I'm going through a proof of differentiability of fourier series on the d-dimensional torus and while proving the following inequality:
$$ \sum_{n\in\mathbb{Z}^d}|a_n(f)|\ll_d\sqrt{\|f\|_2^2+\sum_{i=1}^d \|\partial_{e_i}^k f\|_2^2}$$
My main problem is concerned with the use of the symbol/notation $\ll_d$ which I really don't seem to understand how to interpret. The first time it is used in the proof is while concerned with the following, as to say, 'inequality': (quoting from the proof)
$$\sum_{n\in\mathbb{Z}^d}\left(1+\sum_{j=1}^d(2\pi)^{2k}n_j^{2k}\right)|a_n(f)|^2\gg\sum_{n\in\mathbb{Z}^d}(1+\|n\|_2^{2k})|a_n(f)|^2$$ since $\|n\|_2\le\sqrt{d}\cdot\max_{1\le j\le d }|n_j|$ and hence (?) $\|n\|_2^{2k}\ll\sum_{j=1}^d|n_j|^{2k}$
My professor wrote (quoting from the lecture):
$A\ll B :\Leftrightarrow \exists \text{ a constant } k \text{ s.t } A\le k B \text{ where } k=k(\text{stuff}) $
Can someone help me clarify this passage? Thank you in andvace