# $L\log L$ and Hardy space on the upper half plane

Set $$\mathbb{T}$$ the unit circle, $$dm$$ the Lebesgue measure on $$\mathbb{T}$$ and $$\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$$ the upper half plane.

It is well-known that the Cauchy transform $$K$$ defined by $$(Kf)(z)= \int_{\mathbb{T}} \frac{f(\xi)}{1-z\bar{\xi}} dm(\xi), \quad z\in \mathbb{D}$$ (or equivalently the harmonic conjugate or the Hilbert transform) is bounded from the Zygmund space $$L\log L(\mathbb{T})=\left\{f\in L^1(\mathbb{T})\,s.t.\,\sup_{0\leq r<1}\int_{\mathbb{T}} \left|f(r\xi)\right| \log^+(\left|f(r\xi)\right|)dm(\xi)<\infty\right\}$$ to the classical (complex) Hardy space $$H^1(\mathbb{D})$$ (see the definition).

Now, on the upper plane $$\mathbb{C}^+$$, the Cauchy transform $$K$$ is defined by $$(Kf)(z)= \frac1{2i\pi}\int_{\mathbb{R}} \frac{f(x)}{x-z} dx, \quad z\in \mathbb{\mathbb{C}^+}.$$ Is it bounded from $$L\log L(\mathbb{R})=\left\{f\in L^1(\mathbb{R})\,s.t.\,\int_{\mathbb{R}} \left|f(x)\right|\log^+(\left|f(x)\right|)dx<\infty\right\}$$ to $$H^1(\mathbb{C}^+)$$ (see the definition) ?

I am very grateful for any idea/comment on this matter. References are very welcome.

I am not sure but I would think that your $$L\log L$$ space should be replaced by the space of functions $$f$$ such that $$\int_R f(x)\,dx=0$$ (this is the necessary condition) and $$\int_R |f(x)|\left[\log^+ |f(x)| + \log^+ |x|\right]\, dx <\infty.$$
D. Cruz-Uribe, SFO and A. Fiorenza, $$L\log L$$ results for the maximal operator in variable $$L^p$$ spaces, Trans. Amer. Math. Soc. 361 (2009), 2631-2647, doi:10.1090/S0002-9947-08-04608-4.