# fa.functional analysis – Additivity of squared Schatten \$p\$-norm with respect to spatial partition

Consider a Hilbert-Schmidt operator $$A$$ on $$L^2(mathbb R^d)$$ with integral kernel $$A(x,y)$$. Let $$Omegasubset mathbb R^d$$ and $$1_{Omega}(x)$$ denote its characteristic function as well as the corresponding multiplication operator. Note that by the triangle inequality
$$lVert ArVert_2 leq lVert 1_{Omega} A rVert_2 + lVert 1_{Omega^c} A rVert_2,$$
where $$lVert cdot rVert_2$$ is the Hilbert-Schmidt norm. By the nice property that we can express $$lVert A rVert_2$$ as the $$L^2 times L^2$$ norm of its kernel we also have the (in-)equality
begin{align} lVert A rVert_2^2 & = int_{mathbb R^d} int_{mathbb R^d} lvert A(x,y)rvert^2 , dx , dy \ & = int_{Omega} int_{mathbb R^d} lvert A(x,y)rvert^2 , dx , dy + int_{Omega^c} int_{mathbb R^d} lvert A(x,y)rvert^2 , dx , dy \ & = lVert 1_Omega A rVert_2^2 + lVert 1_{Omega^c} A rVert_2^2 end{align}
with squares on both sides. My question: Does this generalize to higher Schatten $$p$$-norms, $$p>2$$? That is, does
$$lVert A rVert_p^2 leq lVert 1_Omega ArVert_p^2 + lVert 1_{Omega} ArVert_p^2$$ hold? If not, does anyone have a good counterexample? Thanks in advance!