Fractional integral operators on generalized Morrey spaces of non-homogeneous type 1. I. Sihwaningrum and H. Gunawan

Transcription

1 Fractional integral operators on generalized Morrey spaces of non-homogeneous type I. Sihwaningrum and H. Gunawan Abstract We prove here the boundedness of the fractional integral operator I α on generalized Morrey spaces in the non-homogeneous setting, which is analogous to Nakai s result [8] in the homogeneous case. Our proof makes use of the covering lemma of Sawano [2]. As a consequence of our result, we obtain an Olsen inequality for a multiplication operator involving I α in the non-homogeneous setting. Keywords: Fractional integral operators, Olsen inequality, nonhomogeneous spaces, generalized Morrey spaces Mathematics Subect Classification: 42B20, 42B35, 47G0, 3B0, 26A33 Introduction Suppose that R d is equipped with a non-negative Radon measure µ satisfying the growth condition, that is, there exists a constant C > 0 and 0 < n d such that µ) C l n for every d-dimensional cube with center c R d and side length l > 0. Here we consider only the cubes with sides parallel to the coordinate axes. It has been widely known that the growth condition replaces the role of the doubling condition in homogeneous spaces. A non-negative measure µ satisfies the doubling condition if there exists a constant C > 0 such that for every cube with side length l we have µ2) C µ), where 2 denotes the cube concentric to with side length 2l. The analysis on non-homogeneous spaces has been developed since the works of Nazarov et al. [9] and Tolsa [7]. Their success in replacing the doubling condition by the growth condition has inspired other people to work on spaces of non-homogeneous type. Our obect of study here is the fractional integral operator I α, which is defined in the non-homogeneous setting by I α fx) := R d fy) dµy), x y n α This paper was presented at AMC 2009, Kuala Lumpur, June 2009

2 where 0 < α < n d. Notice that when n = d and µ is the usual Lebesgue measure, we recover the classical fractional integral operator introduced by Hardy and Littlewood [5, 6] and Sobolev [6]. See [, 2, 8, ] and many other literatures for various results on fractional integral operators in the classical version. In [3], García-Cuerva and Martell studied the boundedness of I α on Lebesgue spaces of non-homogeneous type. In particular, they obtained the following result. Note that throughout the paper, we denote the norm of f in the space X by f : X.) Theorem.. [3] If < p < n α and q = p α n, then there exists a constant C > 0 such that I α f : L q µ) C f : L p µ), that is, I α is bounded from L p µ) to L q µ). Our goal in this paper is to prove the boundedness of I α on generalized Morrey spaces of non-homogeneous type, which is analogous to Nakai s result [8] in the homogeneous setting. As a consequence, we obtain an Olsen inequality for a multiplication operator involving I α in the non-homogeneous version. For related results, see [4, 3, 4]. The inequality was first introduced in the homogeneous setting by Olsen [0]; and was generalized later by Kurata et al. [7]. They used the inequality to study the behaviour of the solution to a Schrödinger equation with a small perturbed potential W. Notice that I α = ) α/2, where is the Laplacian operator. 2 The boundedness of I α and an Olsen inequality For k >, let k denote a cube concentric to with side length k times side length of ; µ) is the set of all cubes with positive µ-measure; and C stands for a constant which may differ from one line to another. For p < and an almost decreasing function φ : 0, ) 0, ), we define the generalized non-homogeneous Morrey spaces M p,φ k, µ) = M p,φ R d, k, µ) to be the set of all f L p loc µ) such that f : M p,φ k, µ) := sup /p fy) dµy)) p <. φµk)) µk) Another version of such spaces can be found in [5]. Recall that φ : 0, ) 0, ) is almost decreasing [almost increasing] if there exists a constant C such that for s < t, we have φs) C φt) [φs) C φt)]. One may observe that for any two values of k, the norms are mutually equivalent; and thus M p,φ k, µ) M p,φ k 2, µ) for any k, k 2 >. Since the value of k does not matter, we write M p,φ µ) for M p,φ k, µ), where k 2

3 may vary from one part to another. Furthermore, if < p < q <, then f : M,φ µ) f : M p,φ µ) f : M q,φ µ). Accordingly, we have M q,φ µ) M p,φ µ) M,φ µ). We assume, from now on, that r /p φr) is almost increasing, and that there exists a positif constant C such that r t α/n φt)dt Cr α/n φr) for every r > 0. Then we have the following theorem. Theorem 2.. Let < p < n α and q = p α n. Assume that ψ : 0, ) 0, ) is almost decreasing, and there exists a constant C > 0 which is independent of r) such that r α/n φr) Cψr). Then, we have I α f : M q,ψ µ) C f : M p,φ µ), that is, I α is bounded from M p,φ µ) to M q,ψ µ). Proof. Let µ) be fixed. For f M p,φ µ), we set f = f + f 2 = fχ K + fχ R d \K, where K is sufficiently large. It follows that f L p µ) with the norm f : L p µ) [µ2k)] /p φµ2k)) f : M p,φ µ). As a consequence, we have µ2k) ) /q I α f x) q dµy) [µ2k)] /q I αf : L q µ) C [µ2k)] /q f : L p µ) C[µ2K)] /p /q φµ2k)) f : M p,φ µ) Cψµ2K)) f : M p,φ µ). To estimate I α f 2, fix x. Observe that we can find λ > such that for each y R d \ K, we have λr whenever R {x, y, c }. Note that the value of λ is relatively small when K is large.) With this λ, we have x y n α C λ sup [µλr)] α/n R for some constant C λ independent of x and y. Here the supremum is taken over all R µ) such that x, y, c R. Furthermore, let us set { } A := y R d \ K : 2 < inf µλr) 2+. R 3

4 Then, by using the covering lemma of Sawano [2, Lemma 2], there exist N ),..., N) m) such that A λ and 2 < µλ m) ) 2 + ; and m= so we get the pointwise estimate for I α f 2 : fy) I α f 2 x) = dµy) x y n α R d \K C 2 α/n ) fy) dµy) Z A C N 2 α/n ) fy) dµy) m) Z m= λ C Z m= N 2 α/n φ2 ) f : M,φ µ). Recall that µ) µλr) for every R {x, y, c }. So, if A, then µ) inf µλr) R 2+. Since 2 α/n φ2 ) C 2 2 t α/n φt)dt, we obtain 2 α/n φ2 ) C Z Z C µ) 2 t α/n φt) dt t α/n φt) dt C 2 2 µ) ) α/n φ 2 2 µ) ). It then follows that ) /q I α f 2 x) q dµy) ψµ2k)) µ2k) 2 C f : M,φ 2 µ) ) α/n φ 2 2 µ) ) 2 2 µ) ) /q µ) µ2k) α/n φµ2k))µ2k) /q 2 2 µ) ) /p φ 2 2 µ) ) C f : M,φ µ) C f : M p,φ µ) µ2k) /p φµ2k)) because r /p φr) is almost increasing. Now, by the virtue of Minkowski s 4

5 inequality, we get ψµ2k)) µ2k) ψµ2k)) ψµ2k)) C f : M p,φ µ), which yields the desired result. ) /q I α fx) q dµy) µ2k) µ2k) I α f x) q dµy)) /q + ) /q I α f 2 x) q dµy) As a consequence of Theorem 2., we obtain an Olsen inequality involving a multiplication operator W. Theorem 2.2. Let < p < n α and q = p α n. If rα/n φr) is almost decreasing, then W I α f : M p,φ µ) C W : L n/α µ) f : M p,φ µ), provided that W L n/α µ). Proof. For every cube, we use Hölder s inequality to get µk)) µk)) ) /p W I α fx) p dµx) ) α/n W x) n/α dµx) µk)) If ψr) = r α/n φr), then we could apply Theorem 2. to obtain φµk)) µk) ) /p W I α fx) p dµx) α/n W x) dµx)) n/α ψµk)) I α fx) q dµx)) /q. ) /q I α fx) q dµx) µk) C W : L n/α µ) I α f : M p,φ µ) C W : L n/α µ) f : M p,φ µ). Now, by taking the supremum over all, the desired inequality follows. 5

6 Remark. In the case that φt) = t /p, we have M p,φ k, µ) = L p µ). As a result, we obtain the Olsen inequality on Lebesgue spaces, which tells us that W I α is a bounded operator on L p µ). Acknowledgements. The research is supported by Fundamental Research Grant No. 0367/K0.03/Kontr-WRRIM/PL2..5/IV/2008. The authors are grateful to Dr. Yoshihiro Sawano for his ideas about generalized Morrey spaces of non-homogeneous type. Thanks also go to Dr. Yudi Soeharyadi and Dr. Wono Setya-Budhi for valuable discussions on the properties of the generalized non-homogeneous Morrey spaces. References [] D. Adams, A note on Riesz potentials, Duke Math. J ), [2] F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat ), [3] J. García-Cuerva and J.M. Martell, Two weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces, Indiana Univ. Math. J ), no. 3, [4] H. Gunawan and Eridani, Fractional integrals and generalized Olsen inequalities, to appear in Kyungpook Math. J. [5] G.H. Hardy and J.E. Littlewood, Some properties of fractional integrals. I, Math. Zeit ), [6] G.H. Hardy and J.E. Littlewood, Some properties of fractional integrals. II, Math. Zeit ), [7] K. Kurata, S. Nishigaki and S. Sugano, Boundedness of integral operators on generalized Morrey spaces and its application to Schrödinger operators, Proc. Amer. Math. Soc ), [8] E. Nakai, Hardy-Littlewood maximal operator, singular integral operators, and the Riesz potentials on generalized Morrey spaces, Math. Nachr ), [9] F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 998), no. 9, [0] P.A. Olsen, Fractional integration, Morrey spaces and a Schrödinger equation, Comm. Partial Differential Equations ), no. 2,

7 [] J. Peetre, On the theory of L p,λ spaces, J. Func. Anal ), [2] Y. Sawano, l q -valued extention of the fractional maximal operators for non-doubling measures via potential operators, Int. J. Pure Appl. Math ), no. 4, [3] Y. Sawano, T. Sobukawa and H. Tanaka, Limiting case of the boundedness of fractional integral operators on nonhomogeneous space, J. Inequal. Appl. 2006, Art. ID 92470, 6pp. [4] Y. Sawano, Generalized Morrey spaces for non-doubling measures, to appear in Nonlinear Differential Equation Appl. [5] I. Sihwaningrum, H. P. Suryawan and H. Gunawan, Fractional integral operators and Olsen inequalities on non-homogeneous spaces, to appear in Aust. J. Math. Anal. Appl. [6] S.L. Sobolev, On a theorem in functional analysis Russian), Mat. Sob ), [English translation in Amer. Math. Soc. Transl. ser ), 39 68]. [7] X. Tolsa, BMO, H, and Calderón-Zygmund operators for non doubling measures, Math. Ann ), I. Sihwaningrum: Department of Mathematics Bandung Institute of Technology Bandung 4032 Indonesia hanidha@students.itb.ac.id Permanent address: Department of Mathematics General Soedirman University Purwokerto 5322 Indonesia) H. Gunawan: Department of Mathematics Bandung Institute of Technology Bandung 4032 Indonesia hgunawan@math.itb.ac.id 7