arXiv subjects

Md Nurul Molla

Publications and source records attributed to Md Nurul Molla.

7 recordsLinked to original sources

arXiv2026

Bilinear Bochner-Riesz Means on the Complex Sphere

In this paper, we establish the boundedness of the bilinear Bochner-Riesz means $\mathcal{B}^α_R$ on the complex sphere $\mathbb{S}$. More precisely, we prove that $\mathcal{B}^α_R$ is bounded from $L^{p_1}(\mathbb{S}) \times L^{p_2}(\mathbb{S}) \to L^p(\mathbb{S})$ where $1/p_1+1/p_2=1/p$ and $1\leq p_1, p_2 \leq \infty$, for an admissible range of exponents, with the required smoothness parameter $α$ described in terms of the topological dimension of $\mathbb{S}$. To facilitate our proof, we establish several analytic estimates, including restriction-type estimates, weighted Plancherel estimates with large power of weights and bilinear weighted Plancherel estimates, which are derived from the ground up in our setting and may be considered of independent interest.

math.CA↗

arXiv2026

Bochner-Riesz commutators on Métivier groups: boundedness and compactness

In this paper, we prove the boundedness and compactness properties of Bochner-Riesz commutator associated to the sub-Laplacians on Métivier groups. We show that the smoothness parameter can be expressed in terms of the topological dimension rather than the homogeneous dimension of the Métivier groups.

math.CA↗

arXiv2026

Bilinear Bochner-Riesz Means on Métivier groups

In this paper, we study the $L^{p_1}(G) \times L^{p_2}(G)$ to $L^{p}(G)$ boundedness of the bilinear Bochner-Riesz means associated with the sub-Laplacian on Métivier group $G$ under the Hölder's relation $1/p = 1/p_1 + 1/p_2$, $1\leq p_1, p_2 \leq \infty$. Our objective is to obtain boundedness results, analogous to the Euclidean setting, where the Euclidean dimension in the smoothness threshold is possibly replaced by the topological dimension of the underlying Métivier group $G$.

math.AP↗

arXiv2025

Bilinear Bochner-Riesz Means for Grushin Operators

This paper is devoted to the study of $L^{p_1} \times L^{p_2}$ to $L^{p}$ boundedness of the bilinear Bochner-Riesz mean $\mathcal{B}^α$ associated with the Grushin operator $\mathcal{L} = -Δ_{x'} - |x'|^2 Δ_{x''}$ on $\mathbb{R}^{d_1} \times \mathbb{R}^{d_2}$. Our result almost resembles the corresponding Euclidean results, where the Euclidean dimension in the smoothness threshold is replaced by the topological dimension $d$ of the underlying space, except at few cases.

math.AP↗

arXiv2025

Bochner-Riesz commutators for Grushin Operators

In this paper, we study the boundedness of Bochner-Riesz commutator $$[b, S^α(\mathcal{L})](f) = b S^α(\mathcal{L})(f) - S^α(\mathcal{L})(bf)$$ of a $BMO^{\varrho}(\mathbb{R}^d)$ function $b$ and the Bochner-Riesz operator $S^α(\mathcal{L})$ associated to the Grushin operator $\mathcal{L}$ on $\mathbb{R}^d$ with $d:= d_1 +d_2$. We prove that for $1\leq p \leq \min \{2d_1/(d_1 +2), 2(d_2 +1)/(d_2+3)\}$ and $α> d(1/p - 1/2) - 1/2$, if $b \in BMO^{\varrho}(\mathbb{R}^d)$, then $[b, S^α(\mathcal{L})]$ is bounded on $L^q(\mathbb{R}^d)$ whenever $p < q < p'$. Moreover, if $b \in CMO^{\varrho}(\mathbb{R}^d)$, then we show that $[b, S^α(\mathcal{L})]$ is a compact operator on $L^q(\mathbb{R}^d)$ in the same range.

math.AP↗

arXiv2021

Uniform boundedness of the Fourier partial sum operators on the weighted spaces of local fields

Let $S_n f$ be the $n$th partial sum of the Fourier series of a function $f$ in $L^1(\D)$, where $\D$ is the ring of integers of a local field $K$. For $1<p<\infty$, we characterize all weight functions $w$ so that the partial sum operators $S_n$, $n\geq 0$, are uniformly bounded on the weighted space $L^p(\D, w)$ and that $S_n f$ converges to $f$ in $L^p(\D,w)$. This includes the case where $K$ is a $p$-adic number field or a field of formal Laurent series $\mathbb{F}_q((X))$ over a finite field $\mathbb{F}_q$, and in particular, when $\D$ is the Walsh-Paley or dyadic group $2^ω$. As an application, in a local field $K$ of positive characteristic, we provide a necessary and sufficient condition on a function $φ\in L^2(K)$ for which the collection of translates of $φ$ forms a Schauder basis for its closed linear span. Moreover, we establish sharp bounds for the Hardy-Littlewood maximal operator.

math.FA↗

arXiv2020

Pointwise Convergence of Fourier Series on the Ring of Integers of Local Fields with an Application to Gabor Systems

We construct a simple example of an integrable function on the ring of integers of the $p$-adic field $\Q_p$ having an almost everywhere divergent Fourier series. On the other hand, we prove the pointwise convergence of the Fourier series of functions in $L^p(\D,w)$, $1<p<\infty$, where $\D$ is the ring of integers of a local field $K$ and $w$ is a weight in the Muckenhoupt $A_p$ class. This result includes, as special cases, when $\D$ is the ring of integers of $\Q_p$ or the field $\mathbb{F}_q((X))$ of formal Laurent series over a finite field $\mathbb{F}_q$, and in particular, when $\D$ is the Walsh-Paley or dyadic group $2^ω$. To achieve this, we establish a weighted estimate for the maximal operator corresponding to the Fourier partial sum operators for functions in $L^p(\D,w)$. As an application, we characterize the Schauder basis property of the Gabor systems in a local field $K$ of positive characteristic in terms of the $A_2$ weights on $\D\times\D$ and the Zak transform $Zg$ of the window function $g$ that generates the Gabor system. Some examples are given to illustrate this result. In particular, we construct an example of a Gabor system which is complete and minimal, but fails to be a Schauder basis for $L^2(K)$.

math.FA↗