Relationship between LlogL and Hardy spaces

Question

I think that for positive, one-dimensional, periodic functions, the following statement is true:

$$ f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}), $$ where $$ LlogL=\{f\in L^1\,s.t.\,\int_{-\pi}^\pi f(x)\max\{\log(f(x)),0\}dx<\infty\}, $$ $$ H^1=\{f\in L^1\,s.t.\,Hf\in L^1\}, $$ and $Hf$ is the Hilbert tranform of $f$.

Question 1: Am I right?

Question 2: Is the following inequality true? $$ \|f\|_{L^1}+\|H f\|_{L^1}\leq c\int_{-\pi}^\pi f(x)\max\{\log(f(x)),0\}dx $$

PD: References are very welcome.

Answer 1

The implication $\implies$ is true, see, e.g., P.Koosis, Introduction to $H^p$ spaces, section Zigmund's $L\log L$ theorem. The converse is proved there only for positive functions. In general, it is not true. Take an outer function $\varphi$ with modulus $w\in L^1$ (namely, for $u=\log w\in L^1$, define $\varphi=\exp(u+iHu)$; then $\varphi\in H^1$). If $w\log w\not\in L^1$, then at least one of $\Re\varphi, \Im\varphi$ does not belong to $L\log L$. At the same time, we have $H(\Re\varphi)=\Im\varphi$, $H(\Im\varphi)=-\Re\varphi$.