Estimates for the commutators of higher order Schrödinger-type operators

Let m be a non-negative integer and suppose <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mrow> <m:mi>m</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> {n≥ 2ᵐ⁺¹+1} . In this manuscript, the higher order Schrödinger operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">ℋ</m:mi> <m:msup> <m:mn>2</m:mn> <m:mi>m</m:mi> </m:msup> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mn>2</m:mn> <m:mi>m</m:mi> </m:msup> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>V</m:mi> <m:msup> <m:mn>2</m:mn> <m:mi>m</m:mi> </m:msup> </m:msup> </m:mrow> </m:mrow> </m:math> {H_{2ᵐ}=(-Δ)^{2ᵐ}+V^{2ᵐ}} on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {Rⁿ} is considered, the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi>q</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(Lᵖ(Rⁿ),Lq(Rⁿ))} -boundedness of the operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mrow> <m:msup> <m:mn>2</m:mn> <m:mi>m</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:msubsup> <m:mi mathvariant="script">ℋ</m:mi> <m:msup> <m:mn>2</m:mn> <m:mi>m</m:mi> </m:msup> <m:mrow> <m:mo>-</m:mo> <m:mi>β</m:mi> </m:mrow> </m:msubsup> </m:mrow> </m:math> {V^{2ᵐα}H_{2ᵐ}-β} and that of its commutator are established, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>α</m:mi> <m:mo>≤</m:mo> <m:mi>β</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0<α≤β≤ 1} . Here the non-negative potential V belongs to both the reverse Hölder class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>RH</m:mi> <m:mi>s</m:mi> </m:msub> </m:math> {RHₛ} for