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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

We prove optimal embeddings in the subcritical case of the inhomogeneous Sobolev spaces built-up over function spaces in Rn with K−monotone and rearrangement invariant norm into another rearrangement invariant function spaces. The investigation is based on pointwise and integral estimates of the rearrangement or the oscillation of the rearrangement of f in terms of the rearrangement of the derivatives of f . Mathematics subject classification (2010): 46E35.

A conjecture of Grenander and Szego for the traces of Toeplitz matrices generated by the Poisson-Charlier polynomials is proved. The Poisson-Charlier polynomials {pm(x)}, m > 0, are defined on the set X of all nonnegative integers [6, p. 34]. This is a complete system of orthonormal polynomials in the space 12 = 12(X, j(x)), where 1(x) = e-aax(x!)-1 , a > 0, is the Poisson weight: EgX=0 Pk (X)Pm (x)j(x) = Jkm Let q(x) be a real almost periodic function (in the Bohr sense) on X. Then the matrix

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.