Is the unit ball in $L_p$, $1<p<2$, contained in a "compact perturbation" of the unit ball in $L_2$?

Question

For each pair of functions $x,y\in L_2[0,1]$ let us denote by $x\cdot y$ their pointwise product $$ (x\cdot y)(t)=x(t)\cdot y(t),\quad t\in [0,1]. $$ It belongs to $L_1[0,1]$ due to the Cauchy-Bunyakovsky inequality: $$ x,y\in L_2[0,1]\quad\Longrightarrow\quad x\cdot y\in L_1[0,1]. $$

And for each pair of sets $A,B\subseteq L_2[0,1]$ by $A\cdot B$ we denote the corresponding "element-wise product", $$ A\cdot B=\{x\cdot y; \ x\in A, \ y\in B\}, $$ which is contained in $L_1[0,1]$: $$ A,B\subseteq L_2[0,1]\quad\Longrightarrow\quad A\cdot B\subseteq L_1[0,1]. $$ Let us denote by $\overline{\operatorname{absconv}}(A\cdot B)$ the closed absolutely convex hull of $A\cdot B$ in $L_1[0,1]$.

And for each $p>1$ let $B_p$ be the unit ball in $L_p[0,1]$.

I wonder in which case $B_p$ is contained in a set of the form $\overline{\operatorname{absconv}}(K\cdot B_2)$ where $K$ is a compact set in $L_2[0,1]$: $$ B_p\subseteq \underbrace{\overline{\operatorname{absconv}}(K\cdot B_2)}_{\scriptsize\begin{matrix}\text{closed absolutely convex hull in $L_1[0,1]$}\end{matrix}}. $$ For $p\ge 2$ this is trivially true, since in this case we can take $K=\{1\}$, the set consisting of just one function, the constant identity ($1(t)=1$, $t\in[0,1]$): $$ B_p\subseteq B_2\subseteq \overline{\operatorname{absconv}}B_2= \overline{\operatorname{absconv}}(\{1\}\cdot B_2) $$

But for $1<p<2$ this seems to be not true:

If $1<p<2$ then there is no a compact set $K\subseteq L_2[0,1]$ such that $$ B_p\subseteq \underbrace{\overline{\operatorname{absconv}}(K\cdot B_2)}_{\scriptsize\begin{matrix}\text{closed absolutely convex hull in $L_1[0,1]$}\end{matrix}}. $$

Am I right?

Answer 1

You are right, the answer to the question is no: if $1<p<2$, then there is no compact set $K \subset L_2$ such that $$B_p \subset \overline{\textrm{absconv}} (K \cdot B_2).$$

Assume by contradiction that such $K$ exists, and define $C=\max_K \|f\|_2$. By duality (this is an equivalence) we have $$ \|g\|_q \leq \sup_{f \in K} \|fg\|_2$$ for every $g \in L_\infty$, where $q = \frac{p}{p-1}$ is the conjugate exponent of $p$.

The idea is that if $g$ is independant from $K$, this inequality becomes $\|g\|_q \leq C \|g\|_2$, which is not true because $q>2$.

More details : for convenience let me work with $L_p(\{0,1\}^{\mathbf{N}})$ (dyadic coordinates). Let $\varepsilon>0$ be arbitrary. By compactness, there are $f_1,\dots,f_k$ of $L_2$-norm less than $C$ such that $K$ is contained in the $\varepsilon$-neighborhood of $\{f_1,\dots,f_k\}$, and so the previous inequality implies $$ \|g\|_q \leq \max_i \|f_ig\|_2+\varepsilon \|g\|_\infty.$$

By density we can even assume that each $f_i \colon \{0,1\}^{\mathbf N}\to \mathbf C$ depends on finitely many coordinates, say the first $N$. If $g$ does not depend on the first $N$ coordinates, it is then independant from each $f_i$, $\|f_i g\|_2 = \|f_i\|_2 \|g\|_2 \leq C \|g\|_2$, and $$ \|g\|_q \leq C \|g\|_2+\varepsilon \|g\|_\infty.$$ If $g \in L_\infty$ is arbitrary, by applying this to the function $(\omega_n)_{n \in \mathbf N} \mapsto g((\omega_{n+N})_{n \in \mathbf N})$ we obtain that this inequality holds for every $g$. Making $\varepsilon \to 0$ we obtain $\|g\|_q \leq C \|g\|_2$, a contradiction.