arXiv subjects

P. Ivanisvili

Publications and source records attributed to P. Ivanisvili.

3 recordsLinked to original sources

arXiv2022

Strong weighted and restricted weak weighted estimates of the square function

In this note we give a sharp weighted estimate for square function from $L^2(w)$ to $L^2(w)$, $w\in A_2$. This has been known. But we also give a sharpening of this weighted estimate in the spirit of $T1$-type testing conditions. Finally we show that for any weight $w\in A^d_2$ and any characteristic function of a measurable set $\|S_wχ_E\|_{L^{2, \infty}(w^{-1})} \le C \sqrt{[w]_{A^d_2}}\, \|χ_E\|_w$, and this estimate is sharp. So on characteristic functions of measurable sets at least, no logarithmic correction is needed for the weak type of the dyadic square function.The sharp estimate for the restricted weak type is at most $ \sqrt{[w]_{A^d_2}}$.

math.CA↗

arXiv2017

On the failure of lower square function estimates in the non-homogeneous weighted setting

We show that the classical $A_{\infty}$ condition is not sufficient for a lower square function estimate in the non-homogeneous weighted $L^2$ space. We also show that under the martingale $A_2$ condition, an estimate holds true, but the optimal power of the characteristic jumps from $1 / 2$ to $1$ even when considering the classical $A_2$ characteristic. This is in a sharp contrast to known estimates in the dyadic homogeneous setting as well as the recent positive results in this direction on the discrete timenon-homogeneous martingale transforms. Last, we give a sharp $A_{\infty}$ estimate for the $n$-adic homogeneous case, growing with $n$.

math.AP↗

arXiv2014

Hessian of Bellman functions and uniqueness of Brascamp--Lieb inequality

Under some assumptions on the vectors $a_{1},..,a_{n} \in\mathbb{R}^{k}$ and the function $B : \mathbb{R}^{n} \to \mathbb{R}$ we find the sharp estimate of the expression $\int_{\mathbb{R}^{k}} B(u_{1}(a_{1}\cdot x),..., u_{n}(a_{n}\cdot x))dx$ in terms of $\int_{\mathbb{R}}u_{j}(y)dy, j=1,...,n.$ In some particular case we will show that these assumptions on $B$ imply that there is only one Brascamp--Lieb inequality.

math.AP↗