Fourier Analysis for PDEs and Sobolev Embedding

Core Idea

The Fourier transform provides a frequency-space characterization of Sobolev spaces: u ∈ H^s(ℝⁿ) if and only if (1+|ξ|²)^s/2û(ξ) ∈ L²(ℝⁿ). This characterization defines fractional Sobolev spaces H^s for any real s, extends the Sobolev embedding theorems to this general setting, and connects PDE regularity to the decay of Fourier coefficients. The theory of pseudodifferential operators and Fourier multipliers grows from this foundation, providing a systematic calculus for studying variable-coefficient and nonlinear PDEs through their behavior in frequency space.

Explainer

The Fourier transform is the natural lens through which to view Sobolev spaces and regularity for PDEs. A function is smooth if and only if its Fourier transform decays rapidly at high frequencies, and a function is in H^s if its Fourier transform decays fast enough that the weighted L² integral ∫(1+|ξ|²)^s|û|²dξ converges. This frequency-space perspective makes the Sobolev embedding theorems geometrically transparent: embedding into continuous functions requires the Fourier transform to be in L¹, which happens when the H^s norm controls enough frequency decay, specifically when s > n/2.

Fractional Sobolev spaces are indispensable in PDE theory. The trace of an H¹ function on a hypersurface belongs to H^1/2—a fractional space that cannot be defined using classical derivatives. Interpolation between integer-order spaces, regularity results for non-integer gains, and the precise characterization of boundary regularity all require the fractional framework. The Fourier definition makes these spaces effortless to work with: H^s is simply the space where the "frequency weight" (1+|ξ|²)^s/2 times û is square-integrable.

Pseudodifferential operators generalize both differential operators and Fourier multipliers. A differential operator P = Σ a_α(x)D^α has symbol p(x,ξ) = Σ a_α(x)ξ^α, and the pseudodifferential operator P(x,D)u = ∫p(x,ξ)û(ξ)e^iξ·xdξ extends this to symbols p(x,ξ) that need not be polynomial in ξ. The calculus of pseudodifferential operators—composition, adjoint, and parametrix construction—provides a systematic machinery for studying elliptic regularity, wave propagation, and spectral theory. The symbol encodes the microlocal behavior: where in phase space (position × frequency) the operator acts.

The Littlewood-Paley decomposition is another Fourier-analytic tool central to modern PDE theory. It decomposes a function into frequency bands: u = Σ_j Δ_j u, where each Δ_j u has Fourier transform supported in an annulus |ξ| ~ 2^j. This decomposition characterizes Sobolev and Besov spaces through sequence-space conditions on the pieces, and it is the main technical tool for proving nonlinear estimates, establishing well-posedness for dispersive equations, and understanding the cascade of energy across scales in fluid dynamics.