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Let $L$ be a homogeneous sublaplacian on a 2-step stratified Lie group $G$ of topological dimension $d$ and homogeneous dimension $Q$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is bounded on $L^p$…

We improve the $L^p(\mathbb{R}^n)$ bounds on Stein's square function to the best-known range of the Fourier restriction problem when $n\geq4$. Applications including certain local smoothing estimates are also discussed.

The main aim of this paper is to investigate $\left(H_{p},L_{p}\right)$- type inequalities for the the maximal operators of N\"orlund logaritmic means, for $0<p<1.$

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special…

In this article, we address endpoint issues for the bilinear spherical maximal functions. We obtain borderline restricted weak type estimates for the well studied bilinear spherical maximal function…

We define a discrete version of the bilinear spherical maximal function, and show bilinear $l^{p}(\mathbb{Z}^d)\times l^{q}(\mathbb{Z}^d) \to l^{r}(\mathbb{Z}^d)$ bounds for $d \geq 3$, $\frac{1}{p} + \frac{1}{q} \geq \frac{1}{r}$,…

Let $\{u_\lambda\}$ be a sequence of $L^2$-normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold $(M,g)$. We seek to get an $L^p$ restriction bounds of the Neumann data $ \lambda^{-1} \partial_\nu…

In this paper we study perturbed Ornstein-Uhlenbeck operators \begin{align*}[\mathcal{L}_{\infty} v](x)=A\triangle v(x)+\langle Sx,\nabla v(x)\rangle-B v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2,\end{align*} for simultaneously diagonalizable…

We prove resolvent $L_p$ estimates and maximal $L_p$-$L_q$ regularity estimates for the Stokes equations with Dirichlet, Neumann and Robin boundary conditions in the half space. Each solution is constructed by a Fourier multiplier of…

We prove an $L^p$-spectral multiplier theorem under the sharp regularity condition $s > d\left|1/p - 1/2\right|$ for sub-Laplacians on M\'etivier groups. The proof is based on a restriction type estimate which, at first sight, seems to be…

The $R$-boundedness of certain families of vector-valued stochastic convolution operators with scalar-valued square integrable kernels is the key ingredient in the recent proof of stochastic maximal $L^p$-regularity, $2<p<\infty$, for…

This paper investigates the $L^p$-boundedness of wave operators associated with the nonhomogeneous fourth-order Sch\"odinger operator $H = \Delta^2 - \Delta + V(x)$ on $\mathbb{R}^n$. Assuming the real-valued potential $ V $ exhibits…

For any bounded, regulated function $m: [0,\infty) \to \mathbb{C}$, consider the family of operators $\{ T_R \}$ on the sphere $S^d$ such that $T_R f = m(k/R) f$ for any spherical harmonic $f$ of degree $k$. We completely characterize the…

We define a kind of 'operational calculus' for $GL_2(R)$. Namely, the group $GL_2(R)$ can be regarded as an open dense chart in the Grassmannian of 2-dimensional subspaces in $R^4$. Therefore the group $GL_4(R)$ acts in $L^2$ on $GL_2(R)$.…

In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic surfaces. Let $\mathfrak{p}$ be a homogenous…

Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 when p > 2. When the best possible p…

We establish the $L^p$ boundedness of Hilbert transforms and maximal functions along flat curves in the Heisenberg group. This generalizes the $\mathbb{R}^n$ result by Carbery, Christ, Vance, Wainger, and Watson. What is new about our…

We study the infinitesimal generator of the Poisson semigroup in $L^p$ associated with homogeneous, second-order, strongly elliptic systems with constant complex coefficients in the upper-half space, which is proved to be the…

In this paper we study the boundedness of global pseudo-differential operators on smooth manifolds. By using the notion of global symbol we extend a classical condition of H\"ormander type to guarantee the $L^p$-$L^q$-boundedness of global…

In this paper we prove $L^p$ estimates for Stein's square functions associated to Fourier-Bessel expansions. Furthermore we prove transference results for square functions from Fourier-Bessel series to Hankel transforms. Actually, these are…