14 problems

Given $1 \leq p \leq \infty$, define the Bochner–Riesz means … The Bochner–Riesz conjecture. As soon as $\alpha > 0$, there is a range of exponents $p$ for which … for all…

Stein's square function conjecture. For $p>2$, the estimate

For $R>0$, let $S_R^{\alpha}$ be the Bochner–Riesz means on $\mathbb{R}^n$, and let $S_*^{\alpha}f(x)=\sup_{R>0}|S_R^{\alpha}f(x)|$. Tao's maximal Bochner–Riesz conjecture. For…

Let $G$ be a Métivier group with sub-Laplacian $\mathcal{L}$, and let $S_*^{\alpha}(\mathcal{L})$ denote the maximal Bochner–Riesz operator associated with $\mathcal{L}$. Write…

Let $d$ be a positive integer, let $m_\delta$ be the Bochner–Riesz multiplier from the classical formulation, and let $\mathfrak{M}_p({\mathbb R}^{2d})$ denote the class of multipl…

For $R>0$, let $S_R^{\alpha}$ denote the Bochner–Riesz means on $\mathbb{R}^n$. Almost-everywhere convergence conjecture. When $p\geq2$, almost-everywhere convergence … for…

Hörmander dichotomy conjecture. The operators $T_\phi^\lambda$ satisfy the operator boundedness estimate for $p>\frac{2n}{n-1}$ if and only if $\phi$ satisfies Bourgain's condition…

Let $\Sigma\subset\mathbb{R}^d$ be a $C^2$ hypersurface with surface measure $\mathrm{d}\sigma$, let $f\in L^2(\Sigma;\mathrm{d}\sigma)$, and let $w$ be a nonnegative weight. Let…

Let $m$ be a compactly supported radial multiplier on $\mathbb{R}^N$, let $T_m$ be the associated Fourier multiplier operator, and let $\check{m}$ denote its convolution kernel. Ra…

Assume $d\geq 2$, $\alpha\in(0,1)$, and define … For the operator $\widetilde T_\alpha h=\mathcal F_d^{-1}\left(1_B(\cdot)(1-|\cdot|^2)^{-\alpha}\mathcal F_dh\right)$, the set…

Square-function estimate conjecture. For every $\epsilon>0$, there exists $C=C(\epsilon)$ such that

Negative-order Bochner–Riesz conjecture. The operator $\mathcal R^{-\alpha}$ is bounded from $L^p(\mathbb R^d)$ to $L^q(\mathbb R^d)$ if and only if…

Let $d\ge 2$ and define the Bochner–Riesz square function by … where $\mathcal R_t^\alpha$ is the Bochner–Riesz operator. Square-function conjecture. For $p>2$, the operator…

For the Grushin operator, let $B_R^\delta$ denote the Bochner–Riesz means of order $\delta>0$, defined by … The preceding result establishes that the means are uniformly bounded on…