Estimating using ellipticity I am trying to estimate, in either direction, the following integral:
$\int_{B} a^{i,j} X_j D_ig dx$ . We are integrating over a unit ball, $B$, and $[a^{i,j}]$ is positive. 
I have tried to use ellipticity, but it didn't work. I was wondering if some one can show me:
a. How to use ellipticity to estimate the integral, or
b. Is there any other inequality that can be useful to estimating the integral.
Thanks in advance.
 A: Estimating above is just a matter of Cauchy-Schwarz (if you also assume $a^{ij}$ is symmetric). Since $a^{ij}$ is positive, you can define the matrix $[b] = [a]^{1/2}$, and rewrite 
$$ \int_B a^{ij}X_jD_ig dx = \int_B \sum_k b^{ik}b^{kj}X_jD_ig dx $$
apply Cauchy-Schwarz to the terms $V^k = b^{ik}D_ig$ and $W^k = b^{kj}X_j$ you get
$$ \left\lvert\int_B a^{ij}X_jD_ig dx\right\rvert \leq \left(\int_B V^kV^k dx\right)^{1/2}\left(\int_B W^kW^k dx\right)^{1/2} = \left(\int_B a^{ij}X_jX_i dx\right)^{1/2} \left(\int_B a^{ij}D_ig D_jg dx\right)^{1/2} $$
In general: Cauchy-Schwarz can be used for any positive semi-definite symmetric bilinear form.
One direction of the ellipticity definition can then let you control the RHS by a constant factor times
$$ \left(\int_B |X|^2 dx\right)^{1/2} \left(\int_B  |Dg|^2 dx\right)^{1/2} $$

Another thing you can do, if you know the behaviour of $u$ on the boundary of $B$, is to integrate by parts:
$$ \int_{B} a^{ij}X_j D_i g dx = \int_{\partial B} a^{ij}X_j n_i g d\sigma - \int_{B} g D_i( a^{ij}X_j ) dx $$
which can be useful if you know something about the derivatives of the vector $a^{ij}X_j$. $n_i$ in the first term on the right hand side is just the unit out-ward normal on $\partial B$. 

In general you cannot estimate from below. Ellipticity does not rule out the possibility that $X_j$ and $D_i g$ are pointwise orthogonal relative to the bilinear form $a^{ij}$. 

If you say more precisely what kinds of estimates you are looking for, or what possible additional properties about $X_j$ and $D_i g$ that you have, I may be able to give more precise answers. 
