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Proposition 2.1 (Chern-Levine-Nirenberg inequalities) . [032T]

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Proposition 2.1 (Chern-Levine-Nirenberg inequalities).

Let TT be a positive closed current of bidegree (p,p)(p,p) on XX and φ∈P​S​H​(X,ω)∩L∞​(X)\varphi\in PSH(X,\omega)\cap L^{\infty}(X).

Then ‖ωφ∧T‖=‖T‖||\omega_{\varphi}\wedge T||=||T||. Moreover if ψ∈P​S​H​(X,ω)∩L1​(T)\psi\in PSH(X,\omega)\cap L^{1}(T), then ψ∈L1​(T∧ωφ)\psi\in L^{1}(T\wedge\omega_{\varphi}) and

‖ψ‖L1​(T∧ωφ)≤‖ψ‖L1​(T)+[2​supXψ+supXφ−infXφ]​‖T‖.||\psi||_{L^{1}(T\wedge\omega_{\varphi})}\leq||\psi||_{L^{1}(T)}+[2\sup_{X}\psi+\sup_{X}\varphi-\inf_{X}\varphi]||T||.

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