Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces

Abstract

Let L=−△+VL=-\bigtriangleup +V be the Schrödinger operator on Rn{{\mathbb{R}}}^{n}, where V≠0V\ne 0 is a non-negative function satisfying the reverse Hölder class RHq1R{H}_{{q}_{1}} for some q1>n⁄2{q}_{1}\gt n/2. △\bigtriangleup is the Laplacian on Rn{{\mathbb{R}}}^{n}. Assume that bb is a member of the Campanato space Λνθ(ρ){\Lambda }_{\nu }^{\theta }\left(\rho ) and that the fractional integral operator associated with LL is ℐβL{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}. We study the boundedness of the commutators [b,ℐβL]\left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] with b∈Λνθ(ρ)b\in {\Lambda }_{\nu }^{\theta }\left(\rho ) on local generalized mixed Morrey spaces. Generalized mixed Morrey spaces Mp→,φα,V{M}_{\overrightarrow{p},\varphi }^{\alpha ,V}, vanishing generalized mixed Morrey spaces VMp→,φα,VV{M}_{\overrightarrow{p},\varphi }^{\alpha ,V}, and LMp→,φα,V,{x0}L{M}_{\overrightarrow{p},\varphi }^{\alpha ,V,\left\{{x}_{0}\right\}} are related to the Schrödinger operator, in that order. We demonstrate that the commutator operator [b,ℐβL]\left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] is satisfied when bb belongs to Λνθ(ρ){\Lambda }_{\nu }^{\theta }\left(\rho ) with θ>0\theta \gt 0, 0<ν<10\lt \nu \lt 1, and (φ1,φ2)\left({\varphi }_{1},{\varphi }_{2}) satisfying certain requirements are bounded from LMp→,φ1α,V,{x0}L{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V,\left\{{x}_{0}\right\}} to LMq→,φ2α,V,{x0}L{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V,\left\{{x}_{0}\right\}}; from Mp→,φ1α,V{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V} to Mq→,φ2α,V{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V}, and from VMp→,φ1α,VV{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V} to VMq→,φ2α,VV{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V}, ∑i=1n1⁄pi−∑i=1n1⁄qi=β+ν{\sum }_{i=1}^{n}1/{p}_{i}-{\sum }_{i=1}^{n}1/{q}_{i}=\beta +\nu .

Keywords

• schrödinger operator
• fractional integral
• commutator
• lipschitz function
• local generalized mixed morrey space
• 42b35
• 35j10
• 47h50