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Ankit Bhojak

Publications and source records attributed to Ankit Bhojak.

11 recordsLinked to original sources
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

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 Honz\'ik 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↗
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

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 \[\mathfrak{A}_t(f_1,f_2)(x,y)=\int_{\mathbb S^{2d-1}}f_1(x+tz_1,y)f_2(x,y+tz_2)\;d\sigma(z_1,z_2),\;t>0,\] 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↗
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

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↗
arXiv2021

Weak type bounds for rough maximal singular integrals near $L^1$

In this paper it is shown that for $Ω\in L\log L(\mathbb{S}^{d-1})$, the rough maximal singular integral operator $T_Ω^*$ is of weak type $L\log\log L(\mathbb{R}^d)$. Furthermore, for $w\in A_1$ and $Ω\in L^\infty(\mathbb{S}^{d-1})$, it is shown that $T_Ω^*$ is of weak type $L\log\log L(w)$ with weight dependence $[w]_{A_1}[w]_{A_{\infty}}\log([w]_{A_{\infty}}+1),$ which is same as the best known constant for the singular integral $T_Ω$.

math.CA↗