A COUNTEREXAMPLE CONCERNING MAXIMAL ESTIMATES FOR SOLUTIONS TO EQUATIONS OF SCHRODINGER TYPE

LP maximal estimates are considered for solutions to an initial value problem for equations of Schrodinger type. Let f belong to the Schwarz space S(R-n) and assume that Omega :R-n --> R is a measurable functio n. We set S(t)f(x) = u(x,t) = (2 pi)(-n) integral(Rn) e(ix-xi),e(it Om ega(xi))(f) over cap d xi, x is an element of R-n, t is an element of R, where (f) over cap denotes the Fourier transform of f, defined by ( f) over cap(xi) = integral(Rn) e(-i xi-x) f(x)dx It then follows that u(x,0) = f(x) and in the case Omega(xi) = \xi\(2) u is a solution to t he Schrodinger equation i partial derivative u/partial derivative t = Delta u. We shall here consider the maximal function Sf(x)= sup(0<t<1 ) \S(t)f(x)\, x is an element of R-n We also introduce Sobolev spaces H-s by setting H-s = {f is an element of I' ; \\f\\H-s < infinity}, s is an element of R, where \\f\\H-s = ( integral(Rn) (1+\xi\(2))s\(f) o ver cap(xi)\(2)d xi)(1/2).