Trying to solve a conjecture in differential geometry, I am leaded to the following problem (which may seem weird to a analyst). I wonder if anyone know some techniques that happen to solve it.

Let $f$ be $\mathbb{Z}^2$-periodic positive-real-valued function on $\mathbb{R}^2$, which you may assume to be as soomth as you like. Let $h$ be another such function without the positivity assumption. We suppose that both $f$ and $h$ are unit vectors in $L^2([0,1]\times[0,1])$ (defined with the usual Euclidean measure), and they are orthogonal.

**Problem:** Is it always possible to write $h$ as a sum of two functions $a,b\in L^2([0,1]\times[0,1])$ with the following conditions:

(1) $\int_0^1 f(x,y)a(x,y)dx=0$ for any $y$

(2) $\int_0^1 f(x,y)b(x,y)dy=0$ for any $x$

(3) $\int_0^1\int_0^1a(x,y)b(x,y)dxdy\geq 0$

**Remark**: Using Fourier expansion, the proof is easy if $f(x,y)$ is a product such as $\cos(2\pi mx)\sin(2\pi ny)$ (although this $f$ is not positive). In fact in this case we may take $a$ and $b$ to be orthogonal, instead of having inner product $\geq 0$.
But for general $f$, the analysis of Fourier coefficients seems hard.

**Addendum**: As fedja mentioned below, the above assertion is not always true. So I feel like to mention here another assertion which a consequence of the above problem. You can also try to prove it directed. This would provide the last ingredient that I need
to solve some conjecture.

For unit vectors $f, h_1, h_2, h_3, h_4\in L^2([0,1]\times[0,1])$ such that each $h_i$ is orthogonal to $f$, we define $$ I(f,h_1,h_2,h_3,h_4)=\int_0^1[\sum_{i=1}^4(\int_0^1fh_idy)^2]^{1/2}dx\times \int_0^1[\sum_{i=1}^4(\int_0^1fh_idx)^2]^{1/2}dy $$ we consider $I$ as a functional depending on a point $f$ in the unit sphere of $L^2([0,1]\times[0,1])$ and four unit tangent vectors of the unit sphere at $f$. Then the assertion is that the maximal value of $I$ is $2$.

(The value $2$ is achieved when, for instance, $f=1$ and $h_1=\sqrt{2}\cos(2\pi x)$, $h_2=\sqrt{2}\sin(2\pi x)$, $h_3=\sqrt{2}\cos(2\pi y)$, $h_4=\sqrt{2}\sin(2\pi y)$.)

A first attempt to prove this is to apply Cauchy-Schwarz two times, one can then get $I\leq 4$, which is not enough. Then I came up with the idea: If the $h_i$'s can be written as $a_i+b_i$ satisfying above conditions (1), (2) and (3), then the proof goes through.