Trudinger–Moser inequalities on hyperbolic spaces under Lorentz norms

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Abstract

The main purpose of this paper is to establish sharp Trudinger–Moser type inequalities on hyperbolic spaces Bn for functions whose hyperbolic gradient is in the Lorentz space L(n,q), n2. Namely, if ΩBn is bounded and 1<q<, we will show that there exists a constant C=C(n,q) such that for all uC0(Ω) with HuLn,q(Ω)1, we have1|Ω|Ωeβq|u(x)|qdVC, where βq=(nωn1/n)q and H is the hyperbolic gradient; if Ω=Bn, we use only the norm HuLn,q(Bn) rather than HuLn,q(Bn)+uLn,q(Bn) (unlike the case of Euclidean spaces) and will show that there exists a constant C=C(n,q) such that for all uC0(Bn) with HuLn,q(Bn)1, we haveBnΦq(βq|u(x)|q)dVC, whereΦq(t)=etj=0jq2tjj!,jq=min{jN:j1+n/q}.

Keywords

Trudinger–Moser inequality
Lorentz space
Hyperbolic space
Sharp constant

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The work was partially supported by the National Natural Science Foundation of China (No. 11201346).