Related papers: Sharp weak bounds and limiting weak-type behavior …

The soft and hard edge scaling limits of $\beta$-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. It has been shown that by tuning the parameter of the hard edge process one can obtain the soft edge…

In this paper, we will study $n$-dimensional Hardy operator and its dual in mixed radial-angular spaces on Heisenberg groups and obtain their sharp bounds by using the rotation method. Furthermore, the sharp bounds of $n$-dimensional…

We proof pointwise bounds for rough Fourier integral operators by the $L^p$ Hardy-Littlewood maximal function. We assume the Fourier integral operators have amplitudes in $L^\infty S^m_\rho$ and phases $\varphi$ such that $\varphi(x,\xi) -…

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,\Delta^2)$ associated with fourth-order Schr\"odinger operators $H=\Delta^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim…

This paper investigates the $L^p$-bounds of wave operators for higher-order Schr\"odinger operators $H = (-\Delta)^m + V$ on $\mathbb{R}^n$, with $m \ge 2$ and real-valued decaying potentials $V$. Our main objective is to establish the…

We study the behaviour of Hardy-weights for a class of variational quasi-linear elliptic operators of $p$-Laplacian type. In particular, we obtain necessary sharp decay conditions at infinity on the Hardy-weights in terms of their…

In this paper, we investigate the boundedness of bilinear Calder\'on-Zygmund operators $T$ from ${L^{p_1}\left(w_1\right)} \times {L^{p_2}\left(w_2\right)}$ to ${L^{p,\infty}\left(v_{\vec{w}}\right)}$ with the stopping time method, where $1…

Let $p \in (1,\infty)$, $\alpha\in \mathbb{R}$, and $\Omega\subsetneq \mathbb{R}^N$ be a $C^{1,\gamma}$-domain with a compact boundary $\partial \Omega$, where $\gamma\in (0,1]$. Denote by $\delta_{\Omega}(x)$ the distance of a point $x\in…

We investigate a class of Fourier integral operators with weakened symbols, which satisfy a multi-parameter differential inequality in $\R^n$. We establish that these operators retain the classical $L^p$ boundedness and the $H^1$ to $L^1$…

Let $\mathcal{B}$ denote a nonempty translation invariant collection of intervals in $\mathbb{R}^n$ (which we regard as a rare basis), and define the associated geometric maximal operator $M_\mathcal{B}$ by $$M_\mathcal{B}f(x) = \sup_{x \in…

We consider a singular Schr\"odinger operator in $L^2(\mathbb{R}^2)$ written formally as $-\Delta - \beta\delta(x-\gamma)$ where $\gamma$ is a $C^4$ smooth open arc in $\mathbb{R}^2$ of length $L$ with regular ends. It is shown that the…

In this paper we present a proof of sharp boundedness of the discrete 1-dimensional Hardy-Littlewood nontangential maximal operator, when the parameter is in the range $[\frac{1}{3},+\infty)$. This generalizes a theorem by Bober, Carneiro,…

We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator $(P,B)$ with degenerate mixed boundary conditions. By an optimal Hardy-weight for a subcritical operator we mean a nonzero nonnegative…

In this article, the authors consider the Schr\"{o}dinger type operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix $A$ satisfies uniformly elliptic condition and the nonnegative potential $V$ belongs to…

We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator $T$ satisfies a bound like $$ \|T\|_{L^{p}(w)}\le c\, [w]^{\beta}_{A_p} \qquad w \in A_{p}, $$ then the optimal…

We characterize the sufficient conditions which three weight functions $u$ and $v_{1}, v_{2}$ satisfy ensure the boundedness of the Hardy operator with variable limits on product space. The corresponding bound is explicitly worked out.…

Let $L$ be a positive self-adjoint operator on $L^2(X)$, where $X$ is a $\sigma$-finite metric measure space. When $\alpha \in (0,1)$, the subordinated semigroup $\{\exp(-tL^{\alpha}):t \in \mathbb{R}^+\}$ can be defined on $L^2(X)$ and…

Let $\Omega$ be a function of homogeneous of degree zero and vanish on the unit sphere $\mathbb {S}^n$. In this paper, we investigate the limiting weak-type behavior for singular integral operator $T_\Omega$ associated with rough kernel…

We obtain sharp two-sided inequalities between $L^p-$norms $(1<p<\infty)$ of functions $Hf$ and $H^*f$, where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty).$ In an equivalent form,…

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…