Related papers: On a planar Pierce--Yung operator

We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak…

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 prove an extrapolation result for general operators under some weak assumptions on the boundedness of the operator. In particular, we show that if the operator is weakly bounded on some L^{p_{0}}(w), for all "flat" weights, w in…

A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni \cite{LM} of such maps for the space of compact operators on a Hilbert…

Let $T_{a,\varphi}$ be a Fourier integral operator defined with $a\in S^{m}_{0,\delta}(0\leq\delta<1)$ and $\varphi\in \Phi^{2}$ satisfying the strong non-degenerate condition. We demonstrate that when the order satisfies…

In this paper we show that every conjugation $C$ on the Hardy-Hilbert space $H^{2}$ is of type $C=T^{*}C_{1}T$, where $T$ is an unitary operator and $C_{1}f\left(z\right)=\overline{f\left(\overline{z}\right)}$, with $f\in H^{2}$. In the…

For a fixed $1\le p<+\infty$ denote by $\Vert\cdot\Vert_p$ the usual norm in the space $l_p$ (or $L_p$). In this paper we prove that for all real numbers $p$ and $q$ such that $2\le p\le q$ holds $$ 2(\Vert x\Vert_p^q+\Vert y\Vert_p^q)\le…

For $m \geq 2$, let $(\mathbb{Z}_{m+1}^N, |\cdot|)$ denote the group equipped with the so-called $l^0$ metric, \[ |y| = \left| \big( y(1), \dots, y(N) \big) \right| := | \{1 \leq i \leq N : y(i) \neq 0 \} |,\] and define the…

Theorem 1. Given a number c >= 1, there exists a J-unitary operator \hat{V}, such that: (a) r(\hat{V})= r(\hat{V}^{-1})= c ; (b) S(c^{-1}\hat{V})=S(c^{-1}\hat{V}^{-1}) =S(c^{-1}\hat{V}^{*-1}) = S(c^{-1}\hat{V}^*)={0} (c) there exist maximal…

We introduce the Lorentz space $\mathcal{L}^{p(\cdot), q(\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The…

Let $f$ be a function on ${\Bbb R}^2$ in the inhomogeneous Besov space $B_{\infty,1}^1({\Bbb R}^2)$. For a pair $(A,B)$ of not necessarily bounded and not necessarily commuting self-adjoint operators, we define the function $f(A,B)$ of $A$…

A characterization is obtained for those pairs of weights $v$ and $w$ on $\mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(\mathbb{R}^2_+)$ to…

In this paper, we develop boundedness estimates for Fourier integral operators on Fourier Lebesgue spaces when the associated canonical relation is parametrised by a complex phase function. Our result constitutes the complex analogue of…

For $p\in(1,\infty)$, let $u(t,x,v)$ and $f(t,x,v)$ be in $L^p(\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d)$ and satisfy the following nonlocal kinetic Fokker-Plank equation on $\mathbb{R}^{1+2d}$ in the weak sense: $$ \partial_t…

We prove affirmatively the one dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator, for $1<p<\infty$. The proof is based on two new ideas: i) developing a framework for…

Let $C$ be the middle-third Cantor set. Define $C*C=\{x*y:x,y\in C\}$, where $*=+,-,\cdot,\div$ (when $*=\div$, we assume $y\neq0$). Steinhaus \cite{HS} proved in 1917 that \[ C-C=[-1,1], C+C=[0,2]. \] In 2019, Athreya, Reznick and Tyson…

In this paper, it is shown that if F(x , y) is an irreducible binary form with integral coefficients and degree $n \geq 3$, then provided that the absolute value of the discriminant of F is large enough, the equation |F(x , y)| = 1 has at…

Let $L$ be a sectorial operator of type $\alpha$ ($0 \leq \alpha < \pi/2$) on $L^2(\mathbb{R}^d)$ with the kernels of $\{e^{-tL}\}_{t>0}$ satisfying certain size and regularity conditions. Define $$ S_{q,L}(f)(x) =…

To extend the Euclidean operator radius, we define $w_p$ for an $n$-tuples of operators $(T_1,\ldots, T_n)$ in $\mathbb{B}(\mathscr{H})$ by $w_p(T_1,\ldots,T_n):= \sup_{\| x \| =1} \left(\sum_{i=1}^{n}| \langle T_i x, x \rangle |^p…

Let $L^{m,p}(\R^n)$ denote the Sobolev space of functions whose $m$-th derivatives lie in $L^p(\R^n)$, and assume that $p>n$. For $E \subset \R^n$, denote by $L^{m,p}(E)$ the space of restrictions to $E$ of functions $F \in L^{m,p}(\R^n)$.…