A WEAK-(p, q) INEQUALITY FOR FRACTIONAL INTEGRAL OPERATOR ON MORREY SPACES VIA HEDBERG TYPE INEQUALITY

By employing the growth measure, in this paper we prove the weak-(p, q) inequality for fractional integral operator on Morrey spaces via Hedberg type inequality. The proof also needs the weak-(p, p) inequality of the maximal operator in the same spaces.


Introduction
In this paper we prove the weak-(p, q) inequality for fractional integral operator on Morrey spaces, where 1 ≤ p < ∞ and 1 ≤ q < ∞. Fractional integral I α , which is also known as the Riesz potential, is defined by (García- Cuerva and Gatto, 2004). In this definition, X = (X, d, µ) is a metric measure space which is equipped with the metric (distance function) d and Borel measure µ. Particularly, we take the measure to be the growth measure, that is a measure which satisfies the growth condition µ(B(x, r)) ≤ C r n for every ball B(x, r) with center x and radius r. The orde of the measure, n, is less than or equal to the dimension of the metric space. The positive constant C is independent of x and r.
Here and after, C will be used to denote any positive constant which could be vary from line to line. The works using such measure firstly introduced in (Nazarov, et al., 1998) and (Tolsa, 1998). Meanwhile, Verdera (2002) provided some examples on the use of this measure. Other works employing such measure can be found for example in (García-Cuerva and Martell, 2001), (Sihwaningrum, et al., 2012), and (Hakim and Gunawan, 2013). The fractional integral operator in equation (1) is a development of the fractional integral operator given by (Hardy and Littlewood, 1927). Some results on this operator can be found for example in (Adams, 1975), (Nakai, 1994), (Kurata, et al., 2002) and (Eridani, et al., 2014). Other results on the different version of fractional integral operator can be viewed in (Sawano and Shimamura, 2013) and (Sihwaningrum and Sawano, 2013). Meanwhile, the fractional integral operator (equation (1)) has been proved to satisfy the weak-(1, q) inequality (where 1 ≤ q < ∞) in Lebesgue spaces (García-Cuerva and Gatto, 2004), in Morrey spaces (Sihwaningrum, 2016a) and in generalized Morrey spaces (Sihwaningrum, et al., 2015). This fractional integral operator also satisfies the weak-(p, q) inequality (where 1 ≤ p < ∞ and 1 ≤ q < ∞) in Morrey spaces (Sihwaningrum, 2016b). Morrey spaces were first introduced by Morrey (1940); and in this paper, we defined Morrey spaces L p,λ (µ) := L p,λ (X, µ) (for 1 ≤ p < ∞ and 0 ≤ λ < n) to be the set of all functions in L p loc in which The weak inequalities, which measure the size of the distribution function (Duoandikoetxea, 2001), can be proved with or without Hedberg type inequality. The weak-(p, q) inequality in Morrey spaces given by Sihwaningrum (2016b) is proved without Hedberg type inequality. In this paper, we give an alternative proof of this inequality by employing the Hedberg type inequality. The original Hedberg type inequality is available in (Hedberg, 1972). This alternative proof will also need the weak-(p, p) inequality of operator maximal in Morrey spaces. In equation (3), the operator M is defined by Sawano (2005) to be Also, equation (3) is reduced to equation (4) if p = 1. Some properties of M are presented in some papers such as (Terasawa, 2006).

Main Results
The following Hedberg type inequality is an estimate for fractional integral operator I α by the maximal operator M p .
Theorem 2.1. Let 0 < α < n and 1 ≤ p ≤ q < ∞. If there is λ such that 0 ≤ λ < 1 − αp n , then we have inequality Proof. Let r > 0 and x ∈ X. For any function f in L p,λ (µ), it is possible for us to decompose the integral in the definition of the fractional integral operator such that By taking into account the growth condition of µ, we find an estimate for A 1 , that is The growth condition of µ also provides us with

Now, by letting
follows.
As our next result, we have the weak-(p, p) inequality for maximal operator M p . The proof of this weak inequality uses Theorem 2.1 presented in (Sihwaningrum, 2016b).
Theorem 2.2. For 1 ≤ p < ∞, there is a positive constant C such that for every positive γ and every ball B(a, r) in X, the inequality Proof. By using Theorem 2.1 in (Sihwaningrum, 2016b), we find that Therefore, our proof is complete. Now, we state Theorem 2.2 in (Sihwaningrum, 2016b) and present an alternative proof for the theorem, that is by using the Hedberg type inequality (Theorem 2.1).

Concluding Remarks
In this paper we use Hedberg type inequality, which is a common method to prove a weak type inequality. If we take p = 1, our result is reduced to the result in (Sihwaningrum, 2016a). Furthermore, when we take the metric spaces to be the Euclidean spaces, Hakim and Gunawan (2013) found that the fractional integral operator satisfies the weak-(p, q) inequality on the generalized Morrey spaces. Hence, our result is possible to extend into generalized Morrey spaces over metric measure spaces.