mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in the field of harmonic analysis, an oscillatory integral operator is an

integral operator An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms invol ...

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 Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...

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 In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...

. 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 (38 ...

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) transformati ...

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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

) and in physics. Properties of oscillatory integral operators have been studied by

Elias Stein Elias Menachem Stein (January 13, 1931 – December 23, 2018) was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathematics, Emeritus, at Princeton University, whe ...

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. Introd ...

) 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 Medal ...

in his paper on

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 ...

s: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

. 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 vector s ...

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

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...

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