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An explicit formula is developed for Nevanlinna class functions whose behaviour at the boundary is “sufficiently rational” and is then used to deduce the uniqueness of the factorization of such inner functions. A generalization of a theorem of Frostman is given and the above results are then applied to the construction of good and/or irreducible inner functions on a polydisc.

We show that the differential inequality $\left|\Delta u\right|\le v\left|u\right|$ has the unique continuation property relative to the Sobolev space $H_{loc}^{2,1}\left(\Omega \right)$, $\Omega \subset R^n$, $n\le 3$, if $v$ satisfies the condition $$(K_n^{\mathrm{loc}})\underset{r\to 0}{lim}\underset{x\in K}{sup}{\int }_{|x-y|\<r}|x-y{|}^{2-n}v\left(y\right)dy=0$$ for all compact $K\subset \Omega$, where if $n=2$, we replace $|x-y{|}^{2-n}$ by $-log|x-y|$. This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H=-\Delta +v$, in the case $n\le 3$. The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N=0,1,2,...$ and any $u\in H_c^{2,1}({\mathbf{R}}^3-\left\{0\right\}),$ $$\frac{\left|u\right(x\left)\right|}{|x{|}^N}\le C{\int }_{{\mathbf{R}}^3}|x-y{|}^{-1}\frac{\left|\Delta u\right(y\left)\right|}{|y{|}^N}dy\text{for}\text{a.e.}x\text{in}{\mathbf{R}}^3.$$

Suppose $\Phi$ is a nonnegative, locally integrable, radial function on ${\mathbf{R}}^n$, which is nonincreasing in $\left|x\right|$. Set $\left(Tf\right)\left(x\right)={\int }_{R^n}\Phi (x-y)f\left(y\right)dy$ when $f\ge 0$ and $x\in {\mathbf{R}}^n$. Given $1\<p\<\infty$ and $v\ge 0$, we show there exists $C\>0$ so that ${\int }_{{\mathbf{R}}^n}\left(Tf\right)(x{)}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}f(x{)}^{p}dx$ for all $f\ge 0$, if and only if $C^{\text{'}}\>0$ exists with ${\int }_{Q}T(x_{Q}v\left)\right(x{)}^{p^{\text{'}}}dx\le C^{\text{'}}{\int }_{Q}v\left(x\right)dx\<\infty$ for all dyadic cubes Q, where $p^{\text{'}}=p/(p-1)$. This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

G. David, J.-L. Journé and S. Semmes have shown that if b and b are para-accretive functions on R, then the Tb theorem holds: A linear operator T with Calderón-Zygmund kernel is bounded on L if and only if Tb ∈ BMO, T*b ∈ BMO and MTM has the weak boundedness property. Conversely they showed that when b = b = b, para-accretivity of b is necessary for Tb Theorem to hold. In this paper we show that para-accretivity of both b and b is necessary for the Tb Theorem to hold in general. In addition, we...