arXiv subjects
Publications and source records attributed to Dunyan Yan.
This paper is devoted to $L^2$ estimates for trilinear oscillatory integrals of convolution type on $\mathbb{R}^2$. The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp $L^2$ decay estimates of trilinear oscillatory integrals with smooth phases, and then give $L^2$ uniform estimates for these integrals with polynomial phases.
In this paper, we obtain the weighted boundedness for the local multi(sub)linear Hardy-Littlewood maximal operators and local multilinear fractional integral operators associated with the local Muckenhoupt weights on Gaussian measure spaces. We deal with these problems by introducing a new pointwise equivalent "radial" definitions of these local operators. Moreover using a similar approach, we also get the weighted boundedness for the local fractional maximal operators with rough kernel and local fractional integral operators with rough kernel on Gaussian measure spaces.
In this paper, Hardy type operator $H_β$ on $\bR^{n}$ and its adjoint operator $H_β^{*}$ are investigated. We use novel methods to obtain two main results. One is that we obtain the operators $H_β$ and $H_β^{*}$ being bounded from $L^{p}(|x|^α)$ to $L^{q,\infty}(|x|^γ)$, and the bounds of the operators $H_β$ and $H_β^{*}$ are sharp worked out. In particular, when $α=γ=0$, the norm of $H_β$ is equal to $1$. The other is that we study limiting weak-type behavior for the operator $H_β$ and its optimal form was obtained.
In this paper, the authors establish the two-weight boundedness of the local fractional maximal operators and local fractional integrals on Gaussian measure spaces associated with the local weights. More precisely, the authors first obtain the two-weight weak-type estimate for the local-$a$ fractional maximal operators of order $α$ from $L^{p}(v)$ to $L^{q,\infty}(u)$ with $1\leq p\leq q<\infty$ under a condition of $(u,v)\in \bigcup_{b'>a} A_{p,q,α}^{b'}$, and then obtain the two-weight weak-type estimate for the local fractional integrals. In addition, the authors obtain the two-weight strong-type boundedness of the local fractional maximal operators under a condition of $(u,v)\in\mathscr{M}_{p,q,α}^{6a+9\sqrt{d}a^2}$ and the two-weight strong-type boundedness of the local fractional integrals. These estimates are established by the radialization method and dyadic approach.
We consider pointwise convergence of nonelliptic Schrödinger means $e^{it_{n}\square}f(x)$ for $f \in H^{s}(\mathbb{R}^{2})$ and decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, where \[{e^{it_{n}\square }}f\left( x \right): = \int_{{\mathbb{R}^2}} {{e^{i\left( {x \cdot ξ+ t_{n}{{ ξ_{1}ξ_{2} }}} \right)}}\widehat{f}} \left( ξ\right)dξ.\] We prove that when $0<s < \frac{1}{2}$, \[\mathop {\lim }\limits_{n \to \infty} {e^{it_{n}\square }}f\left( x \right) = f(x) \hspace{0.2cm} a.e.\hspace{0.2cm} x\in \mathbb{R}^2\] holds for all $f \in {H^s}\left( {{\mathbb{R}^2}} \right)$ if and only if $\{t_{n}\}_{n=1}^{\infty} \in \ell^{r(s), \infty}(\mathbb{N})$, $r(s)=\frac{s}{1-s}$. Moreover, our result remains valid in general dimensions.
We consider pointwise convergence of Schrödinger means $e^{it_{n}Δ}f(x)$ for $f \in H^{s}(\mathbb{R}^{2})$ and decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero. The main theorem improves the previous results of [Sjölin, JFAA, 2018] and [Sjölin-Strömberg, JMAA, 2020] in $\mathbb{R}^{2}$. This study is based on investigating properties of Schrödinger type maximal functions related to hypersurfaces with vanishing Gaussian curvature.
The goal of this note is to establish non-tangential convergence results for Schrödinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the initial data which implies convergence. As a consequence, we obtain a upper bound for $p$ such that the Schrödinger maximal function is bounded from $H^{s}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$ for any $s > \frac{n}{2(n+1)}$.
We prove a plethora of boundedness property of the Adams type for bilinear fractional integral operators of the form $$B_α(f,g)(x)=\int_{\mathbb{R}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-α}}dy,\qquad 0<α<n.$$ For $1<t\leq s<\infty$, we prove the non-weighted case through the known Adams type result. And we show that these results of Adams type is optimal. For $0<t\leq s<\infty$ and $0<t\leq1$, we obtain new result of a weighted theory describing Morrey boundedness of above form operators if two weights $(v,\vec{w})$ satisfy $$ [v,\vec{w}]_{t,\vec{q}/{a}}^{r,as}=\mathop{\sup_{Q,Q^{\prime}\in\mathscr{D}}}_{Q\subset Q^{\prime}}\left(\frac{|Q|}{|Q^{\prime}|}\right)^{\frac{1-s}{as}}|Q^{\prime}|^{\frac{1}{r}}\left(\fint_{Q}v^{\frac{t}{1-t}}\right)^{\frac{1-t}{t}}\prod_{i=1}^{2}\left(\fint_{Q^{\prime}}w_{i}^{-(q_{i}/a)^{\prime}}\right)^{\frac{1}{(q_{i}/a)^{\prime}}}<\infty,\,\,\, 0<t<s<1 $$ and $$ [v,\vec{w}]_{t,\vec{q}/{a}}^{r,as}:=\mathop{\sup_{Q,Q^{\prime}\in\mathscr{D}}}_{Q\subset Q^{\prime}}\left(\frac{|Q|}{|Q^{\prime}|}\right)^{\frac{1-as}{as}}|Q^{\prime}|^{\frac{1}{r}}\left(\fint_{Q}v^{\frac{t}{1-t}}\right)^{\frac{1-t}{t}}\prod_{i=1}^{2}\left(\fint_{Q^{\prime}}w_{i}^{-(q_{i}/a)^{\prime}}\right)^{\frac{1}{(q_{i}/a)^{\prime}}}<\infty, \,\,\,s\geq1 $$ where $\|v\|_{L^{\infty}(Q)}=\sup_{Q}v$ when $t=1$, $a$, $r$, $s$, $t$ and $\vec{q}$ satisfy proper conditions. As some applications we formulate a bilinear version of the Olsen inequality, the Fefferman-Stein type dual inequality and the Stein-Weiss inequality on Morrey spaces for fractional integrals.
This paper mainly dedicates to prove a plethora of weighted estimates on Morrey spaces for bilinear fractional integral operators and their general commutators with BMO functions of the form $$B_α(f,g)(x)=\int_{\mathbb{R}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-α}}dy,\qquad 0<α<n.$$ We also prove some maximal function control theorems for these operators, that is, the weighted Morrey norm is bounded by the weighted Morrey norm of a natural maximal operator when the weight belongs to $A_{\infty}$. As a corollary, some new weighted estimates for the bilinear maximal function associated to the bilinear Hilbert transform are obtained. Furthermore, we formulate a bilinear version of Stein-Weiss inequality on Morrey spaces for fractional integrals.
In this paper, we investigate sharp damping estimates for a class of one dimensional oscillatory integral operators with real-analytic phases. By establishing endpoint estimates for suitably damped oscillatory integral operators, we are able to give a new proof of the sharp $L^p$ estimates which have been proved by Xiao in Endpoint estimates for one-dimensional oscillatory integral operators, \emph{Advances in Mathematics}, \textbf{316}, 255-291 (2017). The damping estimates obtained in this paper are of independent interest.
In this paper, we study a class of multilinear fractional integral operators which have correlation kernels $\prod_{1\leq i<j \leq k}|x_i-x_j|^{-α_{ij}}$. The necessary and sufficient conditions are obtained under which these oprators are bounded from $L^{p_1}\times \cdots \times L^{p_k}$ into $L^q$. As a consequence, we also get the endpoint estimates from $L^{p_1}\times \cdots \times L^{p_k}$ to $BMO$ of these operators.
We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard $α$-fractional singular integrals, fractional integral operators, Marcinkiewicz integral operators, and square functions. These bounds are knows to be sharp in many cases, and our main new result is the optimal bound $$[w]_{A_{p,q}}^{\frac{1}{q}}[w^{q}]_{A_{\infty}}^{\frac{1}{2}-\frac{1}{p}}\lesssim[w]_{A_{p,q}}^{\frac{1}{2}-\fracα{d}}$$ for proper conditions which satisfy that three index $p$, $q$ and $α$ ensure weak type norm of fractional square functions on $L^{q}(w^{q})$ with $p>2$.
In this paper, we investigate a class of fractional Hardy type operators $\mathscr{H}_{β_{1},\cdots,β_{m}}$ defined on higher-dimensional product spaces $\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}}$. We use novel methods to obtain two main results. One is that we obtain the operator $\mathscr{H}_{β_{1},\cdots,β_{m}}$ is bounded from $L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}},|x|^γ)$ to $L^{q}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}},|x|^α)$ and the bounds of the operator $\mathscr{H}_{β_{1},\cdots,β_{m}}$ is sharp worked out. The other is that when $α=γ=(0,\cdots,0)$, the norm of the operator $\mathscr{H}_{β_{1},\cdots,β_{m}}$ is obtained.
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. Moreover, as application, we can obtain an explicit scale of bound for the Pólya-Knopp type operator with certain weights.
We obtain the $L^p$ decay of oscillatory integral operators $T_λ$ with certain homogeneous polynomial phase of degree $d$ in $(n+n)$-dimensions. In this paper we require that $d>2n$. If $d/(d-n)<p<d/n$, the decay is sharp and the decay rate is related to the Newton distance. In the case of $p=d/n$ or $d/(d-n)$, we also obtain the almost sharp decay, here "almost" means the decay contains a $\log(λ)$ term. For otherwise, the $L^p$ decay of $T_λ$ is also obtained but not sharp. A counterexample also arises in this paper to show that $d/(d-n)\leq p\leq d/n$ is not necessary to guarantee the sharp decay.
We consider the following oscillatory integral operator \begin{equation}\label{opera-defi-1} T_{α,m}f(x)=\int_{\mathbb R^n}e^{i(x_1^{α_1} y_1^m+\cdots+x_n^{α_n} y_n^m)}f(y)dy, \end{equation} where the function $f$ is a Schwartz function. In this paper, the restriction theorem on $\mathbb{S}^{n-1}$ for this operator is obtained. Moreover, we obtain a necessary condition which ensures the restriction theorem hold.
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 when polynomial phases have the form $S(x^{m_{1}},y^{m_{2}})$ with $m_1$ and $m_2$ being positive integers.