The Gagliardo--Slobodeckij space on Euclidean space

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the measure-theoretic interfaces cited below. Let $d\ge 1$, $0<s<1$, $1\le p<\mathrm{\infty }$ and $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. Lebesgue measure on ${\mathbb{R}}^d$ and its sigma-algebra are those of Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$.

For a Lebesgue measurable $g:{\mathbb{R}}^d\to \mathbb{K}$ put $[g{]}_{s,p}:=({\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\mathrm{\mid }g(x)-g(y){\mathrm{\mid }}^p\mathrm{\mid }x-y{\mathrm{\mid }}^{-d-sp}dxdy{)}^{1\mathrm{/}p}\in [0,+\mathrm{\infty }]\mathrm{.}$ The integrand is read as $0$ on the diagonal $x=y$; the integral is the nonnegative extended integral of The nonnegative Lebesgue integral over the completed product measure $dxdy$ of Tonelli and Fubini for the completed product, with only almost-everywhere section measurability. The Gagliardo-- Slobodeckij space of order $s$ and exponent $p$ is $W^{s,p}({\mathbb{R}}^d;\mathbb{K}):=\{g\in L^p({\mathbb{R}}^d;\mathbb{K}):[g{]}_{s,p}<\mathrm{\infty }\},$ where $L^p({\mathbb{R}}^d;\mathbb{K})$ is the quotient of the measurable functions by almost-everywhere equality (The space $L^p(\mu )$ as the quotient by null functions); its elements are classes, and the space is normed by $\mathrm{\parallel }g{\mathrm{\parallel }}_{W^{s,p}({\mathbb{R}}^d)}:=\mathrm{\parallel }g{\mathrm{\parallel }}_{L^p({\mathbb{R}}^d)}+[g{]}_{s,p}\mathrm{.}$ Write $[g{]}_{s,p}$ also for the value on a class, and call $[\cdot {]}_{s,p}$ the Slobodeckij seminorm.

Three conventions are part of the definition. First, the diagonal $\{(x,y):x=y\}$ is a Lebesgue-null subset of ${\mathbb{R}}^d\times {\mathbb{R}}^d$ (its section at every $x$ is a single point, so Tonelli gives product measure zero), and on the diagonal the integrand is declared zero; thus the convention changes the integrand only on a null set and does not affect the integral. Second, the integral is a nonnegative extended integral, so $[g{]}_{s,p}=+\mathrm{\infty }$ is allowed and $W^{s,p}$ is a set of $L^p$ classes with finite seminorm; the difference $g(x)-g(y)$ is taken between representatives, and changing representatives on a null set changes the integrand only on a null subset of the product. Third, the definition is stated on classes but representative independence is not assumed here: it is the first clause of Well-definedness of the Slobodeckij seminorm and norm ↗, the item that establishes that this definition is well posed on $L^p$ classes. That the expression $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{W^{s,p}}$ is a genuine norm, and that $W^{s,p}$ is a vector space, are likewise proved there, not asserted as part of the definition.

On this page the case used is the trace exponent $s=\theta :=1-\frac{1}{p}\in (0,1),$ which is available exactly for $1<p<\mathrm{\infty }$. For that exponent the weight simplifies to $\mathrm{\mid }x-y{\mathrm{\mid }}^{-d-p\theta }=\mathrm{\mid }x-y{\mathrm{\mid }}^{-d-(p-1)}$. The endpoint $p=1$ is treated separately on this page and is never described by a space $W^{0,1}$.

• Constants have zero seminorm: if $g=c$ almost everywhere then the integrand vanishes identically, so $[c{]}_{s,p}=0$. Adding the $L^p$ term therefore removes the ambiguity only where constants are themselves $L^p$ classes. On ${\mathbb{R}}^d$ with $1\le p<\mathrm{\infty }$ no nonzero constant lies in $L^p({\mathbb{R}}^d)$, since Lebesgue measure is infinite, so on the whole space the $L^p$ term is already sensitive to the difference between a constant and the zero class; the definiteness statement is nevertheless proved, not assumed, in Well-definedness of the Slobodeckij seminorm and norm ↗.
• The weight $h\mapsto \mathrm{\mid }h{\mathrm{\mid }}^{-d-sp}$ is not locally integrable at the origin: its radial integral there is ${\int }_0^1{r}^{-1-sp}dr=+\mathrm{\infty }$. Its tail is integrable, since ${\int }_1^{\mathrm{\infty }}{r}^{-1-sp}dr=1\mathrm{/}(sp)$. With density declared zero on the diagonal, the weighted measure $\mathrm{\mid }x-y{\mathrm{\mid }}^{-d-sp}dxdy$ is sigma-finite (exhaust by bounded sets with $\mathrm{\mid }x-y\mathrm{\mid }\ge 1\mathrm{/}k$), but is not finite on compact neighbourhoods of the diagonal. It has the same null sets as Lebesgue product measure because its density is finite and strictly positive off the null diagonal. Finite seminorms depend on cancellation in $g(x)-g(y)$; no equivalence with another function space is asserted here.

Schikorra, Section V.1, printed pp. 96-97, defines $[f{]}_{W^{s,p}}$ by the double integral and $W^{s,p}$ with the sum norm. Gagliardo, definition (1.3) on printed pp. 288-289, uses the equivalent incremental-quotient description on a compact boundary (extended here to ${\mathbb{R}}^d$); Kampanou, printed pp. 18-19, uses the same modular seminorm for the boundary trace space.