It is known that, unlike the one dimensional case it is not possible to find an upper bound for the zeros of an entire map from$C$^{$n$}to$C$^{$n$},$n$≥2, in terms of the growth of the map. However, if we only consider the “non-degenerate” zeros, that is, the zeros where the jacobian is not “too small”, it becomes possible. We give a new proof of this fact.

A new and fairly elementary proof is given of the result by B. Simon [S], that the potential in a Sturm-Liouville operator is determined by the asymptotics of the associated$m$-function near −∞. The proof given is based on relations between the classical transformation operators and the$m$-function.

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We prove a Melin-Hörmander inequality for a Banach algebra of pseudo-differential operators whose calculus was developed by Sjöstrand. The main new difficulties in the proof are settled by a stationary phase method tailored to the low-regularity of the symbols.

Suppose that$X$is a vector field on a manifold$M$whose flow, exp$tX$, exists for all time. If μ is a measure on$M$for which the induced measures$μ$_{$t$}≡(exp$tX$)_{*}$μ$are absolutely continuous with respect to μ, it is of interest to establish bounds on the$L$^{$p$}(μ) norm of the Radon-Nikodym derivative$dμ$_{$t$}/$dμ$. We establish such bounds in terms of the divergence of the vector field$X$. We then specilize$M$to be a complex manifold and derive reverse hypercontractivity bounds and reverse logarithmic Sololev inequalities in some holomorphic function spaces. We give examples on$C$^{m}and on the Riemann surface for$z$^{1/$n$}.