# A parabolic Morrey-Sobolev inequality

Let $$B \subset \mathbb R^2$$ be the unit ball and $$T>0.$$ Let $$u \in W^{2,1}_p(B \times [0,T]),$$ that is $$u \in L^p(B \times [0,T])$$ and we also have, $$\partial_t u, \nabla u, \nabla^2 u \in L^p(B \times [0,T]).$$ Here $$\nabla$$ denotes differentiation in the spacial direction only.

I am looking for a proof of the following result:

If $$p>4,$$ then $$\nabla u \in C^{\alpha,\alpha/2}(\overline B \times [0,T])$$ and there is $$C_p > 0$$ such that $$\sup_{\overline B \times [0,T]} |\nabla u| + \sup_{(x,t) \neq (y,s) \in \overline B \times [0,T]} \frac{|\nabla u(x,t) - \nabla u(y,s)|}{|x-y|^{\alpha} + |t-s|^{\alpha/2}} \leq C_p \left( \lVert u \rVert_{L^p(B \times [0,T])} + \lVert \nabla u \rVert_{L^p(B \times [0,T])} + \lVert \nabla^2 u \rVert_{L^p(B \times [0,T])} + \lVert \partial_t u \rVert_{L^p(B \times [0,T])} \right),$$ or in more compact (but possibly non-standard) notation $$\lVert \nabla u \rVert_{C^{\alpha,\alpha/2}(\overline B \times [0,T])} \leq C_p \lVert u \rVert_{W^{2,1}_p(B \times [0,T])},$$ where $$\alpha = \left(1-\frac4p\right).$$

This result was stated without a proof or reference as lemma 3.1 in the paper "The existence of heat flow of $$H$$-systems" by Chen and Levine. I presume it's well-known, but I've been unable to find a reference for it.

Some thoughts: My initial idea was to try to adapt one of the proofs of the usual Morrey-Sobolev embedding $$W^{1,p} \hookrightarrow C^{1-n/p}$$ separately in $$x$$ and $$t,$$ perhaps by breaking it up as, $$|\nabla u(x,t) - \nabla u(y,s)| = |\nabla u(x,t) - \nabla u(x,s)| + |\nabla u(x,s) - \nabla u(y,s)|.$$ The exponent $$\alpha$$ suggests we apply Sobolev embedding in the $$x$$ variable with exponent $$p/2,$$ but it's not clear to do this uniformly in $$t.$$ Moreover this naiive approach obviously fails because we only have information about $$\partial_t u$$ and not its gradient. So some interpolation argument would be needed, which is where I'm stuck.

• Maybe the following part could be useful for you. I have a similar problem and I was looking for some classic reference. Appendix E of Deterministic and Stochastic optimal control, W.H. Fleming & R. Rishel have a part on Holder estimates of parabolic type. – Cuoredicervo Jun 14 at 8:33
• @Cuoredicervo Thank you for the reference, I will take a look when I have the time. I did end up finding a proof based on establishing a parabolic Poincare-Sobolev inequality (using the standard Rellich compactness theorem) and verifying a Camanato-type characterisation of Holder continuity which I've been meaning to post as an answer, but I haden't gotten around to doing so yet. – ktoi Jun 21 at 13:30