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We use the homogeneous version of the Sobolev embedding theorem in three dimensions:Here homogeneous Sobolev space ${\dot{H}}^1$ is the completion of compactly supported smooth functions in the L2 norm of the gradient, identified with its $L^6$ representative. Apply this Sobolev inequality both to $\varphi$ and to its spatial derivatives.

LetThe Plancherel theorem identifies the L2 norm of the Hessian matrix with $\Vert \varphi {\Vert }_{{\dot{H}}^2}$, because ${\sum }_{i,j}{\xi }_i^2{\xi }_j^2=|\xi {|}^4$. In particular,Differentiating the defocusing semilinear wave equation gives $\square ({\partial }_i\varphi )=3{\varphi }^2{\partial }_i\varphi$. By the Holder inequality with exponents $3$ and $6$, the preceding Sobolev inequality, and conservation of the positive wave energy,Use the inhomogeneous wave energy estimate simultaneously for the three spatial derivatives. It givesThe Gronwall inequality therefore yields $B(t)\le D\exp (CEt)$. We may take the continuous, locally bounded functionThe constant is universal; the dependence on the initial data is only through $D$ and $E$. The H2 bound for the defocusing cubic wave equation holds on every existing smooth interval, without assuming the global conclusion. If $D=0$ or $E=0$, the zero solution satisfies the same estimate.

The wave map scaling symmetry is ${\varphi }_{\lambda }(t,x)=\varphi (t/\lambda ,x/\lambda )$. Its conserved wave map energy isThus energy is subcritical for $n=1$, critical for $n=2$, and supercritical for $n\ge 3$. The scaling-critical homogeneous Sobolev spaces for perturbations of a constant map are ${\dot{H}}^{n/2}\times {\dot{H}}^{n/2-1}$. These relations constitute wave map energy and criticality.

Under ${\varphi }_{\lambda }(t,x)=\varphi (t/\lambda ,x/\lambda )$, the conserved wave map energy scales as ${\lambda }^{n-2}E$. Energy is subcritical for $n=1$, critical for $n=2$, and supercritical for $n\ge 3$. The critical derivative index of the homogeneous Sobolev space is $s_c=n/2$. Small energy alone is not an appropriate general small-data regularity hypothesis in supercritical dimensions.