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.


1 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 $H(\Re\varphi)=\Im\varphi$, $H(\Im\varphi)=-\Re\varphi$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.