The purpose of this paper is to determine the optimal order of
decrease of the L2
norm for the following degenerate and singular oscillatory integral operators
defined
by
formula here
for f∈C∞0(ℝ),
where ψ(x, y) is a smooth compactly supported function
on ℝ2,
S(x, y) is a homogeneous polynomial of degree
n
formula here
and K(x, y) is a distribution kernel which
satisfies the
following hypotheses:
K is a C2 function away from the diagonal
and the estimates
formula here
hold for 0<μ<1 and i=1, or 2.
This paper basically uses the methods of [6], which
are very general and can be
widely used. The main idea of these methods is that the sizes of operator
norms
after the decomposition are estimated in different ways and the decomposition
is
then summed back by balancing the estimates of two types. The first type
estimate
takes into account the support sizes of the integrand after the decomposition.
The
second type estimate is delicate, depending on how to keep track of the
distance to
the singular varieties of phase function. In our case, we have to include
the singular
variety arising from distribution kernel as well. The analytic tool of
the second type
estimate is the operator version of the Van der Corput lemma [7].
We notice that
the crux of matters is to locate the singular varieties of phase function
and kernel.
For higher dimensional situations, the problem is very difficult. One expects
that the
powerful theory of algebraic geometry may play a significant role. Here
we would like
to remark that Mather's results [2] may be useful
although the geometrical shapes
of singular varieties are basically undecidable.