Morrey spaces for Schrodinger operators with nonnegative potentials, fractional integral operators and the Adams inequality on the Heisenberg groups

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:482

Article

Wang, Hua¹ 

[1] Xinjiang Univ, Sch Math & Syst Sci, Urumqi 830046, Peoples R China

关键词: Schrodinger operator;    Reverse Holder class;    Fractional integral operators;    Heisenberg group;    Morrey spaces;    Adams inequality;   

DOI  :  10.1016/j.jmaa.2019.123523

来源: Elsevier

【 摘 要 】

Let L = -Delta(Hn) + V be a Schrodinger operator on the Heisenberg group H-n, where Delta(Hn) is the sublaplacian on H-n and the nonnegative potential V belongs to the reverse Holder class RHs with s is an element of [Q/2, infinity). Here Q = 2n + 2 is the homogeneous dimension of H-n. For given alpha is an element of (0, Q), the fractional integral operator associated with the Schrodinger operator L is defined by ZQ = L'/2. In this article, the author introduces the Morrey space L-rho,infinity(p,k) (H-n) and weak Morrey space w L-rho infinity(p,kappa) (H-n) associated with L, where (p, kappa) is an element of [1, infinity) x [0, 1) and rho(.) is an auxiliary function related to the nonnegative potential V. The relation between the fractional integral operator and the maximal operator on the Heisenberg group is established. From this, the author further obtains the Adams (Morrey-Sobolev) inequality on these new spaces. It is shown that the fractional integral operator I-alpha, = L(-alpha/)2 is bounded from Lf,:':<,(111) to L7a<,(H) with 0 < a < Q, 1 < p < Q/a, 0 < a < 1 (ap)/Q and 1/q = 1/p a/Q(1 a), and bounded from Lip:',(11') to 141L7,:3(1.11.) with 0 < a < Q, 0 < a < 1 a/Q and 1/q = 1 a/Q(1 a). Moreover, in order to deal with the extreme cases a > 1 (ap)/Q, the author also introduces the spaces B) under a = 1 (ap)/Q, and bounded from under a > 1 (ap)/Q and = a (1 tc)Q/p. 0 (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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