This paper is devoted to exploiting the restrictions of Riesz–Morrey potentials on either unbounded or bounded domains in Euclidean spaces.

In this paper we connect Calderón and Zygmund’s notion of Lp-differentiability (Calderón and Zygmund, Proc Natl Acad Sci USA 46:1385–1389, 1960) with some recent characterizations of Sobolev spaces via the asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu (Optimal Control and Partial Differential Equations, pp. 439–455, 2001). We show how the results of the former can be generalized to the setting of the latter, while the latter results can be strengthened in the spirit of the former. As a consequence of these results we give several new characterizations of Sobolev spaces, a novel condition for whether a function of bounded variation is in the Sobolev space W1,1, and complete the proof of a characterization of the Sobolev spaces recently claimed in (Leoni and Spector, J Funct Anal 261:2926–2958, 2011; Leoni and Spector, J Funct Anal 266:1106–1114, 2014).

For the classical space of functions with bounded mean oscillation, it is well known that $\operatorname{VMO}^{**} = \operatorname{BMO}$ and there are many characterizations of the distance from a function$f$in $\operatorname{BMO}$ to $\operatorname{VMO}$ . When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as$Q$_{$K$}-spaces, weighted spaces, Lipschitz–Hölder spaces and rectangular $\operatorname{BMO}$ of several variables.

Asymptotically sharp Bernstein- and Markov-type inequalities are established for rational functions on C2 smooth Jordan curves and arcs. The results are formulated in terms of the normal derivatives of certain Green’s functions with poles at the poles of the rational functions in question. As a special case (when all the poles are at infinity) the corresponding results for polynomials are recaptured.

In this work we introduce a formulation for a non-local Hessian that combines the ideas of higher-order and non-local regularization for image restoration, extending the idea of non-local gradients to higher-order derivatives. By carefully choosing the weights, the model allows to improve on the current state of the art higher-order method, Total Generalized Variation, with respect to overall quality and particularly close to jumps in the data. In the spirit of recent work by Brezis et al., our formulation also has analytic implications: for a suitable choice of weights, it can be shown to converge to classical second-order regularizers, and in fact allows a novel characterization of higher-order Sobolev and BV spaces.