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.