Is this function on antisymmetric 3-tensors (over $\ell^2$) bounded? $\sum_{i,j,k,l,m,n}A_{ijk}A_{imn}A_{ljn}A_{lmk}$

Question

Let $A$ be an antisymmetric 3-tensor over a sequence space (for simplicity, assume finite sequences):

$$A_{ijk}=-A_{jik}=-A_{ikj}\in\mathbb R,\quad(i,j,k)\in\mathbb N^3$$

$$\lVert A\rVert^2=\sum_{i,j,k}|A_{ijk}|^2<\infty$$

and consider the function

$$f(A)=\sum_{i,j,k,l,m,n}A_{ijk}A_{imn}A_{ljn}A_{lmk}$$

$$=\sum_{j,k,m,n}\left(\sum_iA_{ijk}A_{imn}\right)\left(\sum_lA_{ljn}A_{lmk}\right).$$

Is there a constant $C>0$ such that $|f(A)|\leq C\lVert A\rVert^4$ for all $A$?

I tried using the Cauchy-Schwarz inequality in different ways.

Answer 1

First, $$ \left|\sum_i A_{ijk}A_{imn}\right| \le \sum_i |A_{ijk}|\cdot |A_{imn}| \le \left(\sum_i |A_{ijk}|^2\right)^{1/2} \left(\sum_i |A_{imn}|^2\right)^{1/2} . $$ Define $$ a_{jk}^2:= \sum_i |A_{ijk}|^2. $$ Then $$ |f(A)|\le \sum_{j,k,m,n}a_{jk}a_{km}a_{mn} a_{nj} \le \frac12\left(\sum_{j,k,m,n}a_{jk}^2a_{mn}^2 +\sum_{j,k,m,n}a_{km}^2 a_{nj}^2\right) = \left(\sum_{jk}a_{jk}^2\right)^2 = \|A\|^4. $$