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There is an $f$ such that $f\ge 0$, $Hf\ge 0$ however $Hf^{p}$ contains negative values for some $p\ge 1$. Take $$ f = \begin{bmatrix} 727 & 200 & 163 & 234 & 429 & 448 & 437 & 6 \end{bmatrix}^T $$ The vector $Hf$ is positive, however $(Hf^{8/7})_{100}\approx -3.35$. Here the subscript $100\in \mathbb F_2^n$ is the 5th index in lexicographic order.

There is an $f$ such that $f\ge 0$, $Hf\ge 0$ however $Hf^{p}$ contains negative values for some $p\ge 1$. Take $$ f = \begin{bmatrix} 727 & 200 & 163 & 234 & 429 & 448 & 437 & 6 \end{bmatrix}^T $$ The vector $Hf$ is positive, however $(Hf^{8/7})_{100}\approx -3.35$. Here the subscript $100\in \mathbb F_2^3$ is the 5th index in lexicographic order.

There is such an $f$ such that $f\ge 0$, $Hf\ge 0$ however $Hf^{p}$ contains negative values. Take $$ f = \begin{bmatrix} 727 & 200 & 163 & 234 & 429 & 448 & 437 & 6 \end{bmatrix}^T $$ The vector $Hf$ is positive, however $(Hf^{8/7})_{100}\approx -3.35$. Here the subscript $100\in \mathbb F_2^n$ is the 5th index in lexicographic order.

There is an $f$ such that $f\ge 0$, $Hf\ge 0$ however $Hf^{p}$ contains negative values for some $p\ge 1$. Take $$ f = \begin{bmatrix} 727 & 200 & 163 & 234 & 429 & 448 & 437 & 6 \end{bmatrix}^T $$ The vector $Hf$ is positive, however $(Hf^{8/7})_{100}\approx -3.35$. Here the subscript $100\in \mathbb F_2^n$ is the 5th index in lexicographic order.

The answer is a yes also. Consider the vector $$ x = \begin{bmatrix} 727 & 200 & 163 & 234 & 429 & 448 & 437 & 6 \end{bmatrix}^T $$ The vector $Hx$ is positive, however $(Hx^{8/7})_{100}\approx -3.35$. Here the subscript $100\in \mathbb F_2^n$ is the 5th index in lexicographic order.

There is such an $f$ such that $f\ge 0$, $Hf\ge 0$ however $Hf^{p}$ contains negative values. Take $$ f = \begin{bmatrix} 727 & 200 & 163 & 234 & 429 & 448 & 437 & 6 \end{bmatrix}^T $$ The vector $Hf$ is positive, however $(Hf^{8/7})_{100}\approx -3.35$. Here the subscript $100\in \mathbb F_2^n$ is the 5th index in lexicographic order.