In this paper some embedding theorems related to fractional integration and differentiation in harmonic mixed norm spaces $h(p,q,\alpha )$ on the half-space are established. We prove that mixed norm is equivalent to a “fractional derivative norm” and that harmonic conjugation is bounded in $h(p,q,\alpha )$ for the range $0<p\le \infty$, $0<q\le \infty$. As an application of the above, we give a characterization of $h(p,q,\alpha )$ by means of an integral representation with the use of Besov spaces.

In this paper we prove that a subharmonic function in ℝm of finite λ-type can be represented (within some subharmonic function) as the sum of a generalized Weierstrass canonical integral and a function of finite λ-type which tends to zero uniformly on compacts of ℝm. The known Brelot-Hadamard representation of subharmonic functions in ℝm of finite order can be obtained as a corollary from this result. Moreover, some properties of R-remainders of λ-admissible mass distributions are investigated.

We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of $p$-Laplacian type. If $p<\gamma <N$ and the right-hand side is a Radon measure with singularity of order $\gamma$ at $x_0\in \Omega$, then any supersolution in $W_{\mathrm{l}oc}^{1,p}\left(\Omega \right)$ has singularity of order at least $\frac{(\gamma -p)}{(p-1)}$ at $x_0$. In the proof we exploit a pointwise estimate of $𝒜$-superharmonic solutions, due to Kilpeläinen and Malý, which involves Wolff’s potential of Radon’s measure.

The $H$-cone is an abstract model for the cone of positive superharmonic functions on a harmonic space or for the cone of excessive functions with respect to a resolvent family, having sufficiently many properties in order to develop a good deal of balayage theory and also to construct a dual concept which is also an $H$-cone. There are given an integral representation theorem and a representation theorem as an $H$-cone of functions for which fine topology, thinnes, negligible sets and the sheaf property...

A harmonic function in a cylinder with the normal derivative vanishing on the boundary is expanded into an infinite sum of certain fundamental harmonic functions. The growth condition under which it is reduced to a finite sum of them is given.

In questo articolo studieremo le relazioni fra le funzioni armoniche nella palla iperbolica (sia essa reale, complessa o quaternionica), le funzione armoniche euclidee in questa palla, e le funzione pluriarmoniche sotto certe condizioni di crescita. In particolare, estenderemo al caso quaternionico risultati anteriori dell'autore (nel caso reale), e di A. Bonami, J. Bruna e S. Grellier (nel caso complesso).

We consider harmonic Bergman-Besov spaces $b_{\alpha }^p$ and weighted Bloch spaces $b_{\alpha }^{\infty }$ on the unit ball of ${ℝ}^n$ for the full ranges of parameters $0<p<\infty$, $\alpha \in ℝ$, and determine the precise inclusion relations among them. To verify these relations we use Carleson measures and suitable radial differential operators. For harmonic Bergman spaces various characterizations of Carleson measures are known. For weighted Bloch spaces we provide a characterization when $\alpha >0$.

Let $\mathscr{H}$ denote the class of positive harmonic functions on a bounded domain $\Omega$ in ${ℝ}^N$. Let $S$ be a sphere contained in $\overline{\Omega }$, and let $\sigma$ denote the $(N-1)$-dimensional measure. We give a condition on $\Omega$ which guarantees that there exists a constant $K$, depending only on $\Omega$ and $S$, such that ${\int }_{S}ud\sigma \le K{\int }_{\partial \Omega }ud\sigma$ for every $u\in \mathscr{H}\cap C\left(\overline{\Omega }\right)$. If this inequality holds for every such $u$, then it also holds for a large class of non-negative subharmonic functions. For certain types of domains explicit values for $K$ are given. In particular the classical value...

Let ${\Omega }_1,...,{\Omega }_n$ be harmonic spaces of Brelot with countable base of completely determining domains. The elements of a subcone $\mathbf{C}$ of the cone of positive $n$-superharmonic functions in ${\Omega }_1\times ...\times {\Omega }_n$ is shown to have an integral representation with the aid of Radon measures on the extreme elements belonging to a compact base of $\mathbf{C}$. The extreme elements are shown to be the product of extreme superharmonic functions on the component spaces and the measure representing each element is shown to be unique. Necessary and sufficient...