Related papers: Bilinear Sparse Domination for Oscillatory Integra…

We carry on the study of Fourier integral operators of H{\"o}rmander's type acting on the spaces $(\mathcal{F}L^p)_{comp}$, $1\leq p\leq\infty$, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the…

We say that a discrete set $X =\{x_n\}_{n\in\dN_0}$ on the half-line $$0=x_0 < x_1 <x_2 <x_3<... <x_n<... <+\infty$$ is sparse if the distances $\Delta x_n = x_{n+1} -x_n$ between neighbouring points satisfy the condition $\frac{\Delta…

We study singular Schr\"odinger operators on a finite interval as selfadjoint extensions of a symmetric operator. We give sufficient conditions for the symmetric operator to be in the $n$-entire class, which was defined in our previous…

We consider the boundedness of the multilinear pseudo-differential operators with symbols in the multilinear H\"{o}rmander class $S_{0,0}$. The aim of this paper is to discuss smoothness conditions for symbols to assure the boundedness…

This paper investigates the boundedness of a broad class of operators within the framework of generalized Morrey-Banach function spaces. This class includes multilinear operators such as multilinear $\omega$-Calder\'{o}n-Zygmund operators,…

Let $\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\mu_{\Omega}$ be the higher-dimensional Marcinkiewicz integral defined by $$\mu_\Omega(f)(x)= \Big(\int_0^\infty\Big|\int_{|x-y|\leq…

Let $ T_{P } f (x) = \int e ^{i P (y)} K (y) f (x-y) \; dy $, where $ K (y)$ is a smooth Calder\'on-Zygmund kernel on $ \mathbb R ^{n}$, and $ P$ be a polynomial. We show that there is a sparse bound for the bilinear form $ \langle T_P f, g…

In this paper we prove the sharp boundedness for a fractional type operator given by a kernel that satisfy a $L^{\alpha,r'}$-H\"ormander conditions and a fractional size condition, where $0<\alpha<n$ and $1< r'\leq \infty$. To prove this…

How to establish some specific quantitative weighted estimates for the generalized commutator of multilinear fractional singular integral operator $\mathcal{T}_{\eta}^{{\bf b}}$ is the focus of this paper, which is defined by…

In this paper, we consider the boundedness from $H^{1} \times L^{\infty}$ to $L^{1}$ of bilinear Fourier integral operators with non-degenerate phase functions and amplitudes in $BS_{1,0}^{-n/2}$. Our result gives an improvement of…

We represent a general bilinear Calder\'on-Zygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so called sparse T1…

We introduce a new sparse $T1$ theorem that estimates the dual pair associated with a Calderon-Zygmund operator by a sub-bilinear form supported on a sparse family of cubes. The main result in the paper improves previous sparse $T1$…

The paper contains a survey of a class of Fourier integral operators defined by symbols with tempered weight. These operators are bounded (respectively compact) in $L^2$ if the weight of the amplitude is bounded (respectively tends to $0$).

We consider the discrete Schr\"odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of…

For bilinear Fourier multipliers that contain some oscillatory factors, boundedness of the operators between Lebesgue spaces is given including endpoint cases. Sharpness of the result is also considered.

In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{\rho,\delta}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{\rho,\delta}^m) \subset…

For each $ d \geq 2$, the Hilbert transform with a polynomial oscillation as below satisfies a $ (1, r )$ sparse bound, for all $ r>1$ $$ H _{ \ast } f (x) = \sup _{\epsilon } \Bigl\lvert \int_{|y| > \epsilon} f (x-y) \frac { e ^{2 \pi i y…

Let $\mathcal{L}$ be the Schr\"odinger operator with potential $V$, that is, $\mathcal L=-\Delta+V$, where it is assumed that $V$ satisfies a reverse H\"older inequality. We consider weighted Morrey-Campanato spaces $BMO_{\mathcal…

The goal of this expository paper is to give a self-contained introduction to sparse domination. This is a method relying on techniques from dyadic Harmonic Analysis which has received a lot of attention in recent years. Essentially, it…

This paper is devoted to the proof of boundedness of bilinear smooth square functions. Moreover, we deduce boundedness of some bilinear pseudo-differential operators associated with symbols belonging to a subclass of $BS^0_{0,0}$.