Deducing finiteness of the integrand from finiteness of integral

Question

Suppose we know that: $$ \int_{[-\pi, \pi]^d} \, g(k) \, h^2(k) \, dk \leq\int_{ [-\pi, \pi]^d} \, h(k) \, dk < \infty. $$ where $h(k) = \sin(\frac{k_1}{2})^2 + \sum_{i=2}^{d} \sin(k_i)^2.$ Assuming $d > 2$, is it possible from this to deduce that, $$ \int_{[-\pi, \pi]^d} \, g(k) < \infty? $$

Answer 1

The answer to your question is: no, at least for $d<5$.

Consider $g(x)=\frac{\int_{ [-\pi, \pi]^d} \, h(k) \, dk}{h^2(x)}$. Then clearly the desired conditions are satisfied. However, for $x\in\mathbb{R}^d$ such that $x\simeq 0$ one has $$ h(x)\simeq |x|^2, $$ since $$ \sin y\simeq y, \quad \textrm{for}\quad y\sim 0,\, y\in\mathbb{R}. $$ Consequently, $$ g(x)\simeq |x|^{-4}, \quad \textrm{for}\quad x\sim 0, $$ thus $g$ is not integrable for $d<5$.

Answer 2

Not in general. Take the case $d=1$ and $g(k) = 1/h(k) =1/\sin^2(k)$.