Fractional Sobolev spaces and interpolation

For an open set $\Omega\subset\mathbb R^n$, $0<s<1$, and $1\le p<\infty$, define the Gagliardo seminorm by integrating $|u(x)-u(y)|^p/|x-y|^{n+sp}$ over all ordered pairs in the domain. The fractional space Wˢ,ᵖ consists of Lᵖ functions with finite seminorm, with full norm formed by adding the pth power of the Lᵖ norm. Constants have zero seminorm, so the order-zero term is again necessary. The singular kernel does not make every smooth function fail: a Lipschitz difference contributes a compensating power of the distance when s is below one.

The energy is nonlocal because points interact across arbitrary separations. On a bounded domain, the intrinsic seminorm only counts pairs inside the domain. Extending by zero and computing a whole-space seminorm also counts pairs crossing the boundary, producing an additional boundary-layer cost. These are not interchangeable definitions. Even a constant function has zero intrinsic seminorm but its zero extension can have a singular interaction near the boundary. State whether a PDE uses intrinsic differences, an exterior condition, or an interpolation space before comparing its energy to an integer-order norm.

For p equal to two on the whole space and a unitary Fourier transform, the squared Gagliardo seminorm is a positive dimensional constant times $\int|\xi|^{2s}|\widehat u(\xi)|^2\,d\xi$. The full norm is equivalent to the Bessel-potential Hˢ norm with weight $(1+|\xi|^2)^s$. It is equivalent rather than numerically identical with the convention above unless the normalization constant is included. For general p, Slobodeckij spaces and Bessel-potential spaces are not automatically the same; using Hˢ language for the Hilbert case avoids silently identifying different scales.

On a bounded Lipschitz domain and for $1<p<\infty$, real interpolation gives $W^{s,p}(\Omega)=(L^p(\Omega),W^{1,p}(\Omega))_{s,p}$ with equivalent norms. The K-functional measures the cheapest decomposition of a function into a part small in Lᵖ and a part controlled in W¹,ᵖ, with a parameter weighting the latter. Integrating these costs across scales creates fractional regularity. Replacing W¹,ᵖ by its zero-trace subspace changes the boundary behavior of the interpolation space; at critical boundary exponents extra conditions can appear. It is unsafe to identify all spaces with a subscript zero just because their names look similar.

A one-dimensional jump can belong to Wˢ,ᵖ when sp is below one. Near the interface, points on opposite sides give a kernel integral with the same convergence threshold as $\int_0^\varepsilon r^{-sp}\,dr$. At sp equal to one it diverges logarithmically, and above one it also diverges. Thus fractional regularity can admit jumps excluded from first-order Sobolev spaces, but only below the appropriate exponent threshold. This is a local interface calculation, not a claim that every discontinuous function has finite fractional energy.

On a bounded Lipschitz domain, for $0<s<1$ and $1<p<\infty$, fractional embedding has conjugate exponent $np/(n-sp)$ when $sp<n$, while $sp>n$ gives a Hölder representative with exponent $s-n/p$. Compactness is again subcritical under these bounded-domain hypotheses. Smooth-domain trace and zero-extension thresholds are linked to sp crossing one, but the equality case needs special care. For example, the constant function on an interval lies in its intrinsic H¹ᐟ² space while its whole-line zero extension has infinite H¹ᐟ² seminorm. Boundary conventions are part of the mathematical problem and cannot be inferred from an interior formula alone.