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Sharp reiteration theorems for the K-interpolation method in limiting cases are proved using two-sided estimates of the K-functional. As an application, sharp mapping properties of the Riesz potential are derived in a limiting case. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Let L be a differential operator on \({\mathbb R}^5\) whose principal part is of the form \(\!\sum_{\!j,k=1}^2\! \alpha_{jk}\! X_j Y_k\), where \(X_j=\partial_{x_j}\) and \(Y_j=\partial_{y_j}+x_j\partial_u , j=1,2\), are the usual vector fields generating the Lie algebra of the Heisenberg group \({\mathbb H}_2\). We study the problem of local solvability of these doubly characteristic operators. The whole class of operators splits into three subclasses, depending on the sign of a respective determinant. The operators in the first subclass, when the determinant is negative, are generically non-solvable. The operators in the second subclass, when the determinant is positive, are solvable, for arbitrary left-invariant lower order terms, provided that the coefficient matrix \((\alpha_{j,k})\) is non-degenerate. This fact seems remarkable, since many of these operators have the property that the values taken by their principal symbol are not contained in any proper subcone of the complex plane. Under suitable conditions, solvability even holds in the presence of non-invariant lower order terms.

Sharp estimates are obtained for the rates of blow up of the norms of embeddings of Besov spaces bp,qs in Lorentz spaces as the parameters approach critical values. In Edmunds et al. (2006) the case 1 = p < 8 was investigated. The case 0 < p < 1 of the present paper requires different methods as the pointwise estimates established are different and the interpolation argument used in Edmunds et al. (2006) is no longer available.

Sharp estimates are obtained for the rates of blow up of the norms of embeddings of Besov spaces in Lorentz spaces as the parameters approach critical values.

Let (A0, A1) be a compatible pair of quasi-Banach spaces and 1etA be a corresponding space of real interpolation type such that A0∩A1 is not dense in A. Upper and lower estimates are obtained for the distance of any element f of A from A0∩A1. These lead to formulae for the distance in a large number of concrete situations, such as when A0∩A1=L∞ and A is either weak-Lq, a ‘grand’ Lebesgue space or an Orlicz space of exponential type.

In this paper the results from [7, 8], concerning the asymptotic behaviour of the spectral function on the diagonal for Schrodinger operators h 0, are extended to the case of some h-admissible operators, acting in Rn, no 2.

In this paper one obtains a result concerning the asymptotic behaviour of the spectral function on the diagonal for SCHRODINGER operators Ah = –h2/2 λ + V as h 0. This asymptotic change the form on the energy level V(x) = λ.

In this paper we study for small positive e the slow motion of the solution for evolution equations of Burgers’ type with small diffusion, () u t =eu xx +F(u) x , u(x,0)=u 0 (x), u(±1,t)=u ± on the bounded spatial domain [−1,1]; F is a smooth nonpositive function having only a finite number of zeros (at least two) between u − and u + , all of finite order. The initial and boundary value problem (★) has a unique asymptotically stable equilibrium solution that attracts all solutions starting with continuous initial data u 0 . On an interval [−1−c 0 e,1+c 0 e], c 0 >0 the differential equation has slow speed travelling wave solutions generated by profiles that satisfy the boundary conditions of (★). During a long but finite time interval, such travelling waves suitably describe the slow long-term behaviour of the solution of evolution problem (★). Their speed characterizes the local velocity of the slow motion with algebraic precision (w.r.t. e ) in general, and with exponential precision, if F has only two zeros of first order located at u + and u − . A solution that starts near a travelling wave, moves in a small neighbourhood of such a travelling wave during a long time interval (0, T ). If F has zeros of order higher than 1, the equilibrium and the travelling wave are multi-shock solutions of (★). This situation differs strongly from the case where F has only a first-order zero at both u ± , studied by the authors in a previous paper. In this paper we consider multi-shock solutions of (★). Moreover, we improve some results of the previous paper, allowing a larger ball of initial data.