Second order Riesz transforms associated to the Schrödinger operator forp⩽1

Some estimates of Schrödinger type operators on variable Lebesgue and Hardy spaces

“…Case ii): k ∈ N. In this case, by (2.7) and an argument similar to that used in the proof of [16,Proposition 3.3], we conclude that (3.9) and (3.10) hold true, which completes the proof of (i).…”

“…The proof of Theorem 1.7 is in Subsection 3.2. We prove it by borrowing some ideas from [16,17] and using some skills from [26]. The key to the proof is to establish some weighted estimates of the spatial derivatives of the heat kernel of {e −tL } t≥0 (see Lemma 3.6 below).…”

“…Next, inspired by F. K. Ly[16, Proposition 3.3] and [17, Proposition 2.4], we introduce some weighted estimates for the spatial derivatives of the heat kernel of {e −tL } t≥0 , which play a key role in the proof of Theorem 1.7.…”

Generalized Morrey spaces associated to Schrödinger operators and applications

“…In recent years the problem related to Schrödinger operator has attracted a great deal of attention of many mathematicians; see [3], [4], [5], [6], [8], [11], [12], [20], [27], [34] and references therein. In this paper we consider a Schrödinger operator L = −∆ + V on R n , n 3, where the potential V belongs to RH q for some q > n/2 and RH q is the reverse Hölder class defined in Section 2.…”

Several estimates of Musielak--Orlicz--Hardy--Sobolev type for Schröodinger type operators

“…Moreover, the weighted L p (R n )-boundedness of these operators was studied in [37]. Recently, Ly [27] proved that V L −1 and ∇ 2 L −1 are bounded from the Hardy space H p L (R n ), associated with L, to L p (R n ) when p ∈ (0, 1], and ∇ 2 L −1 is also bounded from H p L (R n ) to the classical Hardy space H p (R n ) when p ∈ ( n n+1 , 1], via the boundedness of ∇ 2 L −1 on L p (R n ) with some p ∈ (1, ∞) and some Sobolev type estimates for the heat kernel of L. Moreover, the boundedness of ∇ 2 L −1 and V L −1 on the Musielak-Orlicz-Hardy space H ϕ, L (R n ) was independently studied in [7] by different method. Very recently, the boundedness of V L −1 , V 1/2 (∇ − i a)L −1 and (∇ − i a) 2 L −1 from the Musielak-Orlicz-Hardy space H ϕ, L (R n ), associated with the magnetic Schrödinger operator L, to the Musielak-Orlicz space L ϕ (R n ) was established in [8], where L := −(∇ − i a) • (∇ − i a) + V with a := (a 1 , a 2 , .…”