### A Family of Optimal Conditions for the Absence of Bound States in a Potential

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One gives a general definition of capacity which includes $p$-capacity, extremal length and a quantity defined by N.G. Meyers.

For potentials ${U}_{K}^{T}=K*T$, where $K$ and $T$ are certain Schwartz distributions, an inversion formula for $T$ is derived. Convolutions and Fourier transforms of distributions in $\left({\mathbf{D}}_{L}^{\text{'}}p\right)$-spaces are used. It is shown that the equilibrium distribution with respect to the Riesz kernel of order $\alpha $, $0\<\alpha \<m$, of a compact subset $E$ of ${\mathbf{R}}^{m}$ has the following property: its restriction to the interior of $E$ is an absolutely continuous measure with analytic density which is expressed by an explicit formula.

It is shown that Bessel potentials have a representation in term of measure when the underlying space is Orlicz. A comparison between capacities and Lebesgue measure is given and geometric properties of Bessel capacities in this space are studied. Moreover it is shown that if the capacity of a set is null, then the variation of all signed measures of this set is null when these measures are in the dual of an Orlicz-Sobolev space.

We are concerned with the boundedness of generalized fractional integral operators ${I}_{\rho ,\tau}$ from Orlicz spaces ${L}^{\Phi}\left(X\right)$ near ${L}^{1}\left(X\right)$ to Orlicz spaces ${L}^{\Psi}\left(X\right)$ over metric measure spaces equipped with lower Ahlfors $Q$-regular measures, where $\Phi $ is a function of the form $\Phi \left(r\right)=r\ell \left(r\right)$ and $\ell $ is of log-type. We give a generalization of paper by Mizuta et al. (2010), in the Euclidean setting. We deal with both generalized Riesz potentials and generalized logarithmic potentials.

In this article a general result on smooth truncation of Riesz and Bessel potentials in Orlicz-Sobolev spaces is given and a capacitary type estimate is presented. We construct also a space of quasicontinuous functions and an alternative characterization of this space and a description of its dual are established. For the Riesz kernel Rm, we prove that operators of strong type (A, A), are also of capacitaries strong and weak types (m,A).