For any integer n ≥ 0, the Sobolev space H^{n}(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 ∈ H^{n}(R) ⇐⇒ ∫_{-∞}^{∞} [f^{2} + ∑_{k=1}^{n} (d^{k}f/dx^{k})^{2} dx ≤ ∞

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

∥f∥_{Hn} = ∫_{-∞}^{∞} [f^{2} + ∑_{k=1}^{n} (d^{k}f/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 + y^{2})^{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}_{Hs} = ∫_{-∞}^{∞} [(1 + y^{2})^{s} |fˆ(y)|^{2} dy —– (1)

Clearly, H^{0}(R) = L^{2}(R) and H^{s}(R) ⊂ H^{s′}(R) for s ≥ s′ and in particular H^{s}(R) ⊂ L^{2}(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_{b}^{n}(R) the space of n times continuously differentiable real-valued functions which are bounded together with all their n first derivatives. Let C^{n}_{b0}(R) be the closed subspace of C_{b}^{n}(R) of functions which converges to 0 at ±∞ together with all their n first derivatives. These are Banach spaces for the norm:

∥f∥C_{b}^{n} = max_{0≤k≤n} sup_{x} |f^{(k)}(x)| = max_{0≤k≤n} ∥f(k)∥ C^{0}_{b}

Sobolev embedding:

If s > n + 1/2 and if f ∈ H^{s}(R), then there is a function g in C^{n}_{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∥_{Hs } —– (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∥_{(Hs)}′, 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) ⊂ C^{0}_{b0} (R), so H^{−s}(R) contains all bounded Radon measures.