Oscillatory integral operator
Encyclopedia
In mathematics
, in the field of harmonic analysis
, an oscillatory integral operator is an integral operator of the form
where the function
is called the phase
of the operator
and the function
is called the symbol
of the operator.
is a parameter. One often considers 
to be real-valued and smooth, and 
smooth and compactly supported. Usually one is interested in the behavior of 
for large values of 
.
Oscillatory integral operators often appear in many fields of mathematics (analysis
, partial differential equations, integral geometry
, number theory
) and in physics. Properties of oscillatory integral operators have been studied by E. Stein
and his school.
action of oscillatory integral operators (or 
operator norm
)
was obtained by Lars Hörmander
in his paper on Fourier integral operator
s:
Assume that
, 
.
Let
be real-valued and smooth,
and let
be smooth and compactly supported.
If
everywhere on the support of 
,
then there is a constant
such that 
, which is initially defined on smooth functions, extends to a continuous operator from 
to 
, with the norm
bounded by
, for any 
:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
, an oscillatory integral operator is an integral operator of the form
where the function
is called the phasePhase
-In physics:*Phase , a physically distinctive form of a substance, such as the solid, liquid, and gaseous states of ordinary matter**Phase transition is the transformation of a thermodynamic system from one phase to another*Phase...
of the operator
and the function
is called the symbolSymbol of a differential operator
In mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this...
of the operator.
is a parameter. One often considers to be real-valued and smooth, and smooth and compactly supported. Usually one is interested in the behavior of for large values of .Oscillatory integral operators often appear in many fields of mathematics (analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, partial differential equations, integral geometry
Integral geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant transformations from the space of functions on one geometrical space to the...
, number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
) and in physics. Properties of oscillatory integral operators have been studied by E. Stein
Elias M. Stein
Elias Menachem Stein is a mathematician and a leading figure in the field of harmonic analysis. He is the Albert Baldwin Dod Professor of Mathematics at Princeton University.-Biography:...
and his school.
Hörmander's theorem
The following bound on the action of oscillatory integral operators (or operator normOperator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
)
was obtained by Lars Hörmander
Lars Hörmander
Lars Valter Hörmander is a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". He was awarded the Fields Medal in 1962, and the Wolf Prize in 1988...
in his paper on Fourier integral operator
Fourier integral operator
In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operators as well as classical integral operators as special cases....
s:
Assume that
, .Let
be real-valued and smooth,and let
be smooth and compactly supported.If
everywhere on the support of ,then there is a constant
such that , which is initially defined on smooth functions, extends to a continuous operator from to , with the normOperator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
bounded by
, for any :