Given a function $f\left(t\right)\ge 0$ on $ℝ$ with ${\int }_{-\infty }^{\infty }\left(f\right(t)/(1+t^2\left)\right)dt\<\infty$ and $\left|f\right(t)-f(t^{\text{'}}\left)\right|\le l|t-t^{\text{'}}|$, a procedure is exhibited for obtaining on $ℂ$ a (finite) superharmonic majorant of the function$$F\left(z\right):\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{\left|𝔍z\right|}{|z-t{|}^2}f\left(t\right)dt-Al\left|𝔍z\right|,$$where $A$ is a certain (large) absolute constant. This leads to fairly constructive proofs of the two main multiplier theorems of Beurling and Malliavin. The principal tool used is a version of the following lemma going back almost surely to Beurling: suppose that $f\left(t\right)$, positive and bounded away from 0 on $ℝ$, is such that ${\int }_{-\infty }^{\infty }\left(f\right(t)/(1+t^2)dt\<\infty$ and denote, for any constant $\alpha \>0$ and each $x\in ℝ$, the unique...

A technique is developed for constructing the solution of ${\Delta }^{2}u=F$ in $R=\left\{\right(x,y):|x|\<a,|y|\<b\}$, subject to boundary conditions $u=\phi$, $\frac{\partial u}{\partial n}=\psi$ on $\partial R$. The problem is reduced to that of finding the orthogonal projection $Pw$ of $w$ in $L^2\left(R\right)$ onto the subspace $\mathbf{H}$ of square integrable functions harmonic in $\mathbf{R}$. This problem is solved by decomposition $\mathbf{H}$ into the closed direct (not orthogonal) sum of two subspaces ${\mathbf{H}}^{\left(1\right)},{\mathbf{H}}^{\left(2\right)}$ for which complete orthogonal bases are known. $P$ is expressed in terms of the projections $P^{\left(1\right)}$, $P^{\left(2\right)}$ of $L^2\left(R\right)$ onto ${\mathbf{H}}^{\left(1\right)}$, ${\mathbf{H}}^{\left(2\right)}$ respectively. The resulting construction...

A necessary and sufficient condition for the continuous extendibility of a solution of the Neumann problem for the Laplace equation is given.

A necessary and sufficient condition for the continuous extendibility of a solution of the third problem for the Laplace equation is given.

Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

$M_1$ è un particolare operatore di minimizzazione per forme di Dirichlet definite su un sottoinsieme finito di un frattale $K$ che è, in un certo senso, una sorta di frontiera di $K$. Viene talvolta chiamato mappa di rinormalizzazione ed è stato usato per definire su $K$ un analogo del funzionale $u\mapsto \int {\left|\text{grad}u\right|}^2$ e un moto Browniano. In questo lavoro si provano alcuni risultati sull'unicità dell'autoforma (rispetto a $M_1$ ), e sulla convergenza dell'iterata di $M_1$ rinormalizzata. Questi risultati sono collegati con l'unicità...

We study the boundedness in $L^p{(}^n)$ of the projections onto spaces of functions with spectrum contained in horizontal strips. We obtain some results concerning convergence along nonisotropic regions of harmonic extensions of functions in $L^p{(}^n)$ with spectrum included in these horizontal strips.