Functions of Pairs of Unbounded Noncommuting Self-Adjoint Operators under Perturbation

For a pair $$(A,B)$$ of not necessarily bounded and commuting self-adjoint operators for function f on the Euclidean space $${{\mathbb{R}}^{2}}$$ that belongs to inhomogeneous Besov class $$B_{{\infty ,1}}^{1}({{\mathbb{R}}^{2}})$$ , we define $$f(A,B)$$ these as densely defined operator. We consider problem estimating functions under perturbations . It is established if $$1 \leqslant p 2$$ $$({{A}_{1}},{{B}_{1}})$$ $$({{A}_{2}},{{B}_{2}})$$ are pairs such $${{A}_{1}} - {{A}_{2}}$$ $${{B}_{1}} {{B}_{2}}$$ belong Schatten–von Neumann Sp with $$p \in [1,2]$$ $$f\, \,B_{{\infty then following Lipschitz type estimate holds: $${\text{||}}\,f({{A}_{1}},{{B}_{1}})\, \,f({{A}_{2}},{{B}_{2}}){\text{|}}{{{\text{|}}}_{{{{S}_{p}}}}}\, \,{\text{||}}\,f\,{\text{|}}{{{\text{|}}}_{{B_{{\infty ,1}}^{1}}}}{\text{max}}\{ {\text{||}}{{A}_{1}}\, \,{{A}_{2}}{\text{|}}{{{\text{|}}}_{{_{{{{S}_{p}}}}}}},{\text{||}}{{B}_{1}}\, \,{{B}_{2}}{\text{|}}{{{\text{|}}}_{{_{{{{S}_{p}}}}}}}\} .$$