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In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series., 162 pages

Let Omega subset of Rn+1, n >= 2, be a 1-sided nontangentially accessible domain, that is, a set which is quantitatively open and path-connected. Assume also that Omega satisfies the capacity density condition. Let L(0)u = - div(A(0)del u), Lu = - div(A del u) be two real (not necessarily symmetric) uniformly elliptic operators in Omega, and write at omega(L0),omega(L) for the respective associated elliptic measures. We establish the equivalence between the following properties: (i) omega(L) is an element of A(infinity)(omega(L0)), (ii) L is L-P (omega(L0))-solvable for some p is an element of (1, infinity), (iii) bounded null solutions of L satisfy Carleson measure estimates with respect to omega(L0), (iv) S < Ar (i.e., the conical square function is controlled by the nontangential maximal function) in L-q (omega(L0)) for some (or for all) q is an element of (0, infinity) for any null solution of L, and (v) L is BMO(omega(L0))-solvable. Moreover, in each of the properties (ii)-(v) it is enough to consider the class of solutions given by characteristic functions of Borel sets (i.e, u (X) = omega(X)(L)(S) (S) for an arbitrary Borel set S subset of partial derivative Omega)., The authors would like to thank the anonymous referees for their comments to improve the presentation of the paper. All the authors acknowledge financial support from MCIN/AEI/ 10.13039/501100011033 grants FJC2018-038526-I (first authors), CEX2019-000904-S (first, third and last authors), PID2019-107914GB-I00 (first and third authors), MTM2017-84058- P (second author) and PID2020-116398GB-I00 (last author). The third author also acknowledges that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT.

We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki\u{\i} spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best $n$-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov., Comment: 45 pages

Resumen En este articulo se describen las caracteristicas principales de la ampliacion del puente de Los Santos. El proyecto ha supuesto la ampliacion de un puente de 600 m de longitud, con 3 vanos centrales de 150 m y laterales de 75 m, aumentando el ancho del tablero de 12,00 a 24,60 m, sin recurrir a una estructura independiente. El proyecto se basa en la idea de proveer del menor refuerzo posible a la estructura preexistente. Este hecho ha permitido optimizaciones muy importantes y tambien ha supuesto la realizacion de un proyecto desafiante, algunos de cuyos aspectos se describen en este articulo.