Let $u$ be a solution of the heat equation $$u\_t - u\_{xx} = 0, \quad t>0, x \in \mathbb{R}$$
with initial data $u(0,\cdot) = u\_0$.
Fix $\alpha >0$. How can I estimate (without using explicitly the heat kernel)
$$\sup\_{t>0}\int\_{\mathbb{R}} t^\alpha |u\_x|^2 \ dx,$$
in terms of the initial data? Could you point out a reference where such an estimate is obtained?


Is it fair to call what we obtain a decay estimate?



Decay estimate for the heat equation: $\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2\ dx$
I'm sorry for the late answer, but joined MathOverflow just this week. The Fourier Splitting method, developed by María Elena Schonbek in the 80's asserts that "decay is determined by the low frequencies of the solutions" for many dissipative linear and nonlinear equations (heat, fractional heat, Navier-Stokes, dissipative quase-geostrophic, amongst many others). For $v\_0 \in L^2 (\mathbb{R} ^n)$, assume there is a unique $-\frac{n}{2} < r^{\ast} (v\_0) < \infty$, called the decay character, such that


\begin{equation}
r^{\ast} (v\_0) = \lim \_{\rho \to 0} \rho ^{-2r-n} \int \_{B(\rho)} \large|\widehat{v}\_0 (\xi) \large|^2 \, d \xi
\end{equation}


i.e., the decay character measures the "order" of $v\_0$ at the origin in frequency space (see Bjorland and Schonbek, Adv. Diff. Eq. 2009; Niche and M.E. Schonbek, J. London Math. Soc. 2015). Then, for a large family of dissipative operators, the decay character provides sharp estimates for decay of linear (and some nonlinear) equations. In the specific case of the heat equation in $\mathbb{R}^n$, we have that for the solution $v$ with initial data $v\_0$


\begin{equation}
C\_1 (1 + t)^{- \left( \frac{n}{2} + r^{\ast} \right)} \leq \Vert v(t) \Vert \_{L^2} ^2 \leq C\_2 (1 + t)^{- \left(\frac{n}{2} + r^{\ast} \right)},
\end{equation}
for some absolute constants $C\_1, C\_2 > 0$, see Theorem 6.5 in Bjorland and M.E. Schonbek (op.cit.) and Theorem 2.10 in Niche and M.E. Schonbek (op. cit.).


Now take $u\_0 \in \dot{H} ^1 (\mathbb{R})$, i.e. $\partial\_x u\_0 \in L^2 (\mathbb{R})$. For $r^{\ast} = r^{\ast} (\partial\_x u\_0)$ the estimate above implies


\begin{equation}
C\_1 (1 + t)^{- \left( \frac{1}{2} + r^{\ast} \right)} \leq \Vert u(t) \Vert \_{\dot{H}^1} ^2 \leq C\_2 (1 + t)^{- \left( \frac{1}{2} + r^{\ast} \right)}.
\end{equation}
If $u\_0 \in H^s (\mathbb{R}), s > 0$, then $u\_0 \in L^2 (R)$ as well and from Theorem 2.11 in Niche and M.E. Schonbek we have $r^{\ast} (\partial\_x u\_0) = s + r^{\ast} (u\_0)$ so 


\begin{equation}
C\_1 (1 + t)^{- \left( \frac{1}{2} + s + r^{\ast} (u\_0) \right)} \leq \Vert u(t) \Vert \_{\dot{H}^1} ^2 \leq C\_2 (1 + t)^{- \left( \frac{1}{2} + s + r^{\ast} (u\_0) \right)}.
\end{equation}


Note that the decay character does not always exists, there are $v\_0 \in L^2 (\mathbb{R}^n)$ which oscillate a lot near the origin (in frequency space) for which $r^{\ast} (v\_0)$ is not defined, see the article by Brandolese in SIAM J. Math. Anal. 2016 to find a precise characterization in terms of Besov spaces for when the decay character exists and for other results. 


