相关论文: On the bilinear Bochner-Riesz problem at critical …

We prove the $L^p$-boundedness for all $p \in (1,\infty)$ of the first-order Riesz transforms $X_j \mathcal{L}^{-1/2}$ associated with the Laplacian $\mathcal{L} = -\sum_{j=0}^n X_j^2$ on the $ax+b$-group $G = \mathbb{R}^n \rtimes…

We estimate in Lp the maximal Riesz transform in terms of the Riesz transform itself for p greater than 1. In the limiting case p=1 the weak L1 inequality is shown to fail. Surprisingly, the weak L1 inequality for the maximal Beurling…

We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schr\"{o}dinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the…

We improve the Bochner-Riesz conjecture in $\mathbb{R}^3$ to $\max\{p,p/(p-1)\}\geq3.25$.

Let $P_+$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider the inequalities of the following form $$ \|f\|_{L^p(\mathbb{T})}\leq B_{p,s}\|( |P_ + f | ^s + |P_- f |^s) ^{\frac 1s}\|_{L^p (\mathbb{T})} $$ and prove them…

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain $\Omega$is equivalent to the solvability of the Dirichlet problem for the biharmonic equationin $\Omega$. As a result, we prove that for any given bounded…

We study weighted Poincar\'e and Poincar\'e-Sobolev type inequalities with an explicit analysis on the dependence on the $A_p$ constants of the involved weights. We obtain inequalities of the form $$ \left…

The Bochner-Riesz multipliers $ B_{\delta }$ on $ \mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< \delta < \frac {n-1}2 $. The range of sparse bounds increases to the optimal range, as $ \delta $ increases to the…

In this paper, the authors first consider the bidirectional estimates of several typical integrals. As some applications of these integral estimates, the authors investigate the pointwise multipliers from the normal weight general function…

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends…

The main aim of this paper is to investigate $\left( H_{p},L_{p}\right) $ and $\left( H_{p},L_{p,\infty }\right) $ type inequalities for maximal operators of Riesz logarithmic means of one-dimensional Vilenkin-Fourier series.

We prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two…

We prove $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$, its lacunary version $\mathcal C_{lac}$, and its analogue for the Walsh series $\W$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show…

We present a simple Bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials. The constants are dimension-free. The $p$-range of applicability of this…

In this work we obtain boundedness on weighted variable Lebesgue spaces of some maximal functions that come from the localized analysis considering a critical radius function. This analysis appears naturally in the context of the…

For $f\in {\frak S}({\Bbb R}^d)$, we consider the Bochner-Riesz operator ${\frak R}^{\delta}$ of index $\delta>0$ defined by $$\hat {{\frak R}^{\delta}f}(\xi)=(1-|\xi|^2)^{\delta}_+ \hat f (\xi).$$ Then we prove the Bochner-Riesz conjecture…

A long standing question in the theory of orthogonal matrix polynomials is the matrix Bochner problem, the classification of $N \times N$ weight matrices $W(x)$ whose associated orthogonal polynomials are eigenfunctions of a second order…

We provide elementary proofs that the 2-variation Carleson operator $V_2$ along with explicit bilinear multipliers adapted to $\{\xi_1 + \xi_2 = 0\}$ satisfy no $L^p$ estimates. Furthermore, we obtain $L^p \rightarrow L^p$ estimates when $2…

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…

M. Lacey and C. Thiele proved in [27] (Annals of Math. (1997)) and [28] (Annals of Math. (1999)) that the bilinear Hilbert transform maps $L^{p_1}\times L^{p_2}\rightarrow L^{p}$ boundedly when $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$ with…