Related papers: Superexponential estimates and weighted lower boun…

For any Calder\'on-Zygmund operator $ T$, any weight $ w$, and $ \alpha >1$, the operator $ T$ is bounded as a map from $ L ^{1} (M _{ L \log\log L (\log\log\log L) ^{\alpha } } w )$ into weak-$L^1(w)$. The interest in questions of this…

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of \cite{Magyar_dyadic}, \cite{Magyar_ergodic} and \cite{MSW}. We combine more precise knowledge of…

We prove weighted estimates for rough bilinear singular integral operators with kernel $$K(y_1, y_2) = \frac{\Omega((y_1,y_2)/|(y_1,y_2)|)}{|(y_1, y_2)|^{2d}},$$ where $y_i \in \mathbb{R}^{d}$ and $\Omega \in L^{\infty}(S^{2d-1})$ with…

In this paper, we will obtain the strong type and weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood-Paley $g$-function and $g^*_\lambda$-function on the weighted Herz spaces $\dot…

We study the two-weighted estimate \[ \bigg\|\sum_{k=0}^na_k(x)\int_0^xt^kf(t)dt|L_{q,v}(0,\infty)\bigg\|\leq c\|f|L_{p,u}(0,\infty)\|,\tag{$*$} \] where the functions $a_k(x)$ are not assumed to be positive. It is shown that for $1<p\leq…

Let $\Omega\subset{\mathbb R}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp(u^2)-1]/\delta^2\in L^1(\Omega)$ if $u\in H^1_0(\Omega)$, where $\delta$ denotes the distance to $\partial\Omega$. The…

We prove Strichartz inequalities for the wave and Schr\"odinger equations on noncompact surfaces with ends of finite area, i.e. with ends isometric to $ \big( (r_0,\infty) \times {\mathbb S}^1 , dr^2 + e^{- 2 \phi (r)}d \theta^2 \big) $…

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness $\omega_\alpha(f,t)_q$ and $\omega_\beta(f,t)_p$ for $0<p<q\le \infty$. A similar problem for the generalized $K$-functionals and…

We prove L^1 --> L^\infty estimates for the linear Schroedinger equation in three dimensions. The potential is assumed to belong to certain L^p spaces, but no pointwise decay estimates and no additional regularity is required.

We consider the $1d$ cubic nonlinear Schr\"odinger equation with a large external potential $V$ with no bound states. We prove global regularity and quantitative bounds for small solutions under mild assumptions on $V$. In particular, we do…

Point estimators may not exist, need not be unique, and their distributions are not parameter invariant. Generalized estimators provide distributions that are parameter invariant, unique, and exist when point estimates do not. Comparing…

We show that any Littlewood--Paley square function $S$ satisfying a minimal local testing condition is dominated by a sparse form, \begin{equation*} \langle (Sf)^2,g \rangle\le C \sum_{I \in \mathscr{S}} \langle \lvert f\rvert\rangle_I^2…

We prove $|x|^{-2}$ decay of the critical two-point function for the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$, in the upper critical dimension $d=4$. This is a statement that the critical exponent $\eta$ exists and is…

We provide a purely local computation of the (elliptic) twisted (by "transpose-inverse") character of the representation \pi=I(\1) of PGL(3) over a p-adic field induced from the trivial representation of the maximal parabolic subgroup. This…

We prove novel (local) square function/Carleson measure estimates for non-negative solutions to the evolutionary $p$-Laplace equation in the complement of parabolic Ahlfors-David regular sets. In the case of the heat equation, the Laplace…

The endpoint Strichartz estimate $\|e^{it\Delta} f\|_{L_t^2 L_x^\infty} \lesssim \|f\|_{L^2}$ is known to be false in two space dimensions. Taking averages spherically on the polar coordinates $x=\rho\omega$, $\rho>0$,…

We extend the classical Lyapunov inequality on the measurable space with infinite measure and on the so-called Grand Lebesgue spaces (GLS). We find also the exact value for correspondent constant. Possible applications: Functional Analysis…

In this paper we extend a Calderon-Zygmund commutator-type estimate. This estimate enables us to prove an embedding result concerning weighted function spaces.

The classical Remez inequality bounds the maximum of the absolute value of a polynomial $P(x)$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z$ of positive measure in $[-1,1]$. It was shown in…

In this paper we show the weak differentiability of the unique strong solution with respect to the starting point $x$ as well as Bismut-Elworthy-Li's derivative formula for the following stochastic differential equation in $\mathbb R^d$: $$…