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Robert Kesler

Publications and source records attributed to Robert Kesler.

10 recordsLinked to original sources
arXiv2021

Lacunary Discrete Spherical Maximal Functions

We prove new $\ell ^{p} (\mathbb Z ^{d})$ bounds for discrete spherical averages in dimensions $ d \geq 5$. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In particular, if $ A _{λ} f $ is the spherical average of $ f$ over the discrete sphere of radius $ λ$, we have \begin{equation*} \bigl\lVert \sup _{k} \lvert A _{λ_k} f \rvert \bigr\rVert _{\ell ^{p} (\mathbb Z ^{d})} \lesssim \lVert f\rVert _{\ell ^{p} (\mathbb Z ^{d})}, \qquad \tfrac{d-2} {d-3} < p \leq \tfrac{d} {d-2},\ d\geq 5, \end{equation*} for any lacunary sets of integers $ \{λ_k ^2 \}$. We follow a style of argument from our prior paper, addressing the full supremum. The relevant maximal operator is decomposed into several parts; each part requires only one endpoint estimate.

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arXiv2020

$\ell^p$-improving inequalities for Discrete Spherical Averages

Let $ λ^2 \in \mathbb N $, and in dimensions $ d\geq 5$, let $ A_{λ} f (x)$ denote the average of $ f \;:\; \mathbb Z ^{d} \to \mathbb R $ over the lattice points on the sphere of radius $λ$ centered at $x$. We prove $ \ell ^{p}$ improving properties of $ A_{λ}$. \begin{equation*} \lVert A_{λ}\rVert_{\ell ^{p} \to \ell ^{p'}} \leq C_{d,p, ω(λ^2 )} λ^{d ( 1-\frac{2}p)}, \qquad \tfrac{d-1}{d+1} < p \leq \frac{d} {d-2}. \end{equation*} It holds in dimension $ d =4$ for odd $ λ^2 $. The dependence is in terms of $ ω(λ^2 )$, the number of distinct prime factors of $ λ^2 $. These inequalities are discrete versions of a classical inequality of Littman and Strichartz on the $ L ^{p}$ improving property of spherical averages on $ \mathbb R ^{d}$, in particular they are scale free, in a natural sense. The proof uses the decomposition of the corresponding multiplier whose properties were established by Magyar-Stein-Wainger, and Magyar. We then use a proof strategy of Bourgain, which dominates each part of the decomposition by an endpoint estimate.

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arXiv2019

Sparse Bounds for the Discrete Spherical Maximal Function

We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.

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arXiv2018

$\ell^p(\mathbb{Z}^d)$-Improving Properties and Sparse Bounds for Discrete Spherical Maximal Averages

We exhibit a range of $\ell ^{p}(\mathbb{Z}^d)$-improving properties for the discrete spherical maximal average in every dimension $d\geq 5$. The strategy used to show these improving properties is then adapted to establish sparse bounds, which extend the discrete maximal theorem of Magyar, Stein, and Wainger to weighted spaces. In particular, the sparse bounds imply that the discrete spherical maximal average is a bounded map from $\ell^2(w)$ into $\ell^2(w)$ provided $w^{\frac{d}{d-4}+δ}$ belongs to the Muckenhoupt class $A_2$ for some $δ>0.$

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arXiv2018

Sparse Bounds for the Discrete Cubic Hilbert Transform

Consider the discrete cubic Hilbert transform defined on finitely supported functions $f$ on $\mathbb{Z}$ by \begin{eqnarray*} H_3f(n) = \sum_{m \not = 0} \frac{f(n- m^3)}{m}. \end{eqnarray*} We prove that there exists $r <2$ and universal constant $C$ such that for all finitely supported $f,g$ on $\mathbb{Z}$ there exists an $(r,r)$-sparse form $Λ_{r,r}$ for which \begin{eqnarray*} \left| \langle H_3f, g \rangle \right| \leq C Λ_{r,r} (f,g). \end{eqnarray*} This is the first result of this type concerning discrete harmonic analytic operators. It immediately implies some weighted inequalities, which are also new in this setting.

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arXiv2018

Sparse Endpoint Estimates for Bochner-Riesz Multipliers on the Plane

For $ 0< λ< \frac{1}2$, let $ B_{λ}$ be the Bochner-Riesz multiplier of index $ λ$ on the plane. Associated to this multiplier is the critical index $1 < p_λ= \frac{4} {3+2 λ} < \frac{4}3$. We prove a sparse bound for $ B_{λ}$ with indices $ (p_λ, q)$, where $ p_λ' < q < 4$. This is a further quantification of the endpoint weak $L^{p_λ}$ boundedness of $ B_{λ}$, due to Seeger. Indeed, the sparse bound immediately implies new endpoint weighted weak type estimates for weights in $ A_1 \cap RH_{ρ}$, where $ ρ> \frac4 {4 - 3 p_{λ}}$.

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arXiv2017

Mixed Estimates for Degenerate Multilinear Operators Associated to Simplexes

We prove that the degenerate trilinear operator $C_3^{-1,1,1}$ given by the formula \begin{eqnarray*} C_3^{-1,1,1}(f_1, f_2, f_3)(x)=\int_{x_1 < x_2 < x_3} \hat{f_1}(x_1) \hat{f_2}(x_2) \hat{f_3}(x_3) e^{2πi x (-x_1 + x_2 + x_3)} dx_1dx_2 dx_3 \end{eqnarray*} satisfies the new estimates \begin{eqnarray*} ||C_3^{-1,1,1}(f_1, f_2, f_3)||_{\frac{1}{\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}}} \lesssim_{p_1, p_2, p_3} ||\hat{f}_1||_{p^\prime_1} ||f_2||_{p_2}||f_3||_{p_3} \end{eqnarray*} for all $f_1 \in L^{p_1}(\mathbb{R}): \hat{f}_1 \in L^{p_1^\prime}(\mathbb{R}) , f_2 \in L^{p_2}(\mathbb{R})$, and $f_3 \in L^{p_3}(\mathbb{R})$ such that $2 <p_1 \leq \infty, 1 < p_2, p_3 < \infty, \frac{1}{p_1}+\frac{1}{p_2} <1$, and $\frac{1}{p_2}+\frac{1}{p_3} <3/2$. Mixed estimates for some generalizations of $C_3^{-1,1,1}$ are also shown.

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arXiv2017

Sparse Bounds for Discrete Quadratic Phase Hilbert Transform

Consider the discrete quadratic phase Hilbert Transform acting on $\ell^{2}$ finitely supported functions $$ H^α f(n) : = \sum_{m \neq 0} \frac{e^{2 πiαm^2} f(n - m)}{m}. $$ We prove that, uniformly in $α\in \mathbb{T}$, there is a sparse bound for the bilinear form $\langle H^α f , g \rangle$. The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Hölder classes.

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arXiv2011

Quantum Mechanics on Laakso Spaces

We first review the spectrum of the Laplacian operator on a general Laakso Space before considering modified Hamiltonians for the infinite square well, parabola, and Coulomb potentials. Additionally, we compute the spectrum for the Laplacian and its multiplicities when certain regions of a Laakso space are compressed or stretched and calculate the Casimir force experienced by two uncharged conducting plates by imposing physically relevant boundary conditions and then analytically regularizing the result. Lastly, we derive a general formula for the spectral zeta function and its derivative for Laakso spaces with strict self-similar structure before listing explicit spectral values for cases of interest.

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