Fourier and Pseudodifferential Operators in Differential Equations
Summary
Fourier and pseudodifferential operators lie at the heart of modern analysis of differential equations. The Fourier transform recasts differential problems into algebraic ones by converting derivatives into multiplication by frequency variables. Building on this, Fourier integral operators extend the transform to oscillatory integrals with phase functions, enabling the construction of parametrices for hyperbolic and dispersive equations. Pseudodifferential operators generalise differential operators by allowing symbols that depend smoothly on both position and frequency, classified in Hörmander’s hierarchy of symbol classes. These operators provide a microlocal framework to track singularities, establish regularity estimates in Sobolev, Besov and Triebel–Lizorkin spaces, and to analyse the propagation of waves, quantum scattering and inverse problems. Recent advances have refined mapping properties on weighted spaces, sharpened endpoint estimates, and broadened the scope of symbol regularity. Practical applications range from seismic imaging to signal processing and from numerical analysis of non-linear PDEs to control theory.
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Fourier and Pseudodifferential Operators in Differential Equations publication trend
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Technical terms
Fourier transform: An integral transform converting a function of space into a function of frequency, exchanging differentiation for multiplication.
Fourier integral operator: An operator defined by an oscillatory integral with a phase function and amplitude, used to construct parametrices for hyperbolic PDEs.
Pseudodifferential operator: A generalisation of differential operators whose action is defined via a symbol depending on both position and frequency.
Symbol class (Hörmander class): A classification of functions (symbols) by their growth and smoothness properties in position and frequency variables.
Sobolev space: A function space characterised by integrability and differentiability measured in the Lp sense via the Fourier transform.
Microlocal analysis: A technique that studies the propagation of singularities of solutions to PDEs by combining localisation in both space and frequency.
References
- Regularity of Fourier integral operators with amplitudes in general Hörmander classes. Analysis and Mathematical Physics (2021).
- On L2-boundedness of Fourier integral operators. Journal of Inequalities and Applications (2020).
- Rough Pseudodifferential Operators on Hardy Spaces for Fourier Integral Operators II. Journal of Fourier Analysis and Applications (2022).
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