Is there a constant $C$ such that if $u:[0,1]^2\to \mathbb{R}$ is harmonic with $u\in L^\infty(\partial [0,1]^2)$ (if you prefer you can also assume $\|u\|_\infty = 1$ on the boundary and $u$ smooth on the part of the boundary where $y>0$) one has:

$$ \left| \int_0^{1/2}u(t,0)dt-\int_{1/2}^1u(t,0)dt \right| \le C\int_{[0,1]^4}|u(x_1,y_1)-u(x_2,y_2)|\,dx_1dx_2dy_1dy_2? $$ As a statistician, I look at the left-hand side as a Haar-wavelet coefficient and at the right hand side as an $l^1$-measure of the variance of the distribution $u$.

I have looked at some PDE books and notes but nothing similar popped out. For harmonic functions on $[0,1]$ this works (you want to bound $|u(0)-u(1)|$ from above) just because $u$ is linear.