All Questions

Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...

The problem Consider a $C²$ function $f: X \to \mathbb{R}$, for some compact set $X \subset \mathbb{R}^d$ with $C^1$ boundary, say $\partial X$. I am only interested in $d\in \{2,3\}$. Then, consider ...

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary and $f$ be a harmonic function on $\Omega.$ It is known that $$ \limsup_{\varepsilon\rightarrow0^{+}}\intop_{\partial\Omega_{...

Suppose I have some measurable function $f : B_1(0) \to [0,1)$, which is pointwise very small, i.e. $\|f\|_{\infty} << 1$. I'm looking to construct some kind of dyadic cube decomposition or ...