Let$X$and$Y$be smooth varieties of dimensions$n$−1 and$n$over an arbitrary algebraically closed field,$f: X→Y$a finite map that is birational onto its image. Suppose that$f$is curvilinear; that is, for all$xεX$, the Jacobian ϱ$f(x)$has rank at least$n$−2. For$r$≥1, consider the subscheme$N$_{$r$}of$Y$defined by the ($r$−1)th Fitting ideal of the $$\mathcal{O}_Y $$ -module $$f_ * \mathcal{O}_X $$ , and set$M$_{$r$}∶=$f$^{−1}$N$_{$r$}. In this setting—in fact, in a more general setting—we prove the following statements, which show that$M$_{$r$}and$N$_{$r$}behave like reasonable schemes of source and target$r$-fold points of$f$.

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The Cauchy problem for the Laplace operator $$\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$$ is modified by replacing the Laplace equation by an asymptotic estimate of the form $$\begin{gathered} \Delta u(x,y) = 0, \hfill \\ u(x,0) = f(x),\frac{{\partial u}}{{\partial y}}(x,0) = g(x) \hfill \\ \end{gathered} $$ with a given majorant$h$, satisfying$h$(+0)=0. This$asymptotic Cauchy problem$only requires that the Laplacian decay to zero at the initial submanifold. It turns out that this problem has a solution for smooth enough Cauchy data$f, g$, and this smoothness is strictly controlled by$h$. This gives a new approach to the study of smooth function spaces and harmonic functions with growth restrictions. As an application, a Levinson-type normality theorem for harmonic functions is proved.

Given positive integers$n$_{1}<$n$_{2}<... we show that the Hardy-type inequality $$\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$$ holds true for all$f$∈$H$^{1}, provided that the$n$_{$k$}'s, satisfy an appropriate (and indispensable) regularity condition. On the other hand, we exhibit inifinite-dimensional subspaces of$H$^{1}for whose elements the above inequality is always valid, no additional hypotheses being imposed. In conclusion, we extend a result of Douglas, Shapiro and Shields on the cyclicity of lacunary series for the backward shift operator.

Let$f$be an entire function of order at least 1/2,$M(r)$=max_{|}$z$|=$r$|$f(z)$|, and$n(r, a)$the number of zeros of$f(z)-a$in |$z$|≤$r$. It is shown that lim sup_{$r$→∞}$n(r, a)$/log$M (r)$≥1/2π for all except possibly one$a$∈C.