Abstract
The authors prove that Marcinkiewicz integral operator is not only are bounded onLp, for1<P<∞, but also a bounded map fromL1(Rn)to weakL1(Rn). Meanwhile, theBMOL-boundedness and(HL1,L1)-boundedness are also obtained. Finally, theLp-boundedness and(L∞,BMOL)-boundedness for the commutator of Marcinkiewicz integral of schrödinger type are established.
Highlights
Introduction and NotationLet us consider the Schrodinger operator L = −Δ + V (1)in Rd, d ≥ 3
We introduce the definition of the reverse Holder index of V as q0 = sup{q : V ∈ RHq}
It is known that V ∈ RHq implies V ∈ RHq+ε, for some ε > 0
Summary
Similar to the classical Marcinkiewicz function μ, we define the Marcinkiewicz integral μjL associated with the Schrodinger operator L by μjLf. where KjL(x, y) = KjL(x, y)|x − y| and KJL(x, y) is the kernel of RjL = (∂/∂xj)L−1/2, j = 1, . Tang and Dong [6] have shown that Marcinkiewicz integral μjL is bounded on Lp(Rn), for 1 < p < ∞, and are bounded from L1(Rn) to weak L1(Rn) They proved that μjL are bounded on BMOL(Rn) and are mapped from HL1(Rn) to L1(Rn) under the assumption that KjL satisfy the condition in Lemma 3. Let V ∈ RHd and b ∈ BMO∞(ρ); if b satisfies the stronger condition b ∈ BMOl∞og(ρ), [b, μjL] are bounded from L∞(Rd) to BMOL(Rd). We use the symbol A ≲ B to denote that there exists a positive constant C such that A ≤ CB
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