, 0 < β, γ satisfy θ + γ β < 1, then ϕ given by the formula
ϕ(t ) = t θ lnβ (1 + t γ ),
t > 0,
is a quasi-concave function with αϕ = θ and βϕ = θ + γ β . Let (X , d, µ) be a quasi-metric space and let ϕ ∈ P . In the paper we consider the kernel Kϕ : X × X → [0, ∞) defined by Kϕ (x, y) =
1
ϕ (d(x, y)) 0,∗
,
if x ̸= y, if x = y.
The integral operator generated by Kϕ is denoted by Tϕ . In the case when ϕ(t ) = t α for some α ∈ (0, 1) and all t ≥ 0 we write Tα instead of Tϕ . We will need the following modified maximal operator on a quasi-metric space (X , d, µ) given by
(x) = sup Mf r >0
1
µ(B(x, N0 r ))
B(x,r )
| f | dµ
where N0 = Ct (1 + 2Cs ) with constants Cs and Ct from the definition of a quasi-metric d. It is well-known (see [3, p. 368]) that for any f ∈ L1 (µ),
(x) > λ}) ≤ 1 µ({x ∈ X ; Mf λ
|f (x)| dµ(x),
λ > 0,
X
is of weak type (1, 1). Since M is bounded in L∞ (µ), Marcinkiewicz’s interpolation theorem yields that M is bounded i.e., M in Lp (µ) for every 1 < p < ∞. We are ready to state and prove a result on the boundedness of Tϕ between corresponding Lp (µ) spaces. Theorem 2.1. Let 1 < p < q < ∞ and ϕ ∈ P be such that 0 < αϕ ≤ βϕ < 1. Then the operator Tϕ is bounded from Lp (µ) to Lq (µ) if and only if there exists C > 0 such that ′
µ(B(x, r )) ≤ C ϕ∗ (r )s ,
x ∈ X , r > 0,
(∗)
where 1/s = 1/p − 1/q and 1/s + 1/s = 1. ′
Proof. Assume that Tϕ is bounded from Lp (µ) → Lq (µ). Then there exists C > 0 such that
∥Tϕ f ∥Lq (µ) ≤ C ∥f ∥Lp (µ) ,
f ∈ Lp (µ).
(1)
For f = χB(a,r ) , where a ∈ X , r > 0, we have Tϕ χB(a,r ) (x) =
1 B(a,r )
ϕ∗ (d(x, y))
dµ(y),
x ∈ X.
Observe that for all x, y ∈ B(a, r ) d(x, y) ≤ Ct (d(x, a) + d(a, y)) ≤ Ct (Cs + 1)r. Denoting C0 := Ct (Cs + 1) and combining this estimate with ϕ∗ (C0 r )/C0 r ≤ ϕ∗ (r )/r for all r > 0 we have that 1 µ(B(a, r )) C0
ϕ∗ (r )
≤
µ(B(a, r )) ≤ Tϕ χB(a,r ) , ϕ∗ (C0 r )
x ∈ B(a, r ).
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
151
This shows that 1 µ(B(a, r ))
χB(a,r ) ≤ χB(a,r ) Tϕ χB(a,r ) .
ϕ∗ (r )
C0
Applying inequality (1), we conclude that 1 µ(B(a, r ))
∥χB(a,r ) ∥Lq (µ) ≤ C ∥χB(a,r ) ∥Lp (µ)
ϕ∗ (r )
C0 and whence
1 µ(B(a, r )) 1+1/q
ϕ∗ (r )
C0
≤ C µ(B(a, r ))1/p
gives the required estimate. For sufficiency we claim that the estimate (∗) implies that Tϕ is bounded operator from Lp (µ) to Lq (µ). To see this we define a bounded function ω: X → [0, ∞),
ω(x) := sup r >0
µ(B(x, r )) , ϕ∗ (r )s′
x ∈ X,
which was introduced in [9] in the case of Euclidean spaces and ϕ(t ) = t α , α ∈ (0, 1). Further, for x ∈ X , and f ∈ D(Tϕ ), we have T ϕ f ( x) =
f (y)
B(x,r )
ϕ∗ (d(x, y))
dµ +
f (y)
X \B(x,r )
ϕ∗ (d(x, y))
dµ = I1 (x) + I2 (x).
Set Dk := B(x, 2−k r ) \ B(x, 2−k−1 r ) for each integer k ≥ 0. Then we have
|I1 (x)| ≤
Dk
k=0
≤
|f (y)| dµ(y) ϕ∗ (d(x, y))
∞
µ(B(x, N0 2−k r )) ϕ∗ (2−k r ) µ(B(x, N0 2−k r ))
∞
1
k=0
≤C ≤C
∞
1
k=0
ϕ∗ (2−k r )
∞
B(x,2−k r )
|f (y)| dµ(y)
′
(x) ω(x) ϕ∗ (N0 2−k r )s Mf ′
(x) ω(x). ϕ∗ (2−k r )s −1 Mf
k=0
Here we used the estimate ϕ∗ (N0 s) ≤ C ′ ϕ∗ (s) for some C ′ > 0 and for all s > 0. Now observe that our hypothesis implies 0 < βϕ∗ ≤ αϕ∗ < 1 and so
ϕ∗ (t ) ≤ C2 t αϕ∗ +ε ,
t ∈ (0, 1]
(2)
for some C2 > 0 and 0 < ε < 1 − αϕ∗ . This estimate together with the inequality ϕ∗ (uv) ≤ ϕ∗ (u)ϕ∗ (v), u, v > 0, implies that for all r > 0, ∞
′
ϕ∗ (2−k r )s −1 ≤
k=0
∞ ′ [ϕ∗ (r )ϕ∗ (2−k )]s −1 .
(3)
k=0
Since s′ > 1, the estimates (2) and (3) imply ∞
′
∞
′
ϕ∗ (2−k r )s −1 ≤ (C2 ϕ∗ (r ))s −1
k=0
where C3 :=
2−k(αϕ∗ +ε)(s −1) ≤ C3 ϕ∗ (r )s −1 , ′
′
k=0
∞
k=0
−k(αϕ∗ +ε)
2
s ′ −1
|I1 (x)| ≤ C3 ϕ∗ (r )
< ∞. Combining this fact with the above estimate of I1 we find that
(x) ω(x), Mf
x ∈ X , r > 0.
To estimate I2 (x), we put Ek := B(x, 2 all f ∈ Lp (µ),
k+1
|I2 (x)| ≤ ∥f ∥Lp (µ)
r ) \ B(x, 2k r ) for each integer k ≥ 0. Then the Hölder’s inequality gives that for
1 X \B(x,r )
ϕ∗ (d(x, y))p′
dµ
1/p′
.
152
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
Fix f ∈ Lp (µ) with ∥f ∥Lp (µ) ≤ 1. Then, we have by (∗)
|I2 (x)| ≤
∞
dµ
1/p′
Ek
k=0
≤
ϕ∗ (d(x, y))
−p′
∞
ϕ∗ (2 r ) k
−p′
µ(B(x, 2
k+1
r ))
1/p′
k =0
∞ 1/p′ 1/p′ ′ ′ ≤ C ω(x) ϕ∗ (2k r )−p ϕ∗ (2k+1 r )s k=0
≤ C4 ω(x)1/p
′
′
′
ϕ∗ (2k r )s −p
1/p′
k=0 1/p′
= C4 ω(x)
∞
ϕ(2k r ) p′ −s′
1/p′
2k r
k=0
1/p′ ϕ(r ) ϕ(2k r ) p′ −s′ ≤ C4 ω(x) r 2k ϕ(r ) k=0 1/p′ ∞ 1−s′ /p′ ϕ(2k ) p′ −s′ 1/p′ ϕ(r ) ≤ C4 ω(x) . k 1/p′
∞
r
2
k=0
Here we used the fact that s < p . Our hypothesis on ϕ implies that ′
ϕ(t ) ≤ C5 t βϕ +ε ,
′
t ≥1
with some C5 > 0 and 0 < ε < 1 − βϕ . This inequality and the observation 1 − s′ /p′ = s′ /q give that there is a constant C6 > 0 such that for all r > 0, 1/p′
|I2 (x)| ≤ C6 ω(x)
∞ ′ −1/q ϕ∗ (r )s k=0
1
p′ −s′ 1/p′
2(1−βϕ −ε)k
≤ C7 ω(x)1/p ϕ∗ (r )−s /q . ′
′
The obtained estimates for I1 and I2 imply the following pointwise inequality
′ (x)ω(x) + ω(x)1/p′ ϕ∗ (r )−s′ /q . |Tϕ f (x)| ≤ C ϕ∗ (r )s −1 Mf Since x → ω(x) is a bounded function,
′ (x) + ϕ∗ (r )−s′ /q . |Tϕ f (x)| ≤ C ϕ∗ (r )s −1 Mf Since 0 < αϕ∗ ≤ βϕ∗ < 1, ϕ∗ (s) → 0 as s → 0+ and ϕ∗ (s) → ∞ as s → ∞. This implies that the image of ϕ∗ is [0, ∞).
(x)−p/s′ , we obtain the pointwise inequality true for all f ∈ Lp (µ) with ∥f ∥Lp (µ) = 1, Thus taking r > 0 such that ϕ∗ (r ) = Mf |Tϕ f (x)| ≤ C
1−p(1−1/s′ )
(x) Mf
(x) p/q = 2C Mf (x) p/q , + Mf
x ∈ X.
Lp (µ) → Lp (µ) is bounded, we conclude that for all f ∈ Lp (µ) with ∥f ∥Lp (µ) = 1, Combining with the fact that M: ∥Tϕ f ∥
q Lq (µ)
p
≤ 2C X
(x)p dµ = 2C ∥Mf ∥ p ≤ 2C0 ∥f ∥Lp (µ) ≤ C0 . Mf L (µ)
This shows that Tϕ is bounded from Lp (µ) to Lq (µ), and this completes the proof.
Corollary 2.2 ([10]). Let 1 < p < q < ∞ and 0 < α < 1. Then Tα is a bounded operator from Lp (µ) to Lq (µ) if and only if the pq(1−α) measure µ is upper s-Ahlfors regular, where s = pq−q+p . We note that this result in Euclidean spaces for the first time appeared in [9]. As an application we obtain the following version of the Hardy–Littlewood–Sobolev theorem (see [4,10]) for the fractional integral Iα associated to the nonhomogeneous space (X , d, µ) given by Iα f (x) =
X
where 0 < α < n.
f (y) d(x, y)n−α
dµ(y),
x ∈ X,
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
153
Corollary 2.3. For 1 < p < n/α and 1/q = 1/p − α/n, Iα is a bounded operator from Lp (µ) into Lq (µ) provided that the measure µ is upper n-Ahlfors regular. Before the proof of the next result we need some definition. Let ψ : (0, ∞) → (0, ∞) be nondecreasing function. The (∗) (∗) Marcinkiewicz class Mψ = Mψ (µ) on a measure space (Ω , Σ , µ) is defined to be the set of all measurable functions
f ∈ L0 (µ) such that
∥f ∥ψ = sup ψ(t )f ∗ (t ) < ∞, t >0
where f ∗ denotes the decreasing rearrangement of f , i.e., f ∗ (t ) := inf{λ ≥ 0; µf (λ) ≤ t } for all t > 0, and µf (λ) := µ{t ∈ Ω ; |f (t )| > λ}, λ ≥ 0. In particular, for ψ(t ) = t 1/q , 0 < q < ∞, we obtain the classical spaces Lq,∞ (µ). One can readily show that ∥ · ∥ψ is a quasi-norm (i.e.,
∥f + g ∥ψ ≤ C (∥f ∥ψ + ∥g ∥ψ ),
(∗)
f , g ∈ Mψ ,
for some C > 0) if the function ψ satisfies the ∆2 -condition, i.e., there exists K ≥ 1 such that ψ(2t ) ≤ K ψ(t ) for all t > 0. (∗) It can be easily shown that for any f ∈ Mψ we have
∥f ∥ψ = sup λψ({ω ∈ Ω ; |f (ω)| > λ}). λ>0
Theorem 2.4. Let 1 ≤ p < ∞ and let (X , d, µ) be a quasi-metric space. Let ϕ : [0, ∞) → [0, ∞) be such that the function φ given by φ(t ) = t −1/p ϕ(t ) for all t > 0 be decreasing with φ(R+ ) = R+ . Assume that K is a positive kernel on X which satisfies the following conditions:
B(x,r )
K (x, y) dµ(y) ≤ C1 ϕ(r ),
B(y,r )
K (x, y) dµ(x) ≤ C2 ϕ(r )
and
χX \B(x,r ) K (x, ·) p′
L (µ)
≤ C3 φ(r )
for some Cj > 0 (1 ≤ j ≤ 3) and all x ∈ X and r > 0, where 1/p + 1/p′ = 1. Then the integral operator TK is bounded from (∗)
Lp (µ) into Mψ (µ), where ψ(t ) = φ(1t ) for all t > 0. Proof. We can, and do, assume that Cj = 1 for each 1 ≤ j ≤ 3. Fix f ∈ Lp (µ) with ∥f ∥Lp (µ) ≤ 1. For any ball B(x, r ) in X we have
|TK f |(x) ≤
K (x, y)|f (y)| dµ(y) = I1 (x) + I2 (x), X
where I1 (x) :=
B(x,r )
K (x, y)|f (y)| dµ(y)
and I2 (x) =
X \B(x,r )
K (x, y)|f (y)| dµ(y).
We present the proof for p > 1. Then by Hölder’s inequality, we obtain
|I2 (x)| ≤ ∥f ∥
Lp (µ)
p′
X \B(x,r )
K (x, y) dµ(y)
1/p′
1
≤
ψ(r )
.
For λ > 0, we take r = ψ −1 (2/λ). Then the above inequality yields |I2 (x)| ≤ λ/2 and whence
{x ∈ X ; |TK f |(x) > λ} ⊂ {x ∈ X ; |I1 (x)| > λ/2} ∪ {x ∈ X ; |I2 (x)| > λ/2} = {x ∈ X ; |I1 (x)| > λ/2}. Using Hölder’s inequality we get
|I1 (x)| ≤
B(x,r )
≤ ϕ(r )
1/p′
K (x, y)|f (y)|p dµ(y)
1/p
1/p′
B(x,r )
1/p
B(x,r )
K (x, y)|f (y)|p dµ(y)
K (x, y) dµ(y)
.
154
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
Combining with Tchebyshev’s inequality and Tonelli’s theorem we find that
µ({x ∈ X ; |TK f (x)| > λ}) ≤ µ({x ∈ X ; |I1 (x)| > λ/2}) ′ ≤ ϕ(r )p/p (2/λ)p K (x, y)|f (y)|p dµ(y) dµ(x) X B(x,r ) ′ ′ = ϕ(r )p/p (2/λ)p K (x, y) dµ(x)|f (y)|p dµ(y) ≤ ϕ(r )p/p ϕ(r )(2/λ)p B(y,r )
X
= (2ϕ(r )/λ)p = ψ −1 (2/λ). (∗)
This shows that TK f ∈ Mψ with
∥TK f ∥ψ = sup λψ(µ{x ∈ X ; |TK f (x)| > λ}) ≤ 2. λ>0
To finish we note that the proof for p = 1 is a simple modification of the proof for p > 1.
The following lemma provides examples of generalized fractional operators which satisfies the integral conditions of the above theorem. Lemma 2.5. Let (X , d, µ) be a quasi-metric space and ϕ ∈ P . Suppose that µ(B(x, r )) ≤ Cr for some C > 0 and all 0 < r < diam(X ), x ∈ X . Then we have: (i) If αϕ > 0, then there exists C1 > 0 such that
1 B(x,r )
dµ(y) ≤ C1 ϕ(r ).
ϕ∗ (d(x, y))
(ii) If ϕ is sub-multiplicative (i.e., ϕ(st ) ≤ C ϕ(s)ϕ(t ) for some C > 0 and all s, t > 0) such that 1 ≤ p < 1/βϕ then there exists C2 > 0 for which
1 X \B(x,r )
ϕ∗ (d(x, y))p′
1/p′ ϕ(r ) ≤ C2 1/p . dµ(y) r
Proof. (i) Since α := αϕ > 0, for every ε > 0 there exists C (ε) such that ϕ(s) ≤ C (ε)sα−ε for all s ∈ (0, 1). Fix x ∈ X and r > 0 and put Dj = {y ∈ X ; 2−j−1 r ≤ d(x, y) < 2−j r }; then our hypothesis µ(B(x, r )) ≤ C r, αϕ > 0 implies
1 B(x,r )
ϕ∗ (d(x, y))
dµ(y) =
∞ Dj
j =0
≤
1
ϕ∗ (d(x, y))
∞
1
j =0
ϕ∗ (2−j−1 r )
≤ 2C
∞
dµ(y)
µ(B(x, 2−j r ))
ϕ(2−j−1 )ϕ(r ) ≤ C1 ϕ(r ).
j=0
(ii) Since ϕ is sub-multiplicative, ϕ∗ is a super-multiplicative and βϕ < 1/p,
1 X \B(x,r )
ϕ∗ (d(x, y))p′
dµ(y) =
∞ j =0
≤
2j r ≤d(x,y)<2j+1 r
∞ j =0
p′
1
ϕ∗ (2j r ) ′
X \B(x,r )
ϕ∗ (d(x, y))p′
1/p′
≤ C2
ϕ(r ) r 1/p
∞ ϕ(2j r ) p′ j =0
and so this gives the required estimate dµ(y)
dµ(y)
2j r
p′ ′ ′ ′ 2j(1−p ) ϕ(2j ) ≤ Cϕ ϕ(r )p r 1−p
j =0
1
ϕ∗ (d(x, y))
p′
µ(B(x, 2j+1 r )) ≤ C
∞ ′
≤ C ϕ(r )p r 1−p
1
.
2j r
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
155
3. Fractional integral operators between symmetric spaces An important class of Banach lattices on a measure space (Ω , Σ , µ) are symmetric quasi-Banach spaces. We recall that a Banach lattice E on (Ω , µ) is said to be a symmetric Banach space if g ∈ E and ∥f ∥E = ∥g ∥E whenever f ∈ E, g ∈ L0 (µ) and µf (λ) = µg (λ) for all λ > 0, where µf (λ) = µ({ω ∈ Ω ; |f (ω)| > λ}). Clearly for a symmetric Banach space E, ∥χA ∥E depends only on µ(A) for any measurable set A of finite measure. Thus for every t ∈ {µ(A); A ∈ Σ }, we may define the function ϕE by ϕE (t ) = ∥χA ∥E where A is any measurable set with µ(A) = t. This function is called the fundamental function of E. For the theory of symmetric spaces we refer to [2,11]. We recall some important properties of symmetric spaces which we will need use later. Let E be a symmetric space on nonatomic measure space (Ω , Σ , µ) with 0 < µ(Ω ), and let φ := ϕE be the fundamental function of E. It is well known that φ is a quasi-concave on [0, a), where a = µ(Ω ). Taking φ(t ) = infs∈(0,a) (1 + t /s)φ(s) for all t ∈ [0, a), we obtain a concave function φ on [0, a) such that φ(t ) ≤ φ(t ) ≤ 2φ(t ) for all t ∈ [0, a). In what follows φ is called a concave majorant of φ . For any quasi-concave function ϕ : [0, a) → [0, ∞) the Marcinkiewicz space Mϕ = Mϕ (µ) is defined by the norm
∥f ∥Mϕ = sup
1
0
t
ϕ(t )
f ∗ (s) ds,
0
where f ∗ denotes the decreasing rearrangement of f given by f ∗ (t ) = inf{λ > 0; µ({ω ∈ Ω ; |f (ω)| > λ}) ≤ t },
t > 0.
We note that the largest symmetric space on (Ω , Σ , µ) with the same fundamental ϕ is the Marcinkiewicz space Mϕ∗ . We also note that the fundamental function of a symmetric space E = (E , ∥ · ∥E ) is not necessarily concave, however there exists an equivalent norm under which E is a symmetric space with concave fundamental function. For any symmetric space E with concave fundamental function ϕ = ϕE there is also the smallest symmetric space with the same fundamental function; it is the Lorentz space Λϕ = Λϕ (µ) given by the norm a
∥f ∥Λϕ =
f ∗ (s) dϕ(s). 0
In what follows we will use the following easily verified fact that in the case when µ is a nonatomic measure we have
∥ f ∥ Mϕ =
1
sup
A∈Σ , 0<µ(A)<∞
ϕ(µ(A))
|f | dµ,
f ∈ Mϕ .
A
For the theory of symmetric spaces we refer to [2,11]. To show applications of Lemma 2.5 to fractional integral operators, we define a special class of kernels. Let (X , d, µ) be a metric space. A kernel K ∈ L0 (µ × µ)+ is said to be a Borel kernel whenever there is a constant C ≥ 1 such that for every Borel set A in X with 0 < µ(A) < ∞ and x ∈ X there is a ball B(x, r ) such that µ(B(x, r )) = µ(A) and
K (x, y) dµ(y) ≤ C
A
B(x,r )
K (x, y) dµ(y).
The following proposition gives examples of Borel kernels. Proposition 3.1. Let (X , d, µ) be a nonatomic quasi-metric space. Let φ : [0, ∞) × [0, ∞) → [0, ∞) be a non-increasing function in each variable. Then K : X × X → [0, ∞) defined by the following formula is a Borel kernel: K (x, y) =
φ d(x, y), µ(B(x, d(x, y))) , 0,
if x ̸= y if x = y.
Proof. Let A be an arbitrary Borel set with a finite positive measure. Our hypothesis that µ is a nonatomic measure yields that for any x ∈ A there exists a ball B(x, r ) in X such that µ(B(x, r )) = µ( A). Put B := B(x, r ) ∩ A; then µ(A \ B) = µ(B(x, r ) \ B). Since y ∈ A \ B implies d(x, y) ≥ r, φ d(x, y), µ(B(x, d(x, y))) ≤ φ r , µ(B(x, r )) and so we have
B(x,r )
K (x, y) dµ(y) −
K (x, y) dµ(y) =
A
B(x,r )\B
K (x, y) dµ(y) −
K (x, y) dµ(y) A\B
≥ φ(r , µ(B(x, r ))) (µ(B(x, r ) \ B) − µ(A \ B)) = 0 and this completes the proof.
156
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
Lemma 3.2. Let (S , µ) and (T , ν) be measure spaces, and let TK be an integral operator generated by the kernel K ∈ L0 (µ×ν)+ . Then TK is bounded from a Banach lattice E on (T , ν) to the Marcinkiewicz space Mψ (µ) on (S , µ) if and only if C :=
1
sup
0<µ(A)<∞
ψ(µ(A))
K (s, ·) dµ(s) ′ < ∞. E
A
Proof. Combining Tonelli’s Theorem with the fact that TK is positive we obtain, sup ∥TK f ∥Mψ (µ) = sup
sup
= sup
sup
∥f ∥E ≤1 µ(A)>0
∥f ∥E ≤1
1
ψ(µ(A))
A
1
K (s, t )|f (t )| dν(t ) dµ(s) T
|f | K (s, ·) dµ(s) dν
ψ(µ(A)) T A 1 = sup K (s, ·) dµ(s) ′ . E µ(A)>0 ψ(µ(A)) A µ(A)>0 ∥f ∥E ≤1
Corollary 3.3. Let (X , d, µ) be a nonatomic quasi-metric space. If TK is an integral operator generated by a positive Borel kernel K such that
B(x,r )
K (x, y) dµ(y) ≤ φ(µ(B(x, r )))
(4)
for all balls B(x, r ) in X and some concave function φ : [0, µ(X )) → [0, ∞), then TK is a bounded operator from the Lorentz space Λφ (µ) into L∞ (µ). Proof. Since K is a Borel kernel, for every Borel set A with a finite positive measure and every x ∈ X , there exists a ball B(x, r ) in X such that µ(B(x, r )) = µ(A) and
K (x, y) dµ(y) ≤ A
B(x,r )
K (x, y) dµ(y).
Combining the latter with (4) we have TK χA (x) ≤
B(x,r )
K (x, y) dµ(y) ≤ φ(µ(B(x, r ))) = φ(µ(A)).
It is well-known that this inequality (see [11, Corollary 1, p. 112]) for arbitrary characteristic functions χA implies that TK is bounded from Λφ (µ) into L∞ (µ). As an immediate consequence of the above corollary we have the following result. Theorem 3.4. Let (X , d, µ) be a nonatomic quasi-metric space, such that, for any t > 0, h(t ) := inf{µ(B(x, t )); x ∈ X } > 0. Assume that K : X × X → [0, ∞) is a Borel kernel for which
θ (t ) := sup x∈X
B(x,t )
K (x, y) dµ(y) < ∞,
t > 0.
If t → θ ◦ h−1 (t )/t is almost non-increasing, then the integral operator TK is bounded from the Lorentz space Λφ (µ) into L∞ (µ), where φ is a concave majorant of θ ◦ h−1 . In what follows we will need a general interpolation theorem for operators from Lorentz into Marcinkiewicz spaces from which it follows (see, e.g., [12]): If ϕ0 , ϕ1 and ψ1 , ψ1 be concave functions on (0, a) with 0 < a ≤ ∞ such that
ψ (t ) 1∗ , t ∈ (0, a) ψ0∗ (t ) . Then every bounded operator T from Λϕj into Mψj for j = 0, 1 is also bounded from ΛΦ into MΨ . for some θ ∈ P Φ (t ) = ϕ0 (t )θ
ϕ (t ) 1 , ϕ0 (t )
Ψ∗ (t ) = ψ0∗ (t )θ
Theorem 3.5. Let (X , d, µ) be a nonatomic quasi-metric space. Suppose TK is an integral operator generated by a positive and symmetric Borel kernel K such that the condition (4) holds for some concave function φ : [0, µ(X )) → [0, ∞), and all balls B(x, r ) in X . Suppose further that ϕ and ψ are concave functions such that
ϕ(t ) ≥ C φ(t ), ψ∗ (t )
t ∈ (0, µ(X ))
for some C > 0. Then the integral operator TK is bounded from the Lorentz space Λϕ (µ) into the Marcinkiewicz space Mψ (µ). Proof. From Lemma 3.2 and Corollary 3.3, we have that TK : Λφ (µ) → L∞ (µ) and TK : L1 (µ) → Mφ . Since Mψ0 (µ) = L∞ (µ) and Λϕ1 (µ) = L1 (µ) with equality of norms whenever ψ0 (t ) = ϕ1 (t ) = t for every t ∈ (0, µ(X )), the above mentioned
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
157
interpolation theorem implies that TK : ΛΦ (µ) → MΨ (µ) is bounded for any quasi-concave function ρ , where
t Ψ∗ ( t ) = ρ , φ(t )
t , Φ (t ) = φ(t ) ρ φ(t )
t ∈ (0, µ(X ))
and ρ is a concave majorant of ρ . Now we define ρ ∈ P by
ρ(t ) :=
sup
0
min 1,
t φ(s)
s
s
ψ(s)
,
t ≥ 0.
Our hypothesis on ϕ and ψ implies that Ψ (t ) ≤ c ψ(t ) and ϕ(t ) ≥ c Φ (t ) for all t ∈ (0, µ(X )) and some c > 0. Hence
Λϕ (µ) ↩→ ΛΦ (µ),
MΨ (µ) ↩→ Mψ (µ)
with continuous inclusions and so the desired statement follows.
As an application we obtain the following corollary. Corollary 3.6. Let (X , d, µ) be a nonatomic quasi-metric space, such that the function h: (0, ∞) → (0, µ(X )) given by h(t ) = inf{µ(B(x, t )); x ∈ X } > 0 for all t > 0 is continuous. If TK is an integral operator generated by a positive and symmetric Borel kernel K such that C1 φ(µ(B(x, r ))) ≤
B(x,r )
K (x, y) dµ(y) ≤ C2 φ(µ(B(x, r )))
for some concave function φ : [0, µ(X )) → [0, ∞), some constants C1 , C2 > 0 and all balls B(x, r ) in X , then the integral operator TK is bounded from the Lorentz space Λϕ (µ) into the Marcinkiewicz space Mψ (µ) if and only if there exists C > 0 such that
ϕ(t ) ≥ C φ(t ), ψ∗ (t )
t ∈ (0, µ(X )).
Proof. It follows from Theorem 3.5 that we need to show only necessity; suppose that TK is bounded from Λϕ (µ) into Mψ (µ). Since Λϕ (µ)′ = Mϕ (µ) (see [1]), from Lemma 3.2 we have sup
0<µ(A)<∞
1
ψ(µ(A))
K (·, y) dµ(y) A
Mϕ
< ∞.
Combining the latter with the condition C1 φ(µ(B(x, r ))) ≤ such that for x, y ∈ X and r > 0 we have 1
B(x,r )
K (x, y) dµ(y) yields that there exists a constant C > 0
1
φ(µ(B(x, r ))) dµ(x) ψ(µ(B(x, r ))) ϕ(µ(B(y, r ))) B(y,r ) 1 µ(B(y, r )) ≥ φ(h(r )) ψ(µ(B(x, r ))) ϕ(µ(B(y, r ))) 1 h(r ) φ(h(r )). ≥ ψ(µ(B(x, r ))) ϕ(h(r ))
C ≥
Consequently, there exists C > 0 such that
ϕ(h(r )) ≥ C φ(h(r )), ψ∗ (h(r ))
r >0
and so the required estimate follows by continuity of h.
We conclude with the following result. Corollary 3.7. Let (X , d, µ) be a nonatomic quasi-metric space such that µ is 1-Ahlfors regular measure. Let φ : [0, ∞) → [0, ∞) be a concave function such that αϕ > 0. Suppose that ϕ and ψ are concave functions on [0, µ(X )). Then the integral operator Tφ is bounded from the Lorentz space Λϕ (µ) into the Marcinkiewicz space Mψ (µ) if and only if there exists C > 0 such that
ϕ(t ) ≥ C φ(t ), ψ∗ (t )
t ∈ (0, µ(X )).
Proof. For any ball B(x, r ) in X we have the obvious estimate C1 φ(r ) ≤
1 B(x,r )
φ∗ (d(x, y))
dµ(y).
158
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
Since µ is 1-Ahlfors regular, it follows from Lemma 2.5 that there exist constants C1 , C2 > 0 such that C1 φ(µ(B(x, r ))) ≤
1 B(x,r )
φ∗ (d(x, y))
dµ(y) ≤ C2 φ(µ(B(x, r ))).
Now, applying Corollary 3.6 for the symmetric Borel kernel Kφ yields the required result.
Acknowledgements We thank the referees for careful reading of the paper and useful remarks. The first and third authors were partially supported by the Shota Rustaveli National Science Foundation Grant (Contract Numbers: D/13-23 and 31/47). The second author was partially supported by the National Science Centre (NCN), Poland, grant no. 2011/01/B/ST1/06243. References [1] M.D. Acosta, A. Kamińska, M. Mastyło, The Daugavet property and weak neighborhoods in Banach lattices, J. Convex Anal. 19 (3) (2012) 875–912. [2] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988. [3] D.E. Edmunds, V. Kokilashvili, A. Meskhi, Bounded and Compact Integral Operators, in: Mathematics and its Applications, vol. 543, Kluwer Academic Publishers, Dordrecht, 2002. [4] J. Garcia-Cuerva, A.E. Gatto, Boundedness properties of fractional integral operators associated to non-doubling measures, Studia Math. 162 (2004) 245–261. [5] A.E. Gatto, On fractional calculus associated to doubling and non-doubling measures, in: J.M. Ash, R.L. Jones (Eds.), Harmonic Analysis: Calderón–Zygmund and Beyond, in: Contemporary Math., vol. 411, Amer. Math. Soc., Providence, Chicago, 2002, pp. 15–37. 2006. [6] J. Gustavsson, A function parameter in connection with interpolation of Banach spaces, Math. Scand. 42 (2) (1978) 289–305. [7] A. Kairema, Sharp weighted bounds for fractional integral operators in a space of homogeneous type, Math. Scand. 114 (2) (2014) 226–253. [8] L.V. Kantorovich, G.P. Akilov, Functional Analysis, second ed., Pergamon Press, Oxford-Elmsford, NY, 1982. [9] V. Kokilashvili, Weighted estimates for classical integral operators, in: Nonlinear Analysis, Function Spaces and Applications, Vol. 4 (Roudnice nad Labem, 1990), in: Teubner-Texte Math., vol. 119, Teubner, Leipzig, 1990, pp. 86–103. [10] V. Kokilashvili, A. Meskhi, Fractional integrals on measure spaces, Fract. Calc. Appl. Anal. 4 (2001) 1–24. [11] S.G. Krein, Ju.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, in: Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, RI, 1982. [12] E.I. Pustylnik, Interpolation theorems in massifs of Banach spaces, Dokl. Akad. Nauk SSSR 224 (5) (1975) 1024–1027 (in Russian); English transl.: Soviet. Math. Dokl. 16 (1975) 1359–1363.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Fractional integral operators between Banach function lattices Vakhtang Kokilashvili a,b , Mieczysław Mastyło c,d , Alexander Meskhi a,e,∗ a
Department of Mathematical Analysis, A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, University Str., 0186 Tbilisi, Georgia b
International Black Sea University, Agmashenebeli Ave., Tbilisi 0131, Georgia
c
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
d
Institute of Mathematics, Polish Academy of Science (Poznań branch), Umultowska 87, 61-614 Poznań, Poland
e
Department of Mathematics, Faculty of Informatics and Control Systems, Georgian Technical University, 77, Kostava St., Tbilisi, Georgia
article
abstract
info
Article history: Received 21 October 2014 Accepted 22 January 2015 Communicated by Enzo Mitidieri
We study the generalized fractional integral transforms associated to a measure on a quasimetric space. We give a characterization of those measures for which these operators are bounded between Lp -spaces defined on nonhomogeneous spaces. The key in the proof of one of the main theorems is the boundedness of the modified sublinear Hardy–Littlewood maximal operator in the classical Lebesgue space with general measure. We also provide necessary and sufficient conditions for some classes of integral operators to be bounded from Lorentz to Marcinkiewicz spaces. © 2015 Elsevier Ltd. All rights reserved.
MSC: primary 42B20 26A33 secondary 47B38 47G10 Keywords: Fractional integral operators Non-doubling measure Lorentz spaces Marcinkiewicz spaces
1. Introduction The aim of this paper is to study the boundedness of generalized fractional integral operators between Banach function lattices. The paper is motivated by applications of classical fractional integrals in the theory of Sobolev embeddings. We recall that in the Euclidean space Rn , the fractional integral operator Iα of order α , 0 < α < n, is defined by Iα f (x) =
Rn
f (y) dy, |x − y|n−α
f ∈ L1loc (Rn ).
By the well-known Hardy–Littlewood–Sobolev theorem, Iα is a bounded operator from Lp (Rn ) to Lq (Rn ) if and only if p > 1 and 1/p − 1/q = α/n. In the last two decades potential operators over quasi-metric measure spaces (X , d, µ) are studied intensively; we refer e.g. to the book [3]. We recall that by a quasi-metric we mean a mapping that satisfies the following conditions d(x, y) = 0
∗ Corresponding author at: Department of Mathematical Analysis, A. Razmadze Mathematical Institute, Javakhishvili Tbilisi State University, University Str., 0186 Tbilisi, Georgia. E-mail addresses: [email protected] (V. Kokilashvili), [email protected] (M. Mastyło), [email protected] (A. Meskhi). http://dx.doi.org/10.1016/j.na.2015.01.014 0362-546X/© 2015 Elsevier Ltd. All rights reserved.
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
149
if and only if x = y; d(x, y) ≤ Ct (d(x, z ) + d(z , y)) for some Ct ≥ 1 and all x, y, z ∈ X ; there exists a constant Cs such that d(x, y) ≤ Cs d(y, x) for all x, y ∈ X . As usual, for a ball B = B(x, r ) := {y ∈ X ; d(x, y) < r } with x ∈ X , r > 0 and a > 0, the notation aB := B(x, ar ) stands for the concentric dilation of B. We assume that all balls are measurable with positive finite measure. This implies that our measures are σ -finite and that the set {x ∈ X ; µ({x}) > 0} of atoms is at most countable. We say that the measure µ is lower α -Ahlfors regular (resp., upper β -Ahlfors regular) with α > 0 and β > 0, if there exists C > 0 such that
µ(B(x, r )) ≥ Cr α (resp.,
µ(B(x, r )) ≤ Cr β ), for all x ∈ X and 0 < r < diam(X ). When α = β , the measure µ is called α -Ahlfors regular. We use standard definitions and notation from the theory of Banach lattices (see, e.g., [8]). Let (Ω , Σ , µ) be a complete σ -finite measure space and let L0 (µ) = L0 (Ω , µ) denote the space of all equivalence classes of µ-a.e. finite real-valued measurable functions on Ω with the topology of convergence in measure on µ-finite sets and let L0 (µ, Ω ) denote the space 0 of extended real valued measurable functions on Ω . A linear subspace E of L (µ) is said to be an ideal if f ∈ E and |g | ≤ |f | in L0 (µ), then g ∈ E. By E+ we denote the collection of nonnegative functions in E. Let E and F be two vector subspaces of L0 (S , Σ1 , µ) and L0 (T , Σ2 , ν), respectively. A linear operator T : E → F is said to be an integral operator if there exists a Σ1 × Σ2 measurable function K (called the kernel of the operator) such that we have Tf (x) =
K (x, y)f (y) dν(y) T
for every f ∈ E and for µ-almost all x ∈ S . A Banach (function) lattice (E , ∥·∥E ) on (Ω , Σ , µ) is an ideal of L0 (µ), which is complete with respect to a norm ∥·∥E . We also assume that the support of the space E is Ω (supp(E ) = Ω ), that is, there is an element u ∈ E with u > 0 µ-a.e. on Ω . It is said that E has the Fatou property (or E is maximal) if for any f ∈ L0 , fn ∈ E+ such that fn ↑ f a.e. and sup ∥fn ∥E < ∞, we have that f ∈ E and ∥fn ∥E ↑ ∥f ∥E . The Köthe dual space E ′ of a Banach lattice E on (Ω , Σ , µ) is defined as the space of all f ∈ L0 (µ) such that Ω |fg | dµ < ∞ for every g ∈ E. It is a Banach lattice on (Ω , Σ , µ) when equipped with the norm
∥f ∥E ′ = sup ∥g ∥E ≤1
Ω
|fg | dµ,
f ∈ E′.
Let us remark that E ′ of E is a maximal Banach lattice on (Ω , µ) as well as a number of classical spaces such as Lebesgue spaces Lp , 1 ≤ p ≤ ∞, Orlicz spaces or more general Musielak–Orlicz spaces. Notice that E is a maximal Banach lattice if and only if E = E ′′ := (E ′ )′ with equality of norms (see, e.g., [8, p. 30]). Throughout the paper by c and C we denote various absolute positive constants which may have different values even on the same line. 2. Generalized fractional integral operators Let (X , d, µ) be a quasi-metric space. We say that an integral operator TK is an operator of potential type if it is of the form TK f (x) =
K (x, y)f (y) dµ(y),
x ∈ X,
X
where the kernel K : X × X → [0, ∞] is a function which satisfies the following monotonicity conditions: For every k2 > 1 there exists k1 > 1 such that K (x, y) ≤ k1 K (x′ , y) whenever d(x′ , y) ≤ k2 d(x, y), and K (x, y) ≤ k1 K (x, y′ ) whenever d(x, y′ ) ≤ k2 d(x, y). Important examples of potential type operators are the fractional integrals. We mention that the fractional operators Iα f (x) =
f (y)
X
d(x, y)n−α
dµ(y),
x ∈ X,
where 0 < α < n, has been studied intensively for both the doubling and non-doubling setting. Another type of fractional integral is given by Tf (x) :=
f (y)
X
µ(B(x, d(x, y)))1−γ
dµ(y),
x ∈ X,
150
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
where γ ∈ (0, 1). We refer e.g., to [3] (Chapter 6) and also to the papers [5,7] and references therein where this type operator is studied. Recall that the ϕ : [0, a) → [0, ∞) defined on [0, a) with 0 < a ≤ ∞ is called quasi-concave if it is non-decreasing on [0, a) with ϕ(0) = 0 and t → ϕ(t )/t is non-increasing on [0, a). To simplify presentation we denote the set of quasi-concave . For any quasi-concave function ϕ ∈ P , we functions on [0, ∞) by P . In the case when ϕ ∈ P is concave we write ϕ ∈ P define ϕ∗ (t ) := t /ϕ(t ) for all t > 0. For any ϕ ∈ P , we define ϕ by
ϕ(t ) := sup s>0
ϕ(st ) , ϕ(s)
t > 0.
Since ϕ is sub-multiplicative, it follows that there exist the limits (see, e.g., [11, pp. 52–54]) ln ϕ(t )
αϕ := lim
ln t
t →0+
,
βϕ := lim
ln ϕ(t )
t →∞
ln t
,
which satisfy 0 ≤ αϕ ≤ βϕ ≤ 1. Throughout the paper we consider functions ϕ ∈ P which satisfy 0 < αϕ ≤ βϕ < 1. For general examples of such functions we refer to [6]. For the sake of completeness we include an example; if 0 < θ < 1, 0 < β, γ satisfy θ + γ β < 1, then ϕ given by the formula
ϕ(t ) = t θ lnβ (1 + t γ ),
t > 0,
is a quasi-concave function with αϕ = θ and βϕ = θ + γ β . Let (X , d, µ) be a quasi-metric space and let ϕ ∈ P . In the paper we consider the kernel Kϕ : X × X → [0, ∞) defined by Kϕ (x, y) =
1
ϕ (d(x, y)) 0,∗
,
if x ̸= y, if x = y.
The integral operator generated by Kϕ is denoted by Tϕ . In the case when ϕ(t ) = t α for some α ∈ (0, 1) and all t ≥ 0 we write Tα instead of Tϕ . We will need the following modified maximal operator on a quasi-metric space (X , d, µ) given by
(x) = sup Mf r >0
1
µ(B(x, N0 r ))
B(x,r )
| f | dµ
where N0 = Ct (1 + 2Cs ) with constants Cs and Ct from the definition of a quasi-metric d. It is well-known (see [3, p. 368]) that for any f ∈ L1 (µ),
(x) > λ}) ≤ 1 µ({x ∈ X ; Mf λ
|f (x)| dµ(x),
λ > 0,
X
is of weak type (1, 1). Since M is bounded in L∞ (µ), Marcinkiewicz’s interpolation theorem yields that M is bounded i.e., M in Lp (µ) for every 1 < p < ∞. We are ready to state and prove a result on the boundedness of Tϕ between corresponding Lp (µ) spaces. Theorem 2.1. Let 1 < p < q < ∞ and ϕ ∈ P be such that 0 < αϕ ≤ βϕ < 1. Then the operator Tϕ is bounded from Lp (µ) to Lq (µ) if and only if there exists C > 0 such that ′
µ(B(x, r )) ≤ C ϕ∗ (r )s ,
x ∈ X , r > 0,
(∗)
where 1/s = 1/p − 1/q and 1/s + 1/s = 1. ′
Proof. Assume that Tϕ is bounded from Lp (µ) → Lq (µ). Then there exists C > 0 such that
∥Tϕ f ∥Lq (µ) ≤ C ∥f ∥Lp (µ) ,
f ∈ Lp (µ).
(1)
For f = χB(a,r ) , where a ∈ X , r > 0, we have Tϕ χB(a,r ) (x) =
1 B(a,r )
ϕ∗ (d(x, y))
dµ(y),
x ∈ X.
Observe that for all x, y ∈ B(a, r ) d(x, y) ≤ Ct (d(x, a) + d(a, y)) ≤ Ct (Cs + 1)r. Denoting C0 := Ct (Cs + 1) and combining this estimate with ϕ∗ (C0 r )/C0 r ≤ ϕ∗ (r )/r for all r > 0 we have that 1 µ(B(a, r )) C0
ϕ∗ (r )
≤
µ(B(a, r )) ≤ Tϕ χB(a,r ) , ϕ∗ (C0 r )
x ∈ B(a, r ).
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
151
This shows that 1 µ(B(a, r ))
χB(a,r ) ≤ χB(a,r ) Tϕ χB(a,r ) .
ϕ∗ (r )
C0
Applying inequality (1), we conclude that 1 µ(B(a, r ))
∥χB(a,r ) ∥Lq (µ) ≤ C ∥χB(a,r ) ∥Lp (µ)
ϕ∗ (r )
C0 and whence
1 µ(B(a, r )) 1+1/q
ϕ∗ (r )
C0
≤ C µ(B(a, r ))1/p
gives the required estimate. For sufficiency we claim that the estimate (∗) implies that Tϕ is bounded operator from Lp (µ) to Lq (µ). To see this we define a bounded function ω: X → [0, ∞),
ω(x) := sup r >0
µ(B(x, r )) , ϕ∗ (r )s′
x ∈ X,
which was introduced in [9] in the case of Euclidean spaces and ϕ(t ) = t α , α ∈ (0, 1). Further, for x ∈ X , and f ∈ D(Tϕ ), we have T ϕ f ( x) =
f (y)
B(x,r )
ϕ∗ (d(x, y))
dµ +
f (y)
X \B(x,r )
ϕ∗ (d(x, y))
dµ = I1 (x) + I2 (x).
Set Dk := B(x, 2−k r ) \ B(x, 2−k−1 r ) for each integer k ≥ 0. Then we have
|I1 (x)| ≤
Dk
k=0
≤
|f (y)| dµ(y) ϕ∗ (d(x, y))
∞
µ(B(x, N0 2−k r )) ϕ∗ (2−k r ) µ(B(x, N0 2−k r ))
∞
1
k=0
≤C ≤C
∞
1
k=0
ϕ∗ (2−k r )
∞
B(x,2−k r )
|f (y)| dµ(y)
′
(x) ω(x) ϕ∗ (N0 2−k r )s Mf ′
(x) ω(x). ϕ∗ (2−k r )s −1 Mf
k=0
Here we used the estimate ϕ∗ (N0 s) ≤ C ′ ϕ∗ (s) for some C ′ > 0 and for all s > 0. Now observe that our hypothesis implies 0 < βϕ∗ ≤ αϕ∗ < 1 and so
ϕ∗ (t ) ≤ C2 t αϕ∗ +ε ,
t ∈ (0, 1]
(2)
for some C2 > 0 and 0 < ε < 1 − αϕ∗ . This estimate together with the inequality ϕ∗ (uv) ≤ ϕ∗ (u)ϕ∗ (v), u, v > 0, implies that for all r > 0, ∞
′
ϕ∗ (2−k r )s −1 ≤
k=0
∞ ′ [ϕ∗ (r )ϕ∗ (2−k )]s −1 .
(3)
k=0
Since s′ > 1, the estimates (2) and (3) imply ∞
′
∞
′
ϕ∗ (2−k r )s −1 ≤ (C2 ϕ∗ (r ))s −1
k=0
where C3 :=
2−k(αϕ∗ +ε)(s −1) ≤ C3 ϕ∗ (r )s −1 , ′
′
k=0
∞
k=0
−k(αϕ∗ +ε)
2
s ′ −1
|I1 (x)| ≤ C3 ϕ∗ (r )
< ∞. Combining this fact with the above estimate of I1 we find that
(x) ω(x), Mf
x ∈ X , r > 0.
To estimate I2 (x), we put Ek := B(x, 2 all f ∈ Lp (µ),
k+1
|I2 (x)| ≤ ∥f ∥Lp (µ)
r ) \ B(x, 2k r ) for each integer k ≥ 0. Then the Hölder’s inequality gives that for
1 X \B(x,r )
ϕ∗ (d(x, y))p′
dµ
1/p′
.
152
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
Fix f ∈ Lp (µ) with ∥f ∥Lp (µ) ≤ 1. Then, we have by (∗)
|I2 (x)| ≤
∞
dµ
1/p′
Ek
k=0
≤
ϕ∗ (d(x, y))
−p′
∞
ϕ∗ (2 r ) k
−p′
µ(B(x, 2
k+1
r ))
1/p′
k =0
∞ 1/p′ 1/p′ ′ ′ ≤ C ω(x) ϕ∗ (2k r )−p ϕ∗ (2k+1 r )s k=0
≤ C4 ω(x)1/p
′
′
′
ϕ∗ (2k r )s −p
1/p′
k=0 1/p′
= C4 ω(x)
∞
ϕ(2k r ) p′ −s′
1/p′
2k r
k=0
1/p′ ϕ(r ) ϕ(2k r ) p′ −s′ ≤ C4 ω(x) r 2k ϕ(r ) k=0 1/p′ ∞ 1−s′ /p′ ϕ(2k ) p′ −s′ 1/p′ ϕ(r ) ≤ C4 ω(x) . k 1/p′
∞
r
2
k=0
Here we used the fact that s < p . Our hypothesis on ϕ implies that ′
ϕ(t ) ≤ C5 t βϕ +ε ,
′
t ≥1
with some C5 > 0 and 0 < ε < 1 − βϕ . This inequality and the observation 1 − s′ /p′ = s′ /q give that there is a constant C6 > 0 such that for all r > 0, 1/p′
|I2 (x)| ≤ C6 ω(x)
∞ ′ −1/q ϕ∗ (r )s k=0
1
p′ −s′ 1/p′
2(1−βϕ −ε)k
≤ C7 ω(x)1/p ϕ∗ (r )−s /q . ′
′
The obtained estimates for I1 and I2 imply the following pointwise inequality
′ (x)ω(x) + ω(x)1/p′ ϕ∗ (r )−s′ /q . |Tϕ f (x)| ≤ C ϕ∗ (r )s −1 Mf Since x → ω(x) is a bounded function,
′ (x) + ϕ∗ (r )−s′ /q . |Tϕ f (x)| ≤ C ϕ∗ (r )s −1 Mf Since 0 < αϕ∗ ≤ βϕ∗ < 1, ϕ∗ (s) → 0 as s → 0+ and ϕ∗ (s) → ∞ as s → ∞. This implies that the image of ϕ∗ is [0, ∞).
(x)−p/s′ , we obtain the pointwise inequality true for all f ∈ Lp (µ) with ∥f ∥Lp (µ) = 1, Thus taking r > 0 such that ϕ∗ (r ) = Mf |Tϕ f (x)| ≤ C
1−p(1−1/s′ )
(x) Mf
(x) p/q = 2C Mf (x) p/q , + Mf
x ∈ X.
Lp (µ) → Lp (µ) is bounded, we conclude that for all f ∈ Lp (µ) with ∥f ∥Lp (µ) = 1, Combining with the fact that M: ∥Tϕ f ∥
q Lq (µ)
p
≤ 2C X
(x)p dµ = 2C ∥Mf ∥ p ≤ 2C0 ∥f ∥Lp (µ) ≤ C0 . Mf L (µ)
This shows that Tϕ is bounded from Lp (µ) to Lq (µ), and this completes the proof.
Corollary 2.2 ([10]). Let 1 < p < q < ∞ and 0 < α < 1. Then Tα is a bounded operator from Lp (µ) to Lq (µ) if and only if the pq(1−α) measure µ is upper s-Ahlfors regular, where s = pq−q+p . We note that this result in Euclidean spaces for the first time appeared in [9]. As an application we obtain the following version of the Hardy–Littlewood–Sobolev theorem (see [4,10]) for the fractional integral Iα associated to the nonhomogeneous space (X , d, µ) given by Iα f (x) =
X
where 0 < α < n.
f (y) d(x, y)n−α
dµ(y),
x ∈ X,
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
153
Corollary 2.3. For 1 < p < n/α and 1/q = 1/p − α/n, Iα is a bounded operator from Lp (µ) into Lq (µ) provided that the measure µ is upper n-Ahlfors regular. Before the proof of the next result we need some definition. Let ψ : (0, ∞) → (0, ∞) be nondecreasing function. The (∗) (∗) Marcinkiewicz class Mψ = Mψ (µ) on a measure space (Ω , Σ , µ) is defined to be the set of all measurable functions
f ∈ L0 (µ) such that
∥f ∥ψ = sup ψ(t )f ∗ (t ) < ∞, t >0
where f ∗ denotes the decreasing rearrangement of f , i.e., f ∗ (t ) := inf{λ ≥ 0; µf (λ) ≤ t } for all t > 0, and µf (λ) := µ{t ∈ Ω ; |f (t )| > λ}, λ ≥ 0. In particular, for ψ(t ) = t 1/q , 0 < q < ∞, we obtain the classical spaces Lq,∞ (µ). One can readily show that ∥ · ∥ψ is a quasi-norm (i.e.,
∥f + g ∥ψ ≤ C (∥f ∥ψ + ∥g ∥ψ ),
(∗)
f , g ∈ Mψ ,
for some C > 0) if the function ψ satisfies the ∆2 -condition, i.e., there exists K ≥ 1 such that ψ(2t ) ≤ K ψ(t ) for all t > 0. (∗) It can be easily shown that for any f ∈ Mψ we have
∥f ∥ψ = sup λψ({ω ∈ Ω ; |f (ω)| > λ}). λ>0
Theorem 2.4. Let 1 ≤ p < ∞ and let (X , d, µ) be a quasi-metric space. Let ϕ : [0, ∞) → [0, ∞) be such that the function φ given by φ(t ) = t −1/p ϕ(t ) for all t > 0 be decreasing with φ(R+ ) = R+ . Assume that K is a positive kernel on X which satisfies the following conditions:
B(x,r )
K (x, y) dµ(y) ≤ C1 ϕ(r ),
B(y,r )
K (x, y) dµ(x) ≤ C2 ϕ(r )
and
χX \B(x,r ) K (x, ·) p′
L (µ)
≤ C3 φ(r )
for some Cj > 0 (1 ≤ j ≤ 3) and all x ∈ X and r > 0, where 1/p + 1/p′ = 1. Then the integral operator TK is bounded from (∗)
Lp (µ) into Mψ (µ), where ψ(t ) = φ(1t ) for all t > 0. Proof. We can, and do, assume that Cj = 1 for each 1 ≤ j ≤ 3. Fix f ∈ Lp (µ) with ∥f ∥Lp (µ) ≤ 1. For any ball B(x, r ) in X we have
|TK f |(x) ≤
K (x, y)|f (y)| dµ(y) = I1 (x) + I2 (x), X
where I1 (x) :=
B(x,r )
K (x, y)|f (y)| dµ(y)
and I2 (x) =
X \B(x,r )
K (x, y)|f (y)| dµ(y).
We present the proof for p > 1. Then by Hölder’s inequality, we obtain
|I2 (x)| ≤ ∥f ∥
Lp (µ)
p′
X \B(x,r )
K (x, y) dµ(y)
1/p′
1
≤
ψ(r )
.
For λ > 0, we take r = ψ −1 (2/λ). Then the above inequality yields |I2 (x)| ≤ λ/2 and whence
{x ∈ X ; |TK f |(x) > λ} ⊂ {x ∈ X ; |I1 (x)| > λ/2} ∪ {x ∈ X ; |I2 (x)| > λ/2} = {x ∈ X ; |I1 (x)| > λ/2}. Using Hölder’s inequality we get
|I1 (x)| ≤
B(x,r )
≤ ϕ(r )
1/p′
K (x, y)|f (y)|p dµ(y)
1/p
1/p′
B(x,r )
1/p
B(x,r )
K (x, y)|f (y)|p dµ(y)
K (x, y) dµ(y)
.
154
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
Combining with Tchebyshev’s inequality and Tonelli’s theorem we find that
µ({x ∈ X ; |TK f (x)| > λ}) ≤ µ({x ∈ X ; |I1 (x)| > λ/2}) ′ ≤ ϕ(r )p/p (2/λ)p K (x, y)|f (y)|p dµ(y) dµ(x) X B(x,r ) ′ ′ = ϕ(r )p/p (2/λ)p K (x, y) dµ(x)|f (y)|p dµ(y) ≤ ϕ(r )p/p ϕ(r )(2/λ)p B(y,r )
X
= (2ϕ(r )/λ)p = ψ −1 (2/λ). (∗)
This shows that TK f ∈ Mψ with
∥TK f ∥ψ = sup λψ(µ{x ∈ X ; |TK f (x)| > λ}) ≤ 2. λ>0
To finish we note that the proof for p = 1 is a simple modification of the proof for p > 1.
The following lemma provides examples of generalized fractional operators which satisfies the integral conditions of the above theorem. Lemma 2.5. Let (X , d, µ) be a quasi-metric space and ϕ ∈ P . Suppose that µ(B(x, r )) ≤ Cr for some C > 0 and all 0 < r < diam(X ), x ∈ X . Then we have: (i) If αϕ > 0, then there exists C1 > 0 such that
1 B(x,r )
dµ(y) ≤ C1 ϕ(r ).
ϕ∗ (d(x, y))
(ii) If ϕ is sub-multiplicative (i.e., ϕ(st ) ≤ C ϕ(s)ϕ(t ) for some C > 0 and all s, t > 0) such that 1 ≤ p < 1/βϕ then there exists C2 > 0 for which
1 X \B(x,r )
ϕ∗ (d(x, y))p′
1/p′ ϕ(r ) ≤ C2 1/p . dµ(y) r
Proof. (i) Since α := αϕ > 0, for every ε > 0 there exists C (ε) such that ϕ(s) ≤ C (ε)sα−ε for all s ∈ (0, 1). Fix x ∈ X and r > 0 and put Dj = {y ∈ X ; 2−j−1 r ≤ d(x, y) < 2−j r }; then our hypothesis µ(B(x, r )) ≤ C r, αϕ > 0 implies
1 B(x,r )
ϕ∗ (d(x, y))
dµ(y) =
∞ Dj
j =0
≤
1
ϕ∗ (d(x, y))
∞
1
j =0
ϕ∗ (2−j−1 r )
≤ 2C
∞
dµ(y)
µ(B(x, 2−j r ))
ϕ(2−j−1 )ϕ(r ) ≤ C1 ϕ(r ).
j=0
(ii) Since ϕ is sub-multiplicative, ϕ∗ is a super-multiplicative and βϕ < 1/p,
1 X \B(x,r )
ϕ∗ (d(x, y))p′
dµ(y) =
∞ j =0
≤
2j r ≤d(x,y)<2j+1 r
∞ j =0
p′
1
ϕ∗ (2j r ) ′
X \B(x,r )
ϕ∗ (d(x, y))p′
1/p′
≤ C2
ϕ(r ) r 1/p
∞ ϕ(2j r ) p′ j =0
and so this gives the required estimate dµ(y)
dµ(y)
2j r
p′ ′ ′ ′ 2j(1−p ) ϕ(2j ) ≤ Cϕ ϕ(r )p r 1−p
j =0
1
ϕ∗ (d(x, y))
p′
µ(B(x, 2j+1 r )) ≤ C
∞ ′
≤ C ϕ(r )p r 1−p
1
.
2j r
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
155
3. Fractional integral operators between symmetric spaces An important class of Banach lattices on a measure space (Ω , Σ , µ) are symmetric quasi-Banach spaces. We recall that a Banach lattice E on (Ω , µ) is said to be a symmetric Banach space if g ∈ E and ∥f ∥E = ∥g ∥E whenever f ∈ E, g ∈ L0 (µ) and µf (λ) = µg (λ) for all λ > 0, where µf (λ) = µ({ω ∈ Ω ; |f (ω)| > λ}). Clearly for a symmetric Banach space E, ∥χA ∥E depends only on µ(A) for any measurable set A of finite measure. Thus for every t ∈ {µ(A); A ∈ Σ }, we may define the function ϕE by ϕE (t ) = ∥χA ∥E where A is any measurable set with µ(A) = t. This function is called the fundamental function of E. For the theory of symmetric spaces we refer to [2,11]. We recall some important properties of symmetric spaces which we will need use later. Let E be a symmetric space on nonatomic measure space (Ω , Σ , µ) with 0 < µ(Ω ), and let φ := ϕE be the fundamental function of E. It is well known that φ is a quasi-concave on [0, a), where a = µ(Ω ). Taking φ(t ) = infs∈(0,a) (1 + t /s)φ(s) for all t ∈ [0, a), we obtain a concave function φ on [0, a) such that φ(t ) ≤ φ(t ) ≤ 2φ(t ) for all t ∈ [0, a). In what follows φ is called a concave majorant of φ . For any quasi-concave function ϕ : [0, a) → [0, ∞) the Marcinkiewicz space Mϕ = Mϕ (µ) is defined by the norm
∥f ∥Mϕ = sup
1
0
t
ϕ(t )
f ∗ (s) ds,
0
where f ∗ denotes the decreasing rearrangement of f given by f ∗ (t ) = inf{λ > 0; µ({ω ∈ Ω ; |f (ω)| > λ}) ≤ t },
t > 0.
We note that the largest symmetric space on (Ω , Σ , µ) with the same fundamental ϕ is the Marcinkiewicz space Mϕ∗ . We also note that the fundamental function of a symmetric space E = (E , ∥ · ∥E ) is not necessarily concave, however there exists an equivalent norm under which E is a symmetric space with concave fundamental function. For any symmetric space E with concave fundamental function ϕ = ϕE there is also the smallest symmetric space with the same fundamental function; it is the Lorentz space Λϕ = Λϕ (µ) given by the norm a
∥f ∥Λϕ =
f ∗ (s) dϕ(s). 0
In what follows we will use the following easily verified fact that in the case when µ is a nonatomic measure we have
∥ f ∥ Mϕ =
1
sup
A∈Σ , 0<µ(A)<∞
ϕ(µ(A))
|f | dµ,
f ∈ Mϕ .
A
For the theory of symmetric spaces we refer to [2,11]. To show applications of Lemma 2.5 to fractional integral operators, we define a special class of kernels. Let (X , d, µ) be a metric space. A kernel K ∈ L0 (µ × µ)+ is said to be a Borel kernel whenever there is a constant C ≥ 1 such that for every Borel set A in X with 0 < µ(A) < ∞ and x ∈ X there is a ball B(x, r ) such that µ(B(x, r )) = µ(A) and
K (x, y) dµ(y) ≤ C
A
B(x,r )
K (x, y) dµ(y).
The following proposition gives examples of Borel kernels. Proposition 3.1. Let (X , d, µ) be a nonatomic quasi-metric space. Let φ : [0, ∞) × [0, ∞) → [0, ∞) be a non-increasing function in each variable. Then K : X × X → [0, ∞) defined by the following formula is a Borel kernel: K (x, y) =
φ d(x, y), µ(B(x, d(x, y))) , 0,
if x ̸= y if x = y.
Proof. Let A be an arbitrary Borel set with a finite positive measure. Our hypothesis that µ is a nonatomic measure yields that for any x ∈ A there exists a ball B(x, r ) in X such that µ(B(x, r )) = µ( A). Put B := B(x, r ) ∩ A; then µ(A \ B) = µ(B(x, r ) \ B). Since y ∈ A \ B implies d(x, y) ≥ r, φ d(x, y), µ(B(x, d(x, y))) ≤ φ r , µ(B(x, r )) and so we have
B(x,r )
K (x, y) dµ(y) −
K (x, y) dµ(y) =
A
B(x,r )\B
K (x, y) dµ(y) −
K (x, y) dµ(y) A\B
≥ φ(r , µ(B(x, r ))) (µ(B(x, r ) \ B) − µ(A \ B)) = 0 and this completes the proof.
156
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
Lemma 3.2. Let (S , µ) and (T , ν) be measure spaces, and let TK be an integral operator generated by the kernel K ∈ L0 (µ×ν)+ . Then TK is bounded from a Banach lattice E on (T , ν) to the Marcinkiewicz space Mψ (µ) on (S , µ) if and only if C :=
1
sup
0<µ(A)<∞
ψ(µ(A))
K (s, ·) dµ(s) ′ < ∞. E
A
Proof. Combining Tonelli’s Theorem with the fact that TK is positive we obtain, sup ∥TK f ∥Mψ (µ) = sup
sup
= sup
sup
∥f ∥E ≤1 µ(A)>0
∥f ∥E ≤1
1
ψ(µ(A))
A
1
K (s, t )|f (t )| dν(t ) dµ(s) T
|f | K (s, ·) dµ(s) dν
ψ(µ(A)) T A 1 = sup K (s, ·) dµ(s) ′ . E µ(A)>0 ψ(µ(A)) A µ(A)>0 ∥f ∥E ≤1
Corollary 3.3. Let (X , d, µ) be a nonatomic quasi-metric space. If TK is an integral operator generated by a positive Borel kernel K such that
B(x,r )
K (x, y) dµ(y) ≤ φ(µ(B(x, r )))
(4)
for all balls B(x, r ) in X and some concave function φ : [0, µ(X )) → [0, ∞), then TK is a bounded operator from the Lorentz space Λφ (µ) into L∞ (µ). Proof. Since K is a Borel kernel, for every Borel set A with a finite positive measure and every x ∈ X , there exists a ball B(x, r ) in X such that µ(B(x, r )) = µ(A) and
K (x, y) dµ(y) ≤ A
B(x,r )
K (x, y) dµ(y).
Combining the latter with (4) we have TK χA (x) ≤
B(x,r )
K (x, y) dµ(y) ≤ φ(µ(B(x, r ))) = φ(µ(A)).
It is well-known that this inequality (see [11, Corollary 1, p. 112]) for arbitrary characteristic functions χA implies that TK is bounded from Λφ (µ) into L∞ (µ). As an immediate consequence of the above corollary we have the following result. Theorem 3.4. Let (X , d, µ) be a nonatomic quasi-metric space, such that, for any t > 0, h(t ) := inf{µ(B(x, t )); x ∈ X } > 0. Assume that K : X × X → [0, ∞) is a Borel kernel for which
θ (t ) := sup x∈X
B(x,t )
K (x, y) dµ(y) < ∞,
t > 0.
If t → θ ◦ h−1 (t )/t is almost non-increasing, then the integral operator TK is bounded from the Lorentz space Λφ (µ) into L∞ (µ), where φ is a concave majorant of θ ◦ h−1 . In what follows we will need a general interpolation theorem for operators from Lorentz into Marcinkiewicz spaces from which it follows (see, e.g., [12]): If ϕ0 , ϕ1 and ψ1 , ψ1 be concave functions on (0, a) with 0 < a ≤ ∞ such that
ψ (t ) 1∗ , t ∈ (0, a) ψ0∗ (t ) . Then every bounded operator T from Λϕj into Mψj for j = 0, 1 is also bounded from ΛΦ into MΨ . for some θ ∈ P Φ (t ) = ϕ0 (t )θ
ϕ (t ) 1 , ϕ0 (t )
Ψ∗ (t ) = ψ0∗ (t )θ
Theorem 3.5. Let (X , d, µ) be a nonatomic quasi-metric space. Suppose TK is an integral operator generated by a positive and symmetric Borel kernel K such that the condition (4) holds for some concave function φ : [0, µ(X )) → [0, ∞), and all balls B(x, r ) in X . Suppose further that ϕ and ψ are concave functions such that
ϕ(t ) ≥ C φ(t ), ψ∗ (t )
t ∈ (0, µ(X ))
for some C > 0. Then the integral operator TK is bounded from the Lorentz space Λϕ (µ) into the Marcinkiewicz space Mψ (µ). Proof. From Lemma 3.2 and Corollary 3.3, we have that TK : Λφ (µ) → L∞ (µ) and TK : L1 (µ) → Mφ . Since Mψ0 (µ) = L∞ (µ) and Λϕ1 (µ) = L1 (µ) with equality of norms whenever ψ0 (t ) = ϕ1 (t ) = t for every t ∈ (0, µ(X )), the above mentioned
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
157
interpolation theorem implies that TK : ΛΦ (µ) → MΨ (µ) is bounded for any quasi-concave function ρ , where
t Ψ∗ ( t ) = ρ , φ(t )
t , Φ (t ) = φ(t ) ρ φ(t )
t ∈ (0, µ(X ))
and ρ is a concave majorant of ρ . Now we define ρ ∈ P by
ρ(t ) :=
sup
0
min 1,
t φ(s)
s
s
ψ(s)
,
t ≥ 0.
Our hypothesis on ϕ and ψ implies that Ψ (t ) ≤ c ψ(t ) and ϕ(t ) ≥ c Φ (t ) for all t ∈ (0, µ(X )) and some c > 0. Hence
Λϕ (µ) ↩→ ΛΦ (µ),
MΨ (µ) ↩→ Mψ (µ)
with continuous inclusions and so the desired statement follows.
As an application we obtain the following corollary. Corollary 3.6. Let (X , d, µ) be a nonatomic quasi-metric space, such that the function h: (0, ∞) → (0, µ(X )) given by h(t ) = inf{µ(B(x, t )); x ∈ X } > 0 for all t > 0 is continuous. If TK is an integral operator generated by a positive and symmetric Borel kernel K such that C1 φ(µ(B(x, r ))) ≤
B(x,r )
K (x, y) dµ(y) ≤ C2 φ(µ(B(x, r )))
for some concave function φ : [0, µ(X )) → [0, ∞), some constants C1 , C2 > 0 and all balls B(x, r ) in X , then the integral operator TK is bounded from the Lorentz space Λϕ (µ) into the Marcinkiewicz space Mψ (µ) if and only if there exists C > 0 such that
ϕ(t ) ≥ C φ(t ), ψ∗ (t )
t ∈ (0, µ(X )).
Proof. It follows from Theorem 3.5 that we need to show only necessity; suppose that TK is bounded from Λϕ (µ) into Mψ (µ). Since Λϕ (µ)′ = Mϕ (µ) (see [1]), from Lemma 3.2 we have sup
0<µ(A)<∞
1
ψ(µ(A))
K (·, y) dµ(y) A
Mϕ
< ∞.
Combining the latter with the condition C1 φ(µ(B(x, r ))) ≤ such that for x, y ∈ X and r > 0 we have 1
B(x,r )
K (x, y) dµ(y) yields that there exists a constant C > 0
1
φ(µ(B(x, r ))) dµ(x) ψ(µ(B(x, r ))) ϕ(µ(B(y, r ))) B(y,r ) 1 µ(B(y, r )) ≥ φ(h(r )) ψ(µ(B(x, r ))) ϕ(µ(B(y, r ))) 1 h(r ) φ(h(r )). ≥ ψ(µ(B(x, r ))) ϕ(h(r ))
C ≥
Consequently, there exists C > 0 such that
ϕ(h(r )) ≥ C φ(h(r )), ψ∗ (h(r ))
r >0
and so the required estimate follows by continuity of h.
We conclude with the following result. Corollary 3.7. Let (X , d, µ) be a nonatomic quasi-metric space such that µ is 1-Ahlfors regular measure. Let φ : [0, ∞) → [0, ∞) be a concave function such that αϕ > 0. Suppose that ϕ and ψ are concave functions on [0, µ(X )). Then the integral operator Tφ is bounded from the Lorentz space Λϕ (µ) into the Marcinkiewicz space Mψ (µ) if and only if there exists C > 0 such that
ϕ(t ) ≥ C φ(t ), ψ∗ (t )
t ∈ (0, µ(X )).
Proof. For any ball B(x, r ) in X we have the obvious estimate C1 φ(r ) ≤
1 B(x,r )
φ∗ (d(x, y))
dµ(y).
158
V. Kokilashvili et al. / Nonlinear Analysis 117 (2015) 148–158
Since µ is 1-Ahlfors regular, it follows from Lemma 2.5 that there exist constants C1 , C2 > 0 such that C1 φ(µ(B(x, r ))) ≤
1 B(x,r )
φ∗ (d(x, y))
dµ(y) ≤ C2 φ(µ(B(x, r ))).
Now, applying Corollary 3.6 for the symmetric Borel kernel Kφ yields the required result.
Acknowledgements We thank the referees for careful reading of the paper and useful remarks. The first and third authors were partially supported by the Shota Rustaveli National Science Foundation Grant (Contract Numbers: D/13-23 and 31/47). The second author was partially supported by the National Science Centre (NCN), Poland, grant no. 2011/01/B/ST1/06243. References [1] M.D. Acosta, A. Kamińska, M. Mastyło, The Daugavet property and weak neighborhoods in Banach lattices, J. Convex Anal. 19 (3) (2012) 875–912. [2] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988. [3] D.E. Edmunds, V. Kokilashvili, A. Meskhi, Bounded and Compact Integral Operators, in: Mathematics and its Applications, vol. 543, Kluwer Academic Publishers, Dordrecht, 2002. [4] J. Garcia-Cuerva, A.E. Gatto, Boundedness properties of fractional integral operators associated to non-doubling measures, Studia Math. 162 (2004) 245–261. [5] A.E. Gatto, On fractional calculus associated to doubling and non-doubling measures, in: J.M. Ash, R.L. Jones (Eds.), Harmonic Analysis: Calderón–Zygmund and Beyond, in: Contemporary Math., vol. 411, Amer. Math. Soc., Providence, Chicago, 2002, pp. 15–37. 2006. [6] J. Gustavsson, A function parameter in connection with interpolation of Banach spaces, Math. Scand. 42 (2) (1978) 289–305. [7] A. Kairema, Sharp weighted bounds for fractional integral operators in a space of homogeneous type, Math. Scand. 114 (2) (2014) 226–253. [8] L.V. Kantorovich, G.P. Akilov, Functional Analysis, second ed., Pergamon Press, Oxford-Elmsford, NY, 1982. [9] V. Kokilashvili, Weighted estimates for classical integral operators, in: Nonlinear Analysis, Function Spaces and Applications, Vol. 4 (Roudnice nad Labem, 1990), in: Teubner-Texte Math., vol. 119, Teubner, Leipzig, 1990, pp. 86–103. [10] V. Kokilashvili, A. Meskhi, Fractional integrals on measure spaces, Fract. Calc. Appl. Anal. 4 (2001) 1–24. [11] S.G. Krein, Ju.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, in: Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, RI, 1982. [12] E.I. Pustylnik, Interpolation theorems in massifs of Banach spaces, Dokl. Akad. Nauk SSSR 224 (5) (1975) 1024–1027 (in Russian); English transl.: Soviet. Math. Dokl. 16 (1975) 1359–1363.