2
$\begingroup$

The classical heat semigroup on $\mathbb{R}$ is given by $$ W_t f(x)=\frac{1}{t}\int_{\mathbb{R}}e^{-\pi (\frac{x-y}{t})^2}f(y)dy, \qquad t>0. $$ Then the Littlewood-Paley vertical square function is defined by setting $$ g(f)(x)=\|\partial_t W_t f(x)\|_{L^2((0,\infty),\, tdt)}, \qquad x\in\mathbb{R}. $$ It is known that $g$ is bounded on $L^p(\mathbb{R})$ for $p\in (1,\infty)$ and of weak type (1,1), but it is $\textbf{not}$ bounded on $L^\infty(\mathbb{R})$.

My question is: what is the simplest argument showing that $g$ is not bounded on $L^\infty(\mathbb{R})$.

I would appreciate any hints!

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ Take $f(x) = \int_{-\infty}^{x} e^{-t^{2}}dt \in L^{\infty}$. Now its square function is $+\infty$. $\endgroup$
    – Paata Ivanishvili
    Commented May 1, 2020 at 23:50
  • $\begingroup$ Paata, thanks so much! :] $\endgroup$
    – Tony419
    Commented May 2, 2020 at 1:15

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .