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Saurabh Shrivastava

Publications and source records attributed to Saurabh Shrivastava.

At least 19 recordsLinked to original sources
arXiv2026

On a variant of bilinear spherical maximal function

We obtain $L^p-$estimates for the full and lacunary maximal functions associated to the bilinear spherical averages given by \[\mathfrak{A}_t(f_1,f_2)(x,y)=\int_{\mathbb S^{2d-1}}f_1(x+tz_1,y)f_2(x,y+tz_2)\;dσ(z_1,z_2),\;t>0,\] for all dimensions $d\geq1$. We show that the estimates for such operators in dimensions $d\geq2$ essentially rely on the method of slicing. The bounds for the lacunary maximal function in dimension one are more delicate and require a trilinear smoothing inequality, which is based on an appropriate sublevel set estimate in this context.

math.CA↗
arXiv2026

Bilinear spherical maximal function with fractal dilations

In this paper, we investigate the $L^p-$boundedness of the bilinear spherical maximal function associated with a general set of dilations $E\subset\R_+$. We quantify the range of $L^p-$boundedness in terms of a dilation-invariant notion of the upper Minkowski dimension of the set $E$. A particular case of this study settles an open question of $L^p-$boundedness of the maximal lacunary bilinear spherical function in borderline cases $p_1=1$ or $p_2=1$ in dimension $d\geq4$.

math.CA↗
arXiv2026

An alternate approach to bilinear rough singular integrals

The goal of this paper is to provide a new approach to address the $L^p-$boundedness of bilinear rough singular integral operators. This approach relies on local Fourier series expansion of input functions leading to trilinear estimates with desired decay in the frequency parameter. This approach departs from the existing methods of the wavelet decomposition of the multiplier employed in the work of Grafakos, He and Honzík and in a series of subsequent papers in the context of bilinear rough singular integrals. With this new approach, we prove sharp $L^p-$estimates for maximally truncated bilinear rough singular integrals when the kernel is supported away from the diagonal in the plane. Furthermore, this method allows us to deduce a new and self-contained proof of $L^p-$boundedness of the bilinear rough singular integral operators in all dimensions for the optimal range of exponents.

math.CA↗
arXiv2026

The bilinear cone multiplier on $\mathbb{R}^2\times \mathbb{R}^2$

In this paper, we study the bilinear cone multiplier operator in two dimensions. We establish $L^{p_1}\times L^{p_2}\to L^{p}$ boundedness for a regularized version of this operator over a broad range of exponents satisfying the Hölder scaling condition. Our approach is based on a decomposition of the bilinear operator into square functions associated with linear cone multipliers and their variants. We derive pointwise bounds for these square functions via suitable strong maximal function estimates, and obtain sharp $L^4$ bounds using geometric methods originating in the work of Córdoba and Carbery. The combination of these estimates yields the $L^p$ boundedness for the bilinear cone multiplier.

math.CA↗
arXiv2026

Endpoint Variation and jump inequalities for rough singular integrals

In this article, we prove weak type $(1,1)$ bounds for the variation and jump operators corresponding to the family of truncations of singular integrals with rough kernels. This resolves an open question raised by Jones, Seeger and Wright (Trans. Amer. Math. Soc. (2008)). Moreover, as an immediate consequence of the variational estimate, we recover the weak type $(1,1)$ boundedness of the maximal truncation operator corresponding to singular integrals with rough kernels.

math.CA↗
arXiv2026

$\ell^p(\mathbb{Z}^n)$-estimate for long $r$-variational seminorm of discrete Birch-Magyar averages

We prove $\ell^p(\mathbb{Z}^n)-$estimates for long $r$-variational seminorm of two families of averages: discrete Birch-Magyar averages, for $r>max\{p,p'\}$ with $p>\frac{2c_{\mathfrak{R}}-2}{2c_{\mathfrak{R}}-3}$ and discrete Hardy-Littlewood type averages over certain algebraic varieties, for $r>max\{p,p'\}$ with $p>1$. Further, we discuss an application of these results in ergodic theory.

math.NT↗
arXiv2025

On the bilinear cone multiplier

For $f,g \in \mathscr{S}(\R^n), n\geq 3$, consider the bilinear cone multiplier operator defined by \[{T}^λ_{R}(f,g)(x):=\int_{\mathbb{R}^{2n}}m^λ\left(\frac{ξ'}{Rξ_n},\frac{η'}{Rη_n}\right)\hat{f}(ξ)\hat{g}(η)e^{2πιx\cdot(ξ+η)}~dξdη,\] where $λ>0, R>0$ and \[m^λ\left(\frac{ξ'}{Rξ_n},\frac{η'}{Rη_n}\right)=\Big(1-\frac{|ξ'|^2}{R^2ξ^2_n}-\frac{|η'|^2}{R^2η^2_n}\Big)^λ_{+}φ(ξ_n)φ(η_n),\] $(ξ',ξ_n), (η',η_n)\in\mathbb{R}^{n-1}\times \mathbb{R}$ and $φ\in C_{c}^{\infty}([\frac{1}{2},2])$. We investigate the problem of pointwise almost everywhere convergence of ${T}^λ_{R}(f,g)(x)$ as $R\rightarrow \infty$ for $(f,g)\in L^{p_1}\times L^{p_2}$ for a wide range of exponents $p_1, p_2$ satisfying the Hölder relation $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. This assertion is proved by establishing suitable weighted $L^{2}\times L^{2}\rightarrow L^{1}$--estimates of the maximal bilinear cone multiplier operator \[{T}^λ_{*}(f,g)(x):=\sup_{R>0}|{T}^λ_{R}(f,g)(x)|.\]

math.CA↗
arXiv2024

Sparse Bounds for Discrete Maximal Functions associated with Birch-Magyar averages

In this article, we study discrete maximal function associated with the Birch-Magyar averages over sparse sequences. We establish sparse domination principle for such operators. As a consequence, we obtain $\ell^p$-estimates for such discrete maximal function over sparse sequences for all $p>1$. The proof of sparse bounds is based on scale-free $\ell^p-$improving estimates for the single scale Birch-Magyar averages.

math.NT↗
arXiv2024

$L^{p}-$estimates for uncentered spherical averages and lacunary maximal functions

The primary goal of this paper is to introduce bilinear analogues of uncentered spherical averages, Nikodym averages associated with spheres and the associated bilinear maximal functions. We obtain $L^p$-estimates for uncentered bilinear maximal functions for dimensions $d\geq2$. Moreover, we also discuss the one-dimensional case. In the process of developing these results, we also establish new and interesting results in the linear case. In particular, we will prove $L^p$-improving properties for single scale averaging operators and $L^p$-estimates for lacunary maximal functions in this context.

math.CA↗
arXiv2024

Sharp endpoint $L^p-$estimates for Bilinear spherical maximal functions

In this article, we address endpoint issues for the bilinear spherical maximal functions. We obtain borderline restricted weak type estimates for the well studied bilinear spherical maximal function $$\mathfrak{M}(f,g)(x):=\sup_{t>0}\left|\int_{\mathbb S^{2d-1}}f(x-ty_1)g(x-ty_2)\;dσ(y_1,y_2)\right|,$$ in dimensions $d=1,2$ and as an application, we deduce sharp endpoint estimates for the multilinear spherical maximal function. We also prove $L^p-$estimates for the local spherical maximal function in all dimensions $d\geq 2$, thus improving the boundedness left open in the work of Jeong and Lee (https://doi.org/10.1016/j.jfa.2020.108629). We further study necessary conditions for the bilinear maximal function, \[\mathcal M (f,g)(x)=\sup_{t>0}\left|\int_{\mathbb S^{1}}f(x-ty)g(x+ty)\;dσ(y)\right|\] to be bounded from $L^{p_1}(\mathbb R^2)\times L^{p_2}(\mathbb R^2)$ to $L^p(\mathbb R^2)$ and prove sharp results for a linearized version of $\mathcal M$.

math.CA↗
arXiv2023

Sharp weighted estimates for multi-frequency Calderón-Zygmund operators

In this paper we study weighted estimates for the multi-frequency $ω-$Calderón-Zygmund operators $T$ associated with the frequency set $Θ=\{ξ_1,ξ_2,\dots,ξ_N\}$ and modulus of continuity $ω$ satisfying the usual Dini condition. We use the modern method of domination by sparse operators and obtain bounds $\|T\|_{L^p(w)\rightarrow L^p(w)}\lesssim N^{|\frac{1}{r}-\frac{1}{2}|}[w]_{\mathbb{A}_{p/r}}^{max(1,\frac{1}{p-r})},~1\leq r<p<\infty,$ for the exponents of $N$ and $\mathbb{A}_{p/r}$ characteristic $[w]_{\mathbb{A}_{p/r}}$.

math.CA↗
arXiv2023

Bilinear Bochner-Riesz means for convex domains and Kakeya Maximal function

In this paper we introduce bilinear Bochner-Riesz means associated with convex domains in the plane $\mathbb R^2$ and study their $L^p-$boundedness properties for a wide range of exponents. One of the important aspects of our proof involves the use of bilinear Kakeya maximal function in the context of bilinear Bochner-Riesz problem. This amounts to establish suitable $L^p-$ estimates for the later. We also point out some natural connections between bilinear Kakeya maximal function and Lacey's bilinear maximal function.

math.CA↗
arXiv2023

Sparse bounds for maximal oscillatory rough singular integral operators

We prove sparse bounds for maximal oscillatory rough singular integral operator $$T^{P}_{Ω,*}f(x):=\sup_{ε>0} \left|\int_{|x-y|>ε}e^{ιP(x,y)}\frac{Ω\big((x-y)/|x-y|\big)}{|x-y|^{n}}f(y)dy\right|,$$ where $P(x,y)$ is a real-valued polynomial on $\mathbb{R}^{n}\times \mathbb{R}^{n}$ and $Ω\in L^{\infty}(\mathbb{S}^{n-1})$ is a homogeneous function of degree zero with $\int_{\mathbb{S}^{n-1}}Ω(θ)~dθ=0$. This allows us to conclude weighted $L^p-$estimates for the operator $T^{P}_{Ω,*}$. Moreover, the norm $\|T^P_{Ω,*}\|_{L^p\rightarrow L^p}$ depends only on the total degree of the polynomial $P(x,y)$, but not on the coefficients of $P(x,y)$. Finally, we will show that these techniques also apply to obtain sparse bounds for oscillatory rough singular integral operator $T^{P}_Ω$ for $Ω\in L^{q}(\mathbb{S}^{n-1})$, $1<q\leq\infty$.

math.CA↗
arXiv2022

Bilinear Bochner-Riesz square function and applications

In this paper we introduce Stein's square function associated with bilinear Bochner-Riesz means and investigate its $L^p$ boundedness properties. Further, we discuss several applications of the square function in the context of bilinear multipliers. In particular, we obtain results for maximal function associated with generalised bilinear Bochner-Riesz means. This extends the results proved in~\cite{JS}. Another application concerns the $L^p$ estimates for bilinear fractional Schrödinger multipliers. Finally, we improve upon a result of Grafakos, He and Honzik~\cite{GHH} in the context of bilinear radial multipliers and provide a dimension free sufficient condition on the bilinear multipliers for $L^2\times L^2\rightarrow L^1$ boundedness of the associated maximal function. The generalised bilinear spherical maximal function is a particular example of such maximal functions.

math.CA↗
arXiv2022

On the bilinear Bochner-Riesz problem at critical index

In this paper we study maximal and square functions associated with bilinear Bochner-Riesz means at the critical index. In particular, we prove that they satisfy weighted estimates from $L^{p_1}(w_1)\times L^{p_2}(w_2)\rightarrow L^p(v_w)$ for bilinear weights $(w_1,w_2)\in A_{\vec{P}}$ where $p_1,p_2>1$ and $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. Also, we show that both the operators fail to satisfy weak-type estimates at the end-point $(1,1,\frac{1}{2})$.

math.CA↗
arXiv2021

Maximal estimates for bilinear Bochner-Riesz means

We establish improved and sharp $L^p$ estimates for the maximal bilinear Bochner-Riesz means in all dimensions $n\geq 1$. This work extends the results proved by Jeong and Lee \cite{JL}. We also recover the known results for the bilinear Bochner-Riesz means. The method of proof involves a new decomposition of the bilinear Bochner-Riesz multiplier $(1-|ξ|^2-|η|^2)_+^α$ and delicate analysis in proving $L^p$ estimates for frequency localized square functions.

math.CA↗
arXiv2020

Unimodular bilinear Fourier multipliers on $L^p$ spaces

In this paper we investigate the boundedness properties of bilinear multiplier operators associated with unimodular functions of the form $m(ξ,η)=e^{i ϕ(ξ-η)}$. We prove that if $ϕ$ is a $C^1(\mathbb R^n)$ real-valued non-linear function, then for all exponents $p,q,r$ lying outside the local $L^2-$range and satisfying the Hölder's condition $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, the bilinear multiplier norm $$\|e^{iλϕ(ξ-η)}\|_{\mathcal M_{p,q,r}(\mathbb R^n)}\rightarrow \infty,~ λ\in \mathbb R,~ |λ|\rightarrow \infty.$$ For exponents in the local $L^2-$range, we give examples of unimodular functions of the form $e^{iϕ(ξ-η)}$, which do not give rise to bilinear multipliers. Further, we also discuss the essential continuity property of bilinear multipliers for exponents outside local $L^2-$ range.

math.CA↗