arXiv subjects

Robert M. Kesler

Publications and source records attributed to Robert M. Kesler.

4 recordsLinked to original sources

arXiv2017

L^p Estimates for Semi-Degenerate Simplex Multipliers

Muscalu, Tao, and Thiele prove $L^p$ estimates for the "Biest" operator defined on Schwartz functions by the map \begin{align*} \hspace{5mm} C^{1,1,1}:& (f_1, f_2, f_3) \mapsto \int_{ξ_1 < ξ_2< ξ_3} \left[ \prod_{j=1}^3 \hat{f}_j (ξ_j) e^{2 πi x ξ_j } \right] d \vecξ \end{align*} via a time-frequency argument that produces bounds for all multipliers with non-degenerate trilinear simplex symbols. In this article we prove $L^p$ estimates for a pair of simplex multipliers for which the non-degeneracy condition fails and which are defined on Schwartz functions by the maps \begin{align*} C^{1,1,-2}:& (f_1, f_2, f_3) \mapsto \int_{ξ_1 <ξ_2 < -\frac{ξ_3}{2}}\left[ \prod_{j=1}^3 \hat{f}_j (ξ_j) e^{2 πi x ξ_j } \right] d \vecξ \end{align*} \begin{align*} C^{1,1,1,-2}:& (f_1, f_2, f_3, f_4) \mapsto \int_{ξ_1 <ξ_2 < ξ_3< -\frac{ξ_4}{2}} \left[\prod_{j=1}^4 \hat{f}_j (ξ_j) e^{2 πi x ξ_j} \right] d \vecξ. \end{align*} Our argument combines the standard $\ell^2$-based energy with an $\ell^1$-based energy in order to enable summability over various size parameters. As a consequence, we obtain that $C^{1,1,-2}$ maps into $L^p$ for all $1/2< p < \infty$ and $C^{1,1,1,-2}$ maps into $L^p$ for all $1/3 < p < \infty$. Both target $L^p$ ranges are shown to be sharp.

math.CA↗

arXiv2016

Generic Multilinear Multipliers Associated to Degenerate Simplexes

For each $1 \leq p \leq \infty$, let $W_{p}(\mathbb{R}) = \left\{ f \in L^p(\mathbb{R}): \hat{f} \in L^{p^\prime}(\mathbb{R}) \right\}$ with norm $||f||_{W_{p}(\mathbb{R})} = ||\hat{f}||_{L^{p^\prime}(\mathbb{R})}$. Moreover, let $ Γ= \left\{ ξ_1 + ξ_2 =0\right\} \subset \mathbb{R}^2$ and $a_1,a_2 : \mathbb{R}^2 \rightarrow \mathbb{C}$ satisfy the Hörmander-Mikhlin condition \begin{eqnarray*} \left| \partial^{\vecα} a_j \left(\vecξ\right) \right| \lesssim_{\vecα} \frac{1}{dist(\vecξ, Γ)^{|\vecα|}}~~~\forall \vecξ \in \mathbb{R}^2, j \in \{1, 2\} \end{eqnarray*} for sufficiently many multi-indices $\vecα \in (\mathbb{N} \bigcup \{0\})^2$. Our main result is that the generic degenerate trilinear simplex multiplier defined on $ \mathcal{S}^3(\mathbb{R})$ by \begin{eqnarray*} B[a_1, a_2] : (f_1, f_2, f_3) \rightarrow \int_{\mathbb{R}^3} a_1(ξ_1, ξ_2) a_2(ξ_2, ξ_3) \left[ \prod_{j=1}^3 \hat{f_j} (ξ_j) e^{2 πix ξ_j} \right] dξ_1 dξ_2 dξ_3 \end{eqnarray*} extends to a map $L^{p_1}(\mathbb{R}) \times W_{p_2}(\mathbb{R}) \times L^{p_3}(\mathbb{R}) \rightarrow L^{\frac{1}{\frac{1}{p_1} + \frac{1}{p _2} +\frac{1}{p_3}}}(\mathbb{R})$ provided \begin{eqnarray*} 1 < p_1, p_3 \leq \infty, \frac{1}{p_1} + \frac{1}{p_2} <1, \frac{1}{p_2} + \frac{1}{p_3} <1, 2 < p_2 <\infty. \end{eqnarray*}

math.CA↗

arXiv2016

Unboundedness Theorems for Symbols Adapted to Large Subspaces

For every integer $n \geq 3$, we prove that the n-sublinear generalization of the Bi-Carleson operator of Muscalu, Tao, and Thiele given by nC^{\vecα} :(f_1,..., f_n) \mapsto \sup_{M} \left| \int_{\vecξ \cdot \vecα >0, ξ_n < M} \left[\prod_{j=1}^n \hat{f}_j(ξ_j) e^{2 πi x ξ_j }\right]d\vecξ ~\right|satisfies no $L^p$ estimates provided $\vecα \in \mathbb{Q}^n$ with distinct, non-zero entries. Furthermore, if $n \geq 5$ and $\vecα \in \mathbb{Q}^n$ has distinct, non-zero entries, it is shown that there is a symbol $m:\mathbb{R}^n \rightarrow \mathbb{C}$ adapted to the hyperplane $Γ^{\vec{a}}=\left\{ \vecξ \in \mathbb{R}^n: \sum_{j=1}^n ξ_j \cdot a_j =0 \right\} $ and supported in $\left\{ \vecξ : dist(\vecξ, Γ^{\vecα}) \lesssim 1 \right\}$ for which the associated $n$-linear multiplier also satisfies no $L^p$ estimates. Next, we construct a Hörmander-Marcinkiewicz symbol $Π: \mathbb{R}^2 \rightarrow \mathbb{C}$, which is a paraproduct of $(ϕ, ψ)$ type, such that the trilinear operator $T_m$ whose symbol $m$ is $ sgn(ξ_1 + ξ_2) Π(ξ_2, ξ_3)$ satisfies no $L^p$ estimates. Finally, we state a converse to a theorem of Muscalu, Tao, and Thiele using Riesz kernels in the spirit of Muscalu's recent work: for every pair of integers $(\mathfrak{d},n) $ s.t. $ \frac{n}{2}+\frac{3}{2} \leq \mathfrak{d}<n$ there is an explicit collection $\mathfrak{C}$ of uncountably many $\mathfrak{d}$-dimensional non-degenerate subspaces of $\mathbb{R}^n$ such that for each $Γ\in \mathcal{C}$ there is an associated symbol $m_Γ$ adapted to $Γ$ in the Mikhlin-Hörmander sense and supported in $\left\{ \vecξ : dist(\vecξ, Γ) \lesssim 1 \right\}$ for which the associated multilinear multiplier $T_{m_Γ}$ is unbounded.

math.CA↗

arXiv2016

Restricted Carleson Variations at Endpoint and Discretized Hilbert Transforms in the Plane

We provide elementary proofs that the 2-variation Carleson operator $V_2$ along with explicit bilinear multipliers adapted to $\{ξ_1 + ξ_2 = 0\}$ satisfy no $L^p$ estimates. Furthermore, we obtain $L^p \rightarrow L^p$ estimates when $2 < p <\infty$ for a smooth restricted variant of $V_2$ that is defined a priori on Schwartz functions by the formula \begin{eqnarray*} \mathcal{V}^{res}_2 : f \mapsto \sup_{R \in \mathbb{R}_+} ~~\sup_{0 \leq α< R} ~~\left(\sum_{j \in \mathbb{Z}} \left|f*\mathcal{F}^{-1} \left[ \tilde{1}_{[α+ j R, α+ (j+1)R]}\right] \right|^2 \right)^{1/2} \end{eqnarray*} where $\tilde{1}_{I} (x) := \tilde{1}(|I|^{-1} (x-c_I))$ for all intervals $I = [c_I - |I|/2, c_I + |I|/2] \subset \mathbb{R}$ and $\tilde{1} \in C^\infty([-1/2, 1/2])$. We then study bi-sublinear variants of $\mathcal{V}_2^{res}$ before showing that multipliers, which are adapted to $\{ξ_1 + ξ_2=0\}$ and periodically discretized along each frequency scale, map $L^{p_1}(\mathbb{R}) \times L^{p_2}(\mathbb{R}) \rightarrow L^{p_1 p_2 / (p_1 + p_2)}(\mathbb{R})$ provided $2 \leq p_1, p_2 <\infty$ and $\frac{1}{p_1} + \frac{1}{p_2} <1$.

math.CA↗