Related papers: Improved estimates for bilinear rough singular int…

Let $L$ be a closed, densely defined operator of type $ \omega $ on $ L^2(\mathbb{R}^n)$ with $0 \leq \omega < \pi/2 $. We assume that $ L $ possesses a bounded $ H_\infty $-functional calculus and that its heat kernel satisfies suitable…

We consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups. We prove certain estimates for the operators that imply $L^p$ boundedness of them by an extrapolation argument under a…

We prove a bilinear $L^2(\R^d) \times L^2(\R^d) \to L^2(\R^{d+1})$ estimate for a pair of oscillatory integral operators with different asymptotic parameters and phase functions satisfying a transversality condition. This is then used to…

Let $\Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $\mathcal{M}_{\Omega}$ be the higher-dimensional Marcinkiewicz integral associated with $\Omega$. In this paper, the authors proved that…

Let $r>\frac{4}{3}$ and let $\Omega \in L^{r}(\mathbb{S}^{2n-1})$ have vanishing integral. We show that the bilinear rough singular integral $$T_{\Omega}(f,g)(x)= \textrm{p.v.}…

In this note, we show that the $L\log L$ hypothesis is the strongest size condition on a homogeneous rough function on the sphere which ensures the weak type $(1,1)$ boundedness of the corresponding singular integral $T_\Omega$, provided…

In this article we consider a modification of the Stein's spherical maximal operator of complex order $\alpha$ on ${\mathbb R^n}$: $$ {\mathfrak M}^\alpha_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over \Gamma(\alpha) } \int_{|y|\leq…

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…

Lebesgue space bounds $L^{p_1}({\mathbb R}^1) \times L^{p_2}(^1) \to L^q({\mathbb R}^1)$ are established for certain maximal bilinear operators. The proof combines a trilinear smoothing inequality with Calder\'on-Zygmund theory. A reference…

In this paper, we first introduce $L^{\sigma_1}$-$(\log L)^{\sigma_2}$ conditions satisfied by the variable kernels $\Omega(x,z)$ for $0\leq\sigma_1\leq1$ and $\sigma_2\geq0$. Under these new smoothness conditions, we will prove the…

We introduce the bilinear Nevo-Thangavelu spherical means on the Heisenberg group $\mathbb{H}^n,$ and derive $L^{p_1}(\mathbb{H}^n) \times L^{p_2}(\mathbb{H}^n) \to L^{p}(\mathbb{H}^n)$ estimates for the single-scale bilinear averaging…

We show that the method in recent work of Roncal, Shrivastava, and Shuin can be adapted to show that certain $L^p$-improving bounds in the interior of the boundedness region for the bilinear spherical or triangle averaging operator imply…

We study $L^p\times L^q\to L^r$ bounds for the bilinear Bochner-Riesz operator $\mathcal{B}^\alpha$, $\alpha>0$ in $\mathbb{R}^d,$ $d\ge2$, which is defined by \[ {\mathcal B}^{\alpha}(f,g)=\iint_{\mathbb{R}^d\times\mathbb{R}^d} e^{2\pi i…

We prove the global $L^2 \times L^2 \to L^1$ boundedness of bilinear Fourier integral operators with amplitudes in $S^0_{1,0} (n,2)$. To achieve this, we require that the phase function can be written as $(x,\xi,\eta) \mapsto…

Let $\Omega \subset \mathbb{R}^d$ be bounded open and connected. Suppose that $W^{1,2}(\Omega) \subset L^r(\Omega)$ for some $r > 2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on…

We study the boundary integral operator induced from the fractional Laplace equation in a bounded smooth domain. For $1/2 < \alpha? < 1$, we show the bijectivity of the boundary integral operator $S_{2\alpha} : L^p(\partial \Omega)…

We derive sparse bounds for the bilinear spherical maximal function in any dimension $d\geq 1$. When $d\geq 2$, this immediately recovers the sharp $L^p\times L^q\to L^r$ bound of the operator and implies quantitative weighted norm…

In this paper, we shall prove the $L^{p}$ endpoint decay estimates of oscillatory integral operators with homogeneous polynomial phases $S$ in $\mathbb{R} \times \mathbb{R}$. As a consequence, sharp $L^{p}$ decay estimates are also obtained…

For a limited range of indices $p$, we obtain $L^p(\mathbb{R}^n)$ boundedness for singular integral operators whose kernels satisfy a condition weaker than the typical H\"ormander smoothness estimate. These operators are assumed to be…

We find a minimal notion of non-degeneracy for bilinear singular integral operators $T$ and identify testing conditions on the multiplying function $b$ that characterize the $L^p\times L^q\to L^r,$ $1<p,q<\infty$ and $r>\frac{1}{2},$…