
09/13 5:00pm 
BLOC 220 
Michal Wojciechowski IMPAN, Warsaw 
On yet another analogon of Riesz brothers theorem for Sobolev space
We show that the quotient space BV/W^{1,1} is isomorphic to the space of bounded borel measures. Here BV denotes the space of functions of bounded variation and W^{1,1} the Sobolev space of functions with integrable gradient on regular domain. One can see this as an analogon of Pełczyński's result that dual to the space of C^{1}smooth functions is a separable perturbation of the space of measures. Main ingredients of a proof are G. Alberti rank one theorem and extension/averaging results for Sobolev and BV spaces 

10/26 3:00pm 
BLOC 220 
Jari Taskinen University of Helsinki 
Schauder bases and the decay rate of the heat equation
Joint work with José Bonet (Valencia) and Wolfgang Lusky (Paderborn)
We consider the classical Cauchy problem for the linear heat equation with integrable initial data f = f(x) in the Euclidean space R^N \ni x. As well known, the conventional solution formula implies that for generic f, the supnorm (w.r.t. x) of the solution u(x,t) decays at the speed rate t^{N/2} for large times t. Faster decay rates are possible for special initial data. Our aim is to present a new approach to this phenomenon and also to show that initial data leading to faster decay rates is in a certain sense not so rare.
Accordingly, given a weight function w = w(x) growing rapidly at the infinity, we construct Schauder bases (e_n)_{n=1}^\infty in the Banach space L_w^p (R^N) , 1 < p < \infty or p=1, with the following property: given an arbitrary natural number m, one can find M such that for any initial data f belonging to the closed linear span of (e_n)_{n=M}^\infty , the solution of the Cauchy problem for the heat equation decays at least at the rate t^{m} in the supnorm.
In particular, the subspace of initial data leading to "fast" decay is finite codimensional. Moreover, the mentioned basis can be constructed as a "perturbation" of any given basis. The proof is based on a construction of bases which annihilate an infinite sequence of bounded linear functionals. We also discuss the background of the problem and possible generalizations. 