On the Nazarov--Shcheglova Conjecture for Sharp Sobolev Inequalities: The Case (n,p)=(3,2)

Abstract: For integers $n&gt;k\geq0$ and $1\leq p,q\leq\infty$, let $λ<em>3(n,k,p,q)$ denote the optimal constant in the one-dimensional Sobolev inequality [ |u^{(k)}|{L^q(0,1)} \leq λ3(n,k,p,q) |u^{(n)}|{L^p(0,1)}, \qquad u\in\mathring W_p^n(0,1). ] Nazarov and Shcheglova \cite{NazarovShcheglova} conjectured that [ λ_3(n,1,p,1) = 2λ_3(n,0,p,\infty), \qquad n\geq2,\quad 1\leq p\leq\infty, ] and that the corresponding extremal functions coincide and are symmetric about the midpoint of the interval. We prove this conjecture for $(n,p)=(3,2)$ and, in particular, obtain [ λ_3(3,1,2,1) = \frac{1}{32\sqrt5}. ] The proof is based on a reduction to an operator norm problem with an additional moment constraint. After rescaling to $(-1,1)$, this constraint leads to orthogonality with respect to quadratic polynomials. We extend the inverse of the second derivative through the corresponding orthogonal projection and analyze the adjoint operator by an odd--even decomposition. The odd component is controlled by a positive Gram kernel, while the even component is treated by a weak-$L<sup>2$ estimate. The equality cases yield the characterization of all extremal functions.