I have the following problem:
Let's say $\Omega\subset\mathbb{R}^2$ is a bounded open connected set with Lipschitz boundary and $f\colon\Omega\rightarrow\mathbb{R}$ is a function such that $f\in L^{1+\eta}(l)$ which means that $$ \left( \int_{l} \! |f(x)|^{1+\eta} \, \mathrm{d}s \right)^{\frac{1}{1+\eta}} < \infty \, , $$ where $l$ is any line segment parallel to $x_2$ axis and lying in $\Omega$.
Under what relatively weak assumptions on $f$ will this be true $$ \sup_{l} \int_{l} \! |f(x)|^{1+\eta} \, \mathrm{d}s < \infty \, ? $$