On Bernstein type quantitative estimates for Ornstein non-inequalities

For the sequence of multi-indexes $\{\alpha_i\}_{i=1}^{m}$ and $\beta$ we study the inequality $$\|D^{\beta} f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} \|D^{\alpha_j}f\|_{L_1(\mathbb{T}^d)},$$ where $f$ is a trigonometric polynomial of degree at most $N$ on $d$-dimensional torus. Assuming some natural geometric property of the set $\{\alpha_j\}\cup\{\beta\}$ we show that $$K_{N}\geq C \left(\ln N\right)^{\phi},$$ where $\phi<1$ depends only on the set $\{\alpha_j\}\cup\{\beta\}$.Comment: Presentation improved, typos correcte

On the analytic version of the Mitiagin-DeLeeuw-Mirkhil non-inequality on bi-disc

Using the method of Rudin-Shapiro polynomials we prove the analytic version of the Mitiagin-DeLeeuw-Mirkhil non-inequality for complex partial differential operators with constant coefficients on bi-disc

A conditional regularity result for p-harmonic flows

We prove an $\varepsilon$-regularity result for a wide class of parabolic systems $$u_t-\text{div}\big(|\nabla u|^{p-2}\nabla u) = B(u, \nabla u)$$ with the right hand side $B$ growing like $|\nabla u|^p$. It is assumed that the solution $u(t,\cdot)$ is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo-Nirenberg inequality which has been used earlier in an elliptic context by T. Rivi\`ere and the last named author.Comment: To appear in NoDEA. Referee suggestions implemente