Sobolev Spaces

For any integer n ≥ 0, the Sobolev space Hⁿ(R) is defined to be the set of functions f which are square-integrable together with all their derivatives of order up to n:

f ∈ Hⁿ(R) ⇐⇒ ∫_-∞^∞ [f² + ∑_k=1ⁿ (d^kf/dx^k)² dx ≤ ∞

This is a linear space, and in fact a Hilbert space with norm given by:

∥f∥_Hⁿ = ∫_-∞^∞ [f² + ∑_k=1ⁿ (d^kf/dx^k)²) dx]^1/2

It is a standard fact that this norm of f can be expressed in terms of the Fourier transform fˆ (appropriately normalized) of f by:

∥f∥²_Hⁿ = ∫_-∞^∞ [(1 + y²)ⁿ |fˆ(y)|² dy

The advantage of that new definition is that it can be extended to non-integral and non-positive values. For any real number s, not necessarily an integer nor positive, we define the Sobolev space H^s(R) to be the Hilbert space of functions associated with the following norm:

∥f∥²_H^s = ∫_-∞^∞ [(1 + y²)^s |fˆ(y)|² dy —– (1)

Clearly, H⁰(R) = L²(R) and H^s(R) ⊂ H^s′(R) for s ≥ s′ and in particular H^s(R) ⊂ L²(R) ⊂ H^−s(R), for s ≥ 0. H^s(R) is, for general s ∈ R, a space of (tempered) distributions. For example δ(k), the k-th derivative of a delta Dirac distribution, is in H^{−k−1/2</sup−ε}(R) for ε > 0.

In the case when s > 1/2, there are two classical results.

Continuity of Multiplicity:

If s > 1/2, if f and g belong to H^s(R), then fg belongs to H^s(R), and the map (f,g) → fg from H^s × H^s to H^s is continuous.

Denote by C_bⁿ(R) the space of n times continuously differentiable real-valued functions which are bounded together with all their n first derivatives. Let Cⁿ_b0(R) be the closed subspace of C_bⁿ(R) of functions which converges to 0 at ±∞ together with all their n first derivatives. These are Banach spaces for the norm:

∥f∥C_bⁿ = max_0≤k≤n sup_x |f^(k)(x)| = max_0≤k≤n ∥f(k)∥ C⁰_b

Sobolev embedding:

If s > n + 1/2 and if f ∈ H^s(R), then there is a function g in Cⁿ_b0(R) which is equal to f almost everywhere. In addition, there is a constant c_s, depending only on s, such that:

∥g∥_Cbⁿ ≤ c_s ∥f∥H^s

From now on we shall always take the continuous representative of any function in H^s(R). As a consequence of the Sobolev embedding theorem, if s > 1/2, then any function f in H^s(R) is continuous and bounded on the real line and converges to zero at ±∞, so that its value is defined everywhere.

We define, for s ∈ R, a continuous bilinear form on H^−s(R) × H^s(R) by:

〈f, g〉= ∫_-∞^∞ (fˆ(y))’ gˆ(y)dy —– (2)

where z’ is the complex conjugate of z. Schwarz inequality and (1) give that

|< f , g >| ≤ ∥f∥_H^−s∥g∥_H^s —– (3)

which indeed shows that the bilinear form in (2) is continuous. We note that formally the bilinear form (2) can be written as

〈f, g〉= ∫_-∞^∞ f(x) g(x) dx

where, if s ≥ 0, f is in a space of distributions H^−s(R) and g is in a space of “test functions” H^s(R).

Any continuous linear form g → u(g) on H^s(R) is, due to (1), of the form u(g) = 〈f, g〉 for some f ∈ H^−s(R), with ∥f∥_H^−s = ∥u∥_(H^s)′, so that henceforth we can identify the dual (H^s(R))′ of H^s(R) with H^−s(R). In particular, if s > 1/2 then H^s(R) ⊂ C⁰_b0 (R), so H^−s(R) contains all bounded Radon measures.

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