The coordinate-direction form of the Slobodeckij seminorm

The polar identity and the directional notation. Replace $g$ by a Borel function equal to it almost everywhere; such a function is obtained by replacing the measurable sets in simple approximations by Borel sets modulo null sets. For every fixed increment its difference integral is unchanged, and the double integral is unchanged by Tonelli. Work with that Borel representative below. Substitute $y=x+h$ in [F1] and use translation invariance [F3] to write, for a.e. fixed $h$, ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }g(x)-g(x+h){\mathrm{\mid }}^{p}dx=D_g(\mathrm{\mid }h\mathrm{\mid },h\mathrm{/}\mathrm{\mid }h\mathrm{\mid })$; Tonelli [F2] then gives $[g{]}_{s,p}^p={\int }_{{\mathbb{R}}^d}{D}_g(\mathrm{\mid }h\mathrm{\mid },h\mathrm{/}\mathrm{\mid }h\mathrm{\mid })\mathrm{\mid }h{\mathrm{\mid }}^{-d-sp}dh$. The functions $(h,\omega )\mapsto D_g(h,\omega )$ and $F_g$ are Borel measurable: the first is an iterated integral of the nonnegative measurable function $(x,h,\omega )\mapsto \mathrm{\mid }g(x+h\omega )-g(x){\mathrm{\mid }}^p$ over $x$, and the second is the section integral of $D_g(h,\omega )h^{-1-sp}$, both covered by Tonelli's measurability clause [F2]. Polar coordinates [F4] turn the display into $[g{]}_{s,p}^p={\int }_{S^{d-1}}{F}_g(\omega )d\sigma (\omega )$, and $F_g(e_i)={\int }_0^{\mathrm{\infty }}{h}^{-1-sp}{\int }_{{\mathbb{R}}^d}\mathrm{\mid }g(x+h{e}_i)-g(x){\mathrm{\mid }}^{p}dxdh$ is the $i$-th summand in the statement; all quantities are nonnegative extended integrals, so no convergence hypothesis is needed.

The coordinate increment decomposition. Fix a nonnegative smooth probability density $\rho$ supported in $B(0,1\mathrm{/}2)$ and put $\psi (z)=\rho (z-e_i)$. For every $h>0$, $x$ and $z=e_i+t$, the triangle inequality gives $\mathrm{\mid }g(x+h{e}_i)-g(x){\mathrm{\mid }}^p\le 2^{p-1}(\mathrm{\mid }g(x+h{e}_i)-g(x+hz){\mathrm{\mid }}^p+\mathrm{\mid }g(x+hz)-g(x){\mathrm{\mid }}^p)$. Average this nonnegative inequality against $\psi (z)dz$ and integrate in $x$. Translation invariance and Tonelli yield $D_g(h,e_i)\le 2^{p-1}(\int \rho (t)D_g(h\mathrm{\mid }t\mathrm{\mid },t\mathrm{/}\mathrm{\mid }t\mathrm{\mid })dt+\int \psi (z)D_g(h\mathrm{\mid }z\mathrm{\mid },z\mathrm{/}\mathrm{\mid }z\mathrm{\mid })dz)$, with the zero increment interpreted as zero. This argument remains valid for infinite integrals and requires no integral of $g$ itself.

The sphere comparison. For a bounded nonnegative compactly supported Borel function $\phi$ and a nonnegative Borel measurable $F$ on $S^{d-1}$, Tonelli [F2] and polar coordinates [F4], applied to the nonnegative Borel function $z\mapsto \phi (z)F(z\mathrm{/}\mathrm{\mid }z\mathrm{\mid })$ with the value $0$ prescribed at the origin, give ${\int }_{{\mathbb{R}}^d}\phi (z)F(z\mathrm{/}\mathrm{\mid }z\mathrm{\mid })dz={\int }_{S^{d-1}}F(\omega )({\int }_0^{\mathrm{\infty }}\phi (r\omega )r^{d-1}dr)d\sigma (\omega )\le C_{\phi }{\int }_{S^{d-1}}Fd\sigma$, where $C_{\phi }:={\sup }_{\omega \in S^{d-1}}{\int }_0^{\mathrm{\infty }}\phi (r\omega )r^{d-1}dr$ satisfies $C_{\phi }\le M{R}^d\mathrm{/}d<\mathrm{\infty }$ whenever $\phi \le M$ and its support lies in $B(0,R)$; this applies in particular to $\phi =\rho$ and to $\phi =\psi$.

The upper comparison. Fix $\omega \in S^{d-1}$ and $r>0$ and put $p_k:=r{\sum }_{i\le k}{\omega }_i{e}_i$ for $k=0,\dots ,d$, so that $p_0=0$ and $p_d=r\omega$. Telescoping along the polygonal path $p_0,\dots ,p_d$ gives $g(x+r\omega )-g(x)={\sum }_{k=1}^d(g(x+p_k)-g(x+p_{k-1}))$, and each summand is the translate by $p_{k-1}$ of the increment $g(\cdot +r{\omega }_k{e}_k)-g(\cdot )$; by convexity and [F3], $D_g(r,\omega )\le d^{p-1}{\sum }_{k=1}^d{D}_g(r\mathrm{\mid }{\omega }_k\mathrm{\mid },\mathrm{sgn}({\omega }_k)e_k)$, the term being zero when ${\omega }_k=0$. Multiplying by $r^{-1-sp}$, integrating in $r$, and substituting $h=r\mathrm{\mid }{\omega }_k\mathrm{\mid }$ in the $k$-th term (using $\mathrm{\mid }{\omega }_k{\mathrm{\mid }}^{sp}\le 1$) and using $F_g(-e_k)=F_g(e_k)$ by translation invariance gives $F_g(\omega )\le d^{p-1}{\sum }_{k=1}^d{F}_g(e_k)$ for every $\omega$. Integrating over $S^{d-1}$ with [F4] and step 1.1 yields the upper comparison $[g{]}_{s,p}^p\le d^{p-1}\sigma (S^{d-1}){\sum }_{i=1}^d{F}_g(e_i)$.

The lower comparison. Multiply step 1.2 by $h^{-1-sp}$ and integrate in $h$. Tonelli and the substitutions $u=h\mathrm{\mid }t\mathrm{\mid }$, $u=h\mathrm{\mid }z\mathrm{\mid }$ give $F_g(e_i)\le 2^{p-1}(\int \rho (t)\mathrm{\mid }t{\mathrm{\mid }}^{sp}{F}_g(t\mathrm{/}\mathrm{\mid }t\mathrm{\mid })dt+\int \psi (z)\mathrm{\mid }z{\mathrm{\mid }}^{sp}{F}_g(z\mathrm{/}\mathrm{\mid }z\mathrm{\mid })dz)$. The factors $\mathrm{\mid }t{\mathrm{\mid }}^{sp}$ and $\mathrm{\mid }z{\mathrm{\mid }}^{sp}$ are bounded on the respective supports. Step 1.3 therefore bounds the right-hand side by $C(d,p,s){\int }_{S^{d-1}}{F}_{g}d\sigma =C(d,p,s)[g{]}_{s,p}^p$.

Conclusion. Summing the lower comparison of step 2.2 over $i=1,\dots ,d$ and combining it with the upper comparison of step 2.1 gives $c_1{\sum }_i{F}_g(e_i)\le [g{]}_{s,p}^p\le c_2{\sum }_i{F}_g(e_i)$ with $c_1,c_2$ depending only on $d,p,s$; in particular the two sides are finite simultaneously, since a finite constant times $+\mathrm{\infty }$ is $+\mathrm{\infty }$. Writing out $F_g(e_i)$ as the coordinate-direction integral of the statement and using $1+p\theta =p$ at $s=\theta =1-1\mathrm{/}p$ gives the displayed equivalence and the weight $h^{-p}$.