Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral zero. This is then used to prove a generalization of Furstenberg's theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a “two circles” Morera type theorem (earlier announced by Agranovskii).

Letf 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)/logM (r)≥1/2π for all except possibly one a∈C.

Given positive integers n₁< n₂<... 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∈H¹, provided that the n_k's, satisfy an appropriate (and indispensable) regularity condition. On the other hand, we exhibit inifinite-dimensional subspaces of H¹ 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.

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.

LetX 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 ϱf(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⁻¹N_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.

If each component of M_r, or equivalently of N_r, has the minimal possible dimension n−r, then M_r and N_r are Cohen-Macaulay, and their fundamental cycles satisfy the relation, f_*[M_r]=r[N_r]. Now, suppose that each component of M_s, or of N_s, has dimension n−s for s=1,..., r+1. Then the blowup Bl(N_r, N_r+1) is equal to the Hilbert scheme Hilb ${}_{f}^{r}$ and the blowup Bl(M_r, M_r+1) is equal to the universal subscheme Univ ${}_{f}^{r}$ of Hilb ${}_{f}^{r}$ ×_YX; moreover, Hilb ${}_{f}^{r}$ and Univ ${}_{f}^{r}$ are Gorenstein. In addition, the structure map h:Hilb ${}_{f}^{r}$ →Y is finite and birational onto its image; and its conductor is equal to the ideal $\mathcal{J}_r $ of N_r+1 in N_r, and is locally self-linked. Reciprocally, $h_ * \mathcal{O}_{Hilb_f^r } $ is equal to $\mathcal{H}om(\mathcal{J}_r ,\mathcal{O}_{N_r } )$ . Moreover, h_*[h⁻¹N_r+1]=(r+1)[N_r+1]. Similar assertions hold for the structure map h₁: Univ ${}_{f}^{r}$ →X if r≥2.

We prove L^q-inequalities for the gradient of the Green potential (Gf) in bounded, connected NTA-domains in Rⁿ, n≥2. These domains may have a highly non-rectifiable boundary and in the plane the set of all bounded simply connected NTA-domains coincides with the set of all quasidiscs. We get a restriction on the exponent q for which our inequalities are valid in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.