Usbrth

I am wrestling with the decay of this integral:

$$I := int_mathbbR^3_+ nabla_x^k Gamma(x-y,frac12)(xi(y)f(y))dy,$$

where $f:mathbbR^3_+ rightarrow mathbbR^3$ is a smooth vector field that vanishes on the boundary of the half-space $mathbbR^3_+$, satisfying a decay condition $|f(y)|leqfrac1y$ for all $y$. Here $xi(y) in C_c^infty(mathbbR^3)$ is chosen to be a cut-off function with $xi(y)=1$ on the unit disk centered at the origin.

More importantly, the heat kernel $Gamma$ is defined by
$$ Gamma(z,t)=frac1(4pi t)^3/2e^-frac^24t$$
for all $(z,t)inmathbbR^3 times (0,infty)$.

I am expecting that the quantity $I$ can be controlled by $$C frac1)^2,$$

or possibly, more precisely by $$Ce^-x^2/4$$ for some constant $C$ which is independent of $x$. But I am struggling to prove it rigorously. It's frustrating!!

Could you help me out with this problem, please?

My strategy has worked only to a small extent as follows:

Divide the domain of integral into two cases $|x-y|geq frac12|x|$ and $|x-y|<frac12|x|$ so that $I$ can be decomposed into the sum $I = I_1 + I_2$. Then, for the first case, we have
$$ I_1 leq C e^-fracx4 int_K fracy-xy dy leq C (|x|+1)e^-fracx4,$$
where $K$ is the compact support of $xi$. This seems not optimal but I can take that temporarily because it is enough to obtain polynomial decay.

The second case geometrically implies that $|y|>frac12|x|$, which yields $$ I_2 leq C |x|frac1xint_K e^-frac^22dy leq C int_K e^-fracx4dy leq C e^-fracx4,$$
as desired.

Please, suggest a better idea to investigate the decay or allow me to improve my previous bound for the first case!!

Note: $x$ is an arbitrarily given point in $mathbbR^3_+$.