Kenneth Kuttler

45.3. INTRINSIC NORMS 1499

where dxi refers to the first i entries and dyn−i refers to the remaining entries. From Lemma45.3.9, the complicated expression above equals

1C (n,θ , p)

∑i=1

∫∞

−∞

∫Rn−i

∫Ri

∫Rn−1

1(t2 + |s|2

) 12 (n+pθ)

|u(x1, · · · ,xi−1,xi + t,yi+1, · · · ,yn) − u(x1, · · · ,xi,yi+1, · · · ,yn)|p

dsdxidyn−idt

Now Fubini this to get

1C (n,θ , p)

∑i=1

∫Rn−i

∫Ri

∫Rn−1

∫∞

−∞

1(t2 + |s|2

) 12 (n+pθ)

|u(x1, · · · ,xi−1,xi + t,yi+1, · · · ,yn) − u(x1, · · · ,xi,yi+1, · · · ,yn)|p

dtdsdxidyn−i

Changing the variable in the inside integral to t = yi− xi,this equals

1C (n,θ , p)

∑i=1

∫Rn−i

∫Ri

∫Rn−1

∫∞

−∞

1((yi− xi)

2 + |s|2) 1

2 (n+pθ)

|u(x1, · · · ,xi−1,yi,yi+1, · · · ,yn) − u(x1, · · · ,xi,yi+1, · · · ,yn)|p

dyidsdxidyn−i

Next let

(s1, · · · ,sn−1)

≡ (y1− x1, · · · ,yi−1− xi−1,xi+1− yi+1, · · · ,xn− yn)

where the new variables of integration in the integral corresponding to ds are y1, · · · ,yi−1and xi+1, · · · ,xn. Then changing the variables, the above reduces to

1C (n,θ , p)

∑i=1

∫Rn−i

∫Ri

∫Rn−1

∫∞

−∞

|x−y|(n+pθ)

|u(x1, · · · ,xi−1,yi,yi+1, · · · ,yn) − u(x1, · · · ,xi,yi+1, · · · ,yn)|p

dyidy1 · · ·dyi−1dxi+1 · · ·dxndx1 · · ·dxidyi+1 · · ·dyn

Then if you Fubini again, it reduces to the expression

1C (n,θ , p)

∑i=1∫

∫Rn

|u(x1, · · · ,xi−1,yi,yi+1, · · · ,yn) − u(x1, · · · ,xi,yi+1, · · · ,yn)|p

|x−y|(n+pθ)dxdy

45.3. INTRINSIC NORMS 1499where dx; refers to the first i entries and dy,_; refers to the remaining entries. From Lemma45.3.9, the complicated expression above equalscmon yl. be bo ewlu(x1,-°° M1, Xi +t, Vi41,°°° i. _ *+el) Xi, Vi4+1,°°° Yn)?dsdxjdy,_;dtNow Fubini this to getlyROE 7B es FF ye(?+\s”)\u(x1,-°° Xi-1, Xi +t, Vi41,°°° Yn) —_ u(xX{,-°° Xi, Vi+15°°° Yn)?dtdsdx;dyy_iChanging the variable in the inside integral to t = y; — x;,this equalsLhe hiher tl.(n, ole 8a} P) 2h — n-i i Jipn-l 4 (n+p@)- (ix)? +|s!”)|u(x1,-°° »Xi-15 Vis Vi+15°°* Yn) _ u(x|,-°° Xi, Vi+1,°°° Yn)?dy;dsdxjdyn_iNext let(Si,-° Sn—1)S (YX RL WX, XEHI Vit 11 Xn — Yn)where the new variables of integration in the integral corresponding to ds are y,,--- ,yj-1and x;41,°*: ,Xn. Then changing the variables, the above reduces to1 ycue? Yhelebert. |x — arrlu(x1,°°° »Xi-1, Vi, Vi4+15°°° 5Yn) ~ U(x1,°°* Mis Vit15° °° Yn)?dyjdy, +++ dyi—1dxj41 +++ AXpdx, ---dxidyiz1+++dynThen if you Fubini again, it reduces to the expression1 1C(n,0,p) 2 Li=1|u(x1,-°° »Xi—15,Yi,sVi+1,°°° Yn) _ u(X1,°°° Xi, Vit15°°° al tedn n (n+p®) yRv JR Ix—y|