We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the $GL$-energy $E_{\epsilon }$ and the parameter $\epsilon$. These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.

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...

We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set $E$ whose supports are compact polar subsets of $E$. We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set $E$, we prove...

We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x&gt;0, γ≥0 and fa smooth function on ${\Re }^{+}$, $$\begin{array}{c}\hfill {𝐋}^{\left(\gamma \right)}f\left(x\right)=x^{-\alpha }(\frac{\sigma }{2}{x}^2{f}^{\text{'}\text{'}}\left(x\right)+(\sigma \gamma +b)x{f}^{\text{'}}\left(x\right)\\ \hfill +{\int }_0^{\infty }\left(f\left({\mathrm{e}}^{-r}x\right)-f\left(x\right)\right){\mathrm{e}}^{-r\gamma }+x{f}^{\text{'}}\left(x\right)r{𝕀}_{\{r\le 1\}}\nu \left(\mathrm{d}r\right)),(0.1)\end{array}$$ where the coefficients $b\in \Re$,σ≥0 and the measure ν, which satisfies the integrability condition ∫0∞(1∧r2)ν(dr)&lt;+∞, are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent ψ. L(γ) is known to be the infinitesimal generator of a positive...