In mathematics, in the field of

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an e ...

, an oscillatory integral operator is an integral operator of the form :

T_\lambda u(x)=\int_e^ a(x, y) u(y)\,dy, \qquad x\in\R^m, \quad y\in\R^n,

where the function ''S''(''x'',''y'') is called the phase of the operator and the function ''a(x,y)'' is called the

symbol A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...

of the operator. ''λ'' is a parameter. One often considers ''S''(''x'',''y'') to be real-valued and smooth, and ''a''(''x'',''y'') smooth and compactly supported. Usually one is interested in the behavior of ''T''_λ for large values of ''λ''. Oscillatory integral operators often appear in many fields of mathematics (

analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

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 (or equivariant) transformat ...

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

) and in physics. Properties of oscillatory integral operators have been studied by Elias Stein and his school.

Hörmander's theorem

The following bound on the ''L''² → ''L''² action of oscillatory integral operators (or ''L''² → ''L''²

operator norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Intr ...

) was obtained by

Lars Hörmander Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". Hörmander was awarded the Fields Meda ...

in his paper on Fourier integral operators:L. Hörmander ''Fourier integral operators'', Acta Math. 127 (1971), 79–183. doi 10.1007/BF02392052, https://doi.org/10.1007%2FBF02392052 Assume that ''x,y'' ∈ R^''n'', ''n'' ≥ 1. Let ''S''(''x'',''y'') be real-valued and smooth, and let ''a''(''x'',''y'') be smooth and compactly supported. If

\det_ \frac(x,y)\ne 0

everywhere on the support of ''a''(''x'',''y''), then there is a constant ''C'' such that ''T''_λ, which is initially defined on

smooth functions In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

, extends to a

continuous operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vect ...

from ''L''²(R^''n'') to ''L''²(R^''n''), with the norm bounded by

C \lambda^

, for any ''λ'' ≥ 1: :

\, T_\lambda\, _\le C\lambda^{-n/2}.

References

Microlocal analysis Harmonic analysis Singular integrals Fourier analysis Integral transforms Inequalities