Some Estimates of Higher Order Riesz Transform Related to Schrödinger Type Operators

Abstract

In this paper we consider the Schrödinger type operators H 2 = (− ) 2 + V 2 , where the nonnegative potential V belongs to the reverse Hölder class B q 1 for q 1 ≥ n 2 , n ≥ 5. The L p and weak type (1, 1) estimates of higher order Riesz transform ∇ 2 H − 1 2 2 related to Schrödinger type operators H 2 are obtained. In particular, ∇ 2 H − 1 2 2 is a Calderón-Zygmund operator if V ∈ B 2n or V ∈ B n 2 and there exists a constant C such that V(x) ≤ Cm(x, V) 2 .

Some estimates for commutators of Riesz transform associated with Schrödinger type operators

“…When we consider the term A 2 (x), we note that ̺(x j ) > r j ̺(x j )/4. Similar to the proof of Lemma 9 in [12], by Lemma 3.3 we obtain…”

“…These two operators have been investigated in [19], [18], [12], [3]. We denote by K and K * the kernels of R and R * , respectively.…”

“…When k = 1, L 1 is exactly the Schrödinger operator. In this paper, we focus on the operator L 2 , which has been studied in [19], [18], [3], [12], [11] and [14]. Let T be a bounded operator on L p (R n ) for some p with p ∈ (1, ∞).…”

“…In recent years, many scholars have investigated the singular integral operators related to Schrödinger operators L 1 and their commutators, see, for example, [17], [9], [7], [12], [1], [10], [16], [13], [15] and their references. Especially, when b ∈ BMO, Guo, Li and Peng [7] investigated the boundedness of the commutators of the Riesz transform ∇L −1/2 1 .…”

Boundedness of higher order Riesz transforms associated with Schrodinger type operator on generalized Morrey spaces

“…The L p boundedness of the higher Riesz transforms has been obtained in [4] by Liu and Dong. Theorem 1.1.…”