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.



Relationship between LlogL and Hardy spaces
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 $H(\Re\varphi)=\Im\varphi$, $H(\Im\varphi)=-\Re\varphi$.


