Related papers: Properties of singular integral operators $S_{\alp…

In this paper, we prove that for $\frac{n}{2}+\frac{1}{4}<\alpha \leq\frac{n+1}{2} $, the convolution operator $$S_{\alpha} f(x)=\int_{|y| \geq 1} f(x-y)\left(|y|^{2}-1\right)^{-\alpha} d y$$ is bounded from $L^p$ to $L^q$ for certain…

We present a uniqueness result in dimensions $2$ and $3$ for the inverse fixed angle scattering problem associated to the Schr\"odinger operator $-\Delta+q$, where $q$ is a small real valued potential with compact support in the Sobolev…

Suppose that $d\geq 1$ and $0<\beta<2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to the operator $\mathcal{L}^b=\Delta+S^b$, where $$S^bf(x) := \int_{\mathbb{R}^d} \left( f(x+z) - f(x) - \nabla…

This note deals with the operator $T^*T$, where $T$ is a densely defined operator on a complex Hilbert space. We reprove a recent result of Z. Sebesty\'en and Zs. Tarcsay [13]: If $T^*T$ and $TT^*$ are self-adjoint, then $T$ is closed. In…

The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the…

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 present complete characterizations of Toeplitz operators that are complex symmetric. This follows as a by-product of characterizations of conjugations on Hilbert spaces. Notably, we prove that every conjugation admits a canonical…

In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of $L^2 \mapsto L^2$ type for the operator, as well as for the corresponding maximal function. If…

We study one-dimensional Schr\"{o}dinger operators $\mathrm{S}(q)$ on the space $L^{2}(\mathbb{R})$ with potentials $q$ being complex-valued generalized functions from the negative space $H_{unif}^{-1}(\mathbb{R})$. Particularly the class…

We consider self-adjoint Schr\"odinger operators in $L^2 (\mathbb{R}^d)$ with a $\delta$-interaction of strength $\alpha$ and a $\delta'$-interaction of strength $\beta$, respectively, supported on a hypersurface, where $\alpha$ and…

This paper deals with the approximation of a magnetic Schr\"odinger operator with a singular $\delta$-potential that is formally given by $(i \nabla + A)^2 + Q + \alpha \delta_\Sigma$ by Schr\"odinger operators with regular potentials in…

In this article we introduce a stochastic counterpart of the H\"ormander condtion on the kernel $K(r,t,x,y)$: there exists a pseudo-metric $\rho$ on $(0,\infty)\times R^d$ and a positive constant $C_0$ such that for $X=(t,x), Y=(s,y),…

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…

We first consider some questions raised by N. Zorboska in her thesis. In particular she asked for which sequences $\beta$ every symbol $\varphi \colon \mathbb{D} \to \mathbb{D}$ with $\varphi \in H^2 (\beta)$ induces a bounded composition…

The one-dimensional oscillatory integral operator associated to a real analytic phase $S$ is given by $$ T_\lambda f(x) =\int_{-\infty}^\infty e^{i\lambda S(x,y)} \chi(x,y) f(y) dy. $$ In this paper, we obtain a complete characterization…

We consider a generalized Schr\"odinger operator in $L^2(\R^2)$ with an attractive strongly singular interaction of $\delta'$ type characterized by the coupling parameter $\beta>0$ and supported by a $C^4$-smooth closed curve $\Gamma$ of…

In this note, we present a characterization of semistable unitary operators on $L^2(\mathbb{R})$, under the assumption that the operator is (i) translation-invariant, (ii) symmetric, and (iii) locally uniformly continuous (LUC) under…

We study singular integral operators induced by $3$-dimensional Calder\'on-Zygmund kernels in the Heisenberg group. We show that if such an operator is $L^{2}$ bounded on vertical planes, with uniform constants, then it is also $L^{2}$…

In the first part, we obtain sharp results for L^2 boundedness of strongly singular operators on the Heisenberg group. We also define the oscillating convolution operators on the Heisenberg group and study their boundedness properties. In…

In this article we consider the classical singular integral operator over a local field with rough kernels. We study the boundedness of such an operator on different function spaces by relaxing the smoothness condition on kernels.