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Kalachand Shuin

Publications and source records attributed to Kalachand Shuin.

14 recordsLinked to original sources
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 H\"older 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 C\'ordoba and Carbery. The combination of these estimates yields the $L^p$ boundedness for the bilinear cone multiplier.

math.CA↗
arXiv2026

Helical maximal function and weighted estimates

In this article, we characterize the range of $\alpha$ for which the helical maximal function is bounded from $L^p(|x|^\alpha)$ to itself for $3<p<\infty$. Our result is optimal for $4\leq p<\infty,$ except possibly at end-points.

math.CA↗
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 \[{T}^{\lambda}_{R}(f,g)(x):=\int_{\mathbb{R}^{2n}}m^{\lambda}\left(\frac{\xi'}{R\xi_n},\frac{\eta'}{R\eta_n}\right)\hat{f}(\xi)\hat{g}(\eta)e^{2\pi\iota x\cdot(\xi+\eta)}~d\xi d\eta,\] where $\lambda>0, R>0$ and \[m^{\lambda}\left(\frac{\xi'}{R\xi_n},\frac{\eta'}{R\eta_n}\right)=\Big(1-\frac{|\xi'|^2}{R^2\xi^2_n}-\frac{|\eta'|^2}{R^2\eta^2_n}\Big)^{\lambda}_{+}\varphi(\xi_n)\varphi(\eta_n),\] $(\xi',\xi_n), (\eta',\eta_n)\in\mathbb{R}^{n-1}\times \mathbb{R}$ and $\varphi\in C_{c}^{\infty}([\frac{1}{2},2])$. We investigate the problem of pointwise almost everywhere convergence of ${T}^{\lambda}_{R}(f,g)(x)$ 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 H\"{o}lder 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 \[{T}^{\lambda}_{*}(f,g)(x):=\sup_{R>0}|{T}^{\lambda}_{R}(f,g)(x)|.\]

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arXiv2024

$L^{p}$ estimates for multilinear maximal Bochner--Riesz means and square function

In this article we have investigated $L^{p}$ boundedness of the multilinear maximal Bochner--Riesz means and the corresponding square function. We have exploited the ideas given in the paper "Maximal estimates for bilinear Bochner--Riesz means" (Adv. Math. 395(2022) 108100) by Jotsaroop and Shrivastava, in order to prove our results.

math.CA↗
arXiv2024

${l}^2$ Decoupling for certain degenerate surfaces in $\mathbb{R}^4$

In this article, we aim to study decoupling inequality for a specific degenerate hypersurface in $\mathbb{R}^4$. Inspired by the work of Bourgain--Demeter and Li--Zheng, we consider the hypersurface $\mathcal{S}^3_{4}:=\{(\xi_1,\xi_2,\xi_3,\xi^4_1+\xi^4_2+\xi^4_3):0\leq \xi_j\leq1, \text{for}~j=1,2,3\}$ in $\mathbb{R}^4$ and study decoupling estimates.

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arXiv2024

Improved curvature conditions on $L^2\times\cdots\times L^2 \to L^{2/m}$ bounds for multilinear maximal averages

In this article, we focus on $L^{2}(\mathbb{R}^d)\times\cdots\times L^{2}(\mathbb{R}^d)\rightarrow L^{2/m}(\mathbb{R}^d)$ estimates for multilinear maximal averages over non-degenerate hypersurfaces. Our findings is new for $m$-linear averages with $m\geq3$, and represent a reproof of the recent result of T. Borges, B. Foster, and Y. Ou on the curvature conditions of the hypersurfaces required in establishing $L^{2}(\mathbb{R}^d)\times L^{2}(\mathbb{R}^d)\rightarrow L^{1}(\mathbb{R}^d)$ estimates of bilinear maximal functions.

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arXiv2023

$L^{p}-$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↗
arXiv2023

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 $$\mathfrak{M}(f,g)(x):=\sup_{t>0}\left|\int_{\mathbb S^{2d-1}}f(x-ty_1)g(x-ty_2)\;d\sigma(y_1,y_2)\right|,$$ 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, \[\mathcal M (f,g)(x)=\sup_{t>0}\left|\int_{\mathbb S^{1}}f(x-ty)g(x+ty)\;d\sigma(y)\right|\] 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

$L^p$ improving properties and maximal estimates for certain multilinear averaging operators

In this article we focus on $L^{p}$ estimates for two types of multilinear lacunary maximal averages over hypersurfaces with curvature conditions. Moreover, we give a different proof for the bilinear lacunary spherical maximal functions. To obtain our results, we make use of the $L^1$-improving estimates of multilinear averaging operators. We also obtain $L^p$-improving estimates for certain multilinear averages by means of the nonlinear Brascamp-Lieb inequality.

math.CA↗
arXiv2023

Sparse bounds for maximal oscillatory rough singular integral operators

We prove sparse bounds for maximal oscillatory rough singular integral operator $$T^{P}_{\Omega,*}f(x):=\sup_{\epsilon>0} \left|\int_{|x-y|>\epsilon}e^{\iota P(x,y)}\frac{\Omega\big((x-y)/|x-y|\big)}{|x-y|^{n}}f(y)dy\right|,$$ where $P(x,y)$ is a real-valued polynomial on $\mathbb{R}^{n}\times \mathbb{R}^{n}$ and $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$ is a homogeneous function of degree zero with $\int_{\mathbb{S}^{n-1}}\Omega(\theta)~d\theta=0$. This allows us to conclude weighted $L^p-$estimates for the operator $T^{P}_{\Omega,*}$. Moreover, the norm $\|T^P_{\Omega,*}\|_{L^p\rightarrow L^p}$ 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 $T^{P}_{\Omega}$ for $\Omega\in L^{q}(\mathbb{S}^{n-1})$, $1<q\leq\infty$.

math.CA↗
arXiv2022

Bilinear maximal functions associated with degenerate surfaces

We study $L^{p}\times L^{q}\rightarrow L^{r}$-boundedness of (sub)bilinear maximal functions associated with degenerate hypersurfaces. First, we obtain the maximal bound on the sharp range of exponents $p,q,r$ (except some border line cases) for the bilinear maximal functions given by the model surface $\big\{(y,z)\in\mathbb{R}^{n}\times \mathbb{R}^{n}:|y|^{l_{1}}+|z|^{l_{2}}=1\big\}$, $(l_{1},l_{2})\in [1,\infty)^2$, $n\ge 2$. Our result manifests that nonvanishing Gaussian curvature is not good enough, in contrast with $L^p$-boundedness of the (sub)linear maximal operator associated to hypersurfaces, to characterize the best possible maximal boundedness. Secondly, we consider the bilinear maximal function associated to the finite type curve in $\mathbb R^2$ and obtain a complete characterization of the maximal bound. We also prove multilinear generalizations of the aforementioned results.

math.CA↗
arXiv2021

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 Schr\"{o}dinger 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.

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arXiv2020

$L^p$ estimates for multilinear convolution operators defined with spherical measure

Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{n})\in \mathbb{S}^{n-1}$ and $d\sigma$ denote the normalised Lebesgue measure on $\mathbb{S}^{n-1},~n\geq 2$. For functions $f_1, f_2,\dots,f_n$ defined on $\R$ consider the multilinear operator given by $$T(f_{1},f_{2},\dots,f_{n})(x)=\int_{\mathbb{S}^{n-1}}\prod^{n}_{j=1}f_{j}(x-\sigma_j)d\sigma, ~x\in \R.$$ In this paper we obtain necessary and sufficient conditions on exponents $p_1,p_2,\dots,p_n$ and $r$ for which the operator $T$ is bounded from $\prod_{j=1}^n L^{p_j}(\R)\rightarrow L^r(\R),$ where $1\leq p_j,r\leq \infty, j=1,2,\dots,n.$ This generalizes the results obtained in~\cite{jbak,oberlin}.

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