## A Phragmen-Lindelof and Saint Venant principle in harmonic analysis

- Date: 03/30/2007

Peter Lax (Courant Institute)

University of Washington

Let S be a linear space of vector valued functions u(y) on the

half-line whose values belong to some Banach space. We suppose that S

is translation invariant; that is, if u(y) belongs to S, so does u(y+t)

for all t>0. S is called 'interior compact' if the unit ball of S in

the L1 norm over a y-interval [a,b] is precompact in the L1 norm over any proper subinterval [a',b'].

THEOREM: Any function u(y) in a translation invariant, interior compact space that is L1

on y>0 decays exponentially as y tends to infinity, and has an

asymptotic expansion near infinity in terms of exponential functions in

y contained in S.

This result can be applied to solutions of elliptic equations in a half cylinder.

10th Anniversary Speaker Series 2007