Fractional maximal and integral operators in variable exponent Morrey spaces

In this paper, we study the boundedness of the fractional maximal operator and fractional integral operator on the variable exponent Morrey spaces defined over spaces (X,d,μ) of homogeneous type.


Introduction
A quasi-metric d on a set X is a function d : X × X → [0, ∞] satisfying (i) d(x, y) = 0 if and only if x = y; (ii) d(x, y) = d(y, x) for all x, y ∈ X; (iii) There exists a constant A < ∞ such that d(x, y) < A(d(y, z) + d(x, z)) for x, y, z ∈ X.
The space of homogeneous type (X, d, µ) in the sense of Coifman and Weiss [4] is a topological space X defined by d with nonnegative measure µ which is defined on the σ-algebra generated by quasi-metric balls and open sets such that 0 < µ(B(x, r)) < ∞ for all x ∈ X and arbitrary r > 0, and so that there exists a constant b > 0 such that µ(B(x, 2r)) bµ(B(x, r)) < ∞, where B(x, r) is the ball centred at x with radius r.Iterating (1) we obtain that there exists a positive constant C µ such that for all x ∈ X, 0 < r < R and y ∈ B(x, R), µ(B(y, r)) µ(B(x, R)) If b is the smallest constant for the measure µ satisfying (2), we call the number Q = log 2 b the doubling order of µ.Obviously, in the case of R n with the Lebesgue measure, Q = n.In addition, we say (X, d, µ) is a reverse doubling spaces if there exists a constant γ, 0 < γ < 1 such that for every x ∈ X and r > 0 such that B(x, r) ⊂ X, µ(B(x, r/2)) γµ(B(x, r)).
For any spaces of homogeneous type (X, d, µ), Macías and Segovia [8] prove that there exists an equivalent quasi-metric ρ such that all balls with respect to ρ are open in the topology induced by ρ.As in [12], the definition of the reverse doubling condition would need to be changed slightly: there exist constants C and γ, 0 < γ < 1 such that for any ball B(x, r) ⊂ X and any i 1 For more details on this perspective, see [5].From [10], we know that any doubling measure on any metric space which is connected is reverse doubling.
It is valid on any space of homogeneous type that satisfies a non-empty annuli condition, the details to see [14].The similar conclusions on space of homogeneous type can be seen in [15] and [4].Let p : X → [1, ∞) be a measurable function.We suppose that where p − = ess inf x∈X p(x), p + = ess sup x∈X p(x).We let L p(•) (X) be the set of functions f such that It is a Banach space equipped with the norm We denote the conjugate exponent by p (x) = p(x) p(x) − 1 for x ∈ X.The Hölder inequality is valid in the form The variable Morrey spaces over a bounded open set Ω ⊂ R n were introduced in [10].In [11] the authors introduced the following variable Morrey spaces on the space of homogeneous type with diam(X) < ∞.Definition 1 Let 1 < q − q(•) p(•) p + < ∞.We say that a measurable locally integrable function f on X belongs to the class M It is obvious that M p(•) q(•) = L p(•) when p = q; when p, q are constants, the space M p(•) q(•) coincides with the classical Morrey space M p q .The definition and some properties of M p(•) q(•) we can see from [16,1,9] and so on.As in the Euclidean case we know the log-Hölder continuity condition has play an important role, for details see [17].On unbounded spaces, it was used in [3,12] the similar condition to control the continuity of p(•) locally and at infinity.Definition 2 Given a function r(•) : X → [0, ∞), we say that r(•) satisfies the local log-Hölder condition, and denote this by r(•) ∈ LH 0 , if there exists a constant C 0 such that for all x, y ∈ X, d(x, y) < 1/2, The constant C 0 is called the LH 0 constant of r(•).
Definition 3 Given a function r(•) : X → [0, ∞), we say that r(•) satisfies the log-Hölder condition with respect to a base point x 0 ∈ X, and denote this by r( For η, 0 η < 1, the fractional maximal operator is given by When η = 0 this reduces to the Hardy-Littlewood maximal operator, denoted by M .On the classical Morrey spaces over R n , the weighted norm inequalities for M η were proved in [18].For variable spaces over spaces of homogeneous type, norm inequalities was proved in [12] extending the work on the Hardy-Littlewood maximal operator.We state our result on M η on the variable exponent Morrey spaces defined over spaces of homegeneous type as following. Theorem 1 Let (X, d, µ) be a space of homogeneous type.For η, The definition of the fractional integral operator is The results about fractional integrals defined on quasi-metric measure space but without doubling condition discussed in the monograph [24].When µ(X) < ∞ these operators were considered in [19,20] on spaces of homogeneous type.In [12] the boundedness of them were discussed when µ(X) = ∞ on reverse doubling space of homogeneous type.We have the following result on the variable exponent Morrey spaces defined over spaces of homogeneous type.
Theorem 2 Let (X, d, µ) be a reverse doubling space of homogeneous type.For η, 0 Next we introduce two classes of operators which were applied to study the Sobolev and Poincaré inequalities over metric spaces, for details see [21,22].Given 0 < α < Q, define the operators It is immediate that these operators are pointwise equivalent to I η with η = α/Q, if (X, d, µ) be an Ahlfors regular space of homogeneous type.
Based on Theorem 1 and Theorem 2, we obtain the following conclusion at once.
Corollary 1 Let (X, d, µ) be an Ahlfors regular space of homogeneous type.For α, 0 q 2 (x) , and I * * α is also bounded from M to M p 2 (x) 1 Preliminaries Lemma 1 [7] Assume µ(X) < ∞ and let p ∈ LH 0 .Then there exists a positive constant C such that for all balls B ⊂ X the inequality holds.
There seems to require a stronger hypothesis in more general setting of spaces of homogeneous type.
Lemma 3 [11] Let β be a measurable function on X satisfying β(x) < −1 for all X.Suppose that r is a small positive number.Then there exists a positive constant C independent of r and x such that where Lemma 4 [12] If (X, d, µ) is a reverse doubling spaces, then for all x ∈ X, µ{x} = 0.
Comparing Lemmas 3 and 4 we have the following conclusion.
Let β be a measurable function on X satisfying β(x) < −1 for all X.Suppose that r is a small positive number.Then there exists a positive constant C independent of r and x such that where Lemma 6 Assume (X, d, µ) is a reverse doubling spaces.Let β be a measurable function on X satisfying β(x) < −1 for all X.Suppose that r is a small positive number.Then there exists a positive constant C C µ independent of r and x such that where Proof: For i 1 we define R i = {y ∈ X : 2 i−1 r d(x, y) < 2 i r}.
Since the measure µ is both doubling and reverse doubling, we get Lemma 7 [13] Let f be a measurable function on X and let E be a measurable subset of X.Then the following inequalities hold: Lemma 8 [12] Let (X, d, µ) be a space of homogeneous type.For η, 0 Lemma 9 [12] Let (X, d, µ) be a space of homogeneous type.For η, 0

Proof of the Main Results
Proof of Theorem 1 : Let r be a small positive number.Decompose f as follows: By Lemma 8 and doubling condition we have Since B(x, r) ⊂ B(y, 2Ar) ⊂ B(x, A(2A + 1)r), we have By Lemma 2 and doubling condition, we get q 1 (x) (X) (µ(B(x, d(x, t))) −1/p 2 (x) .Since 1 + p 2 (x) > 1, by Lemma 6 we have x) .Then by Lemma 2 and Lemma 9, we get

Application
Let Γ = {t ∈ C : t = t(s), 0 s ∞} be a connected rectifiable curve and and let ν be an arc-length measure on Γ, that is ν(t) = s.We denote Γ(t, r) = Γ ∩ B(t, r), t ∈ Γ, r > 0, where B(t, r) = {z ∈ C : |z − t| < r} is a disc in C with center t and radius r.Γ is called a Carleson curve (regular curve), if there exists a constant C not depending on t and r, such that ν(Γ(t, r)) Cr.
If we equip Γ with the measure ν and the Euclidean metric, the regular curve becomes a space of homogeneous type.
The maximal operator is and the potential type operator is The Cauchy integral is The associated kernel is and it is a Calderón-Zygmund kernel in the case of regular curves.Definition 4 Let 1 < q − q(•) p(•) p + < ∞.We say that a measurable locally integrable function f on Γ belongs to the class M Theorem 1 implies the following statement.
Theorem 2 implies the following statement.
Let Γ be a subset of R n which is an s-set (0 s n) in the sense that there is a Borel measure µ in R n such that (i) supp µ = Γ; (ii) there are positive constants C 1 and C 2 such that for all z ∈ Γ and all r ∈ (0, 1), C 1 r s µ(B(x, r) ∩ Γ) C 2 r s .