Related papers: Singular integrals along variable codimension one …
In this paper we investigate the boundedness of classical operators, namely the Hardy-Littlewood maximal operator, fractional integral operators, and Calderon-Zygmund operators, on generalized weighted Morrey spaces and generalized weighted…
The reappearance of a sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be present in a recently introduced class of bilinear pseudodifferential operators…
In theory and practice of inverse problems, linear operator equations $Tx=y$ with compact linear forward operators $T$ having a non-closed range $\mathcal{R}(T)$ and mapping between infinite dimensional Hilbert spaces plays some prominent…
Let $\L$ be a Schr\"odinger operator of the form $\L=-\Delta+V$ acting on $L^2(\mathbb R^n)$, $n\geq3$, where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ Let ${\rm BMO}_{{\mathcal{L}}}(\RR)$…
This is a continuation of recent work on the general definition of pseudo-differential operators of type $1,1$, in H\"ormander's sense. Continuity in $L_p$-Sobolev spaces and H\"older--Zygmund spaces, and more generally in Besov and…
A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the…
The local maximal inequality for the Schr\"{o}dinger operators of order $\a>1$ is shown to be bounded from $H^s(\R^2)$ to $L^2$ for any $s>\frac38$. This improves the previous result of Sj\"{o}lin on the regularity of solutions to…
Let $T$ be a bilinear Calder\'on-Zygmund singular integral operator and $T^*$ be its corresponding truncated maximal operator. For any $b\in\text{BMO}(\mathbb {R}^n)$ and $\vec{b}=(b_1,\ b_2)\in\text{BMO}(\mathbb {R}^n)\times\text…
We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic…
It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate them usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient…
Consider the multidimensional Bessel operator $$B f(x) = -\sum_{j=1}^N \left(\partial_j^2 f(x) +\frac{\alpha_j}{x_j} \partial_j f(x)\right), \quad x\in(0,\infty)^N. $$ Let $d = \sum_{j=1}^N \max(1,\alpha_j+1)$ be the homogeneous dimension…
Given a hypersurface $S\subset \mathbb{R}^{2d}$, we study the bilinear averaging operator that averages a pair of functions over $S$, as well as more general bilinear multipliers of limited decay and various maximal analogs. Of particular…
The late time limit of the power spectrum for heavy (principal series) fields in de Sitter space yields a series of polynomial terms with complex scaling dimensions. Such scaling behavior is expected to result from an associated operator…
Motivated by the study of the maximal operator for the Schr\"{o}dinger equation on the one-dimensional torus $ \mathbb{T}^1 $, it is conjectured that for any complex sequence $ \{b_n\}_{n=1}^N $, $$ \left\| \sup_{t\in [0,N^2]}…
Let $\nu = (\nu_1, \ldots, \nu_n) \in (-1/2, \infty)^n$, with $n \ge 1$, and let $\Delta_\nu$ be the multivariate Bessel operator defined by \[ \Delta_{\nu} = -\sum_{j=1}^n\left( \frac{\partial^2}{\partial x_j^2} - \frac{\nu_j^2 -…
We prove $L^p$ bounds, $\frac{d^2 + 4d + 2}{(d+1)^2} < p < 2(d+1)$, for maximal linear modulations of singular integrals along paraboloids with frequencies in certain subspaces of $\mathbb{R}^{d+1}$, for $d \geq 2$. This generalizes…
We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of…
We consider complex resonances for discrete and continuous Schr\"odinger operators, and we show that the resonances of discrete models converge to resonances of continuous models in the continuum limit. The potential is supposed to be a sum…
We extend the recent results concerning boundedness of the maximal regularity operator on tent spaces. This leads us to develop a singular integral operator theory on tent spaces. Such operators have operator-valued kernels. A seemingly…
Motivated by the results of Korry and Kinnunen and Saksman, we study the behaviour of the discrete fractional maximal operator on fractional Hajlasz spaces, Hajlasz-Besov and Hajlasz-Triebel-Lizorkin spaces on metric measure spaces. We show…