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K. S. Senthil Raani

Publications and source records attributed to K. S. Senthil Raani.

4 recordsLinked to original sources
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↗
arXiv2017

$L^p$ Fourier asymptotics, Hardy type inequality and fractal measures

Suppose $μ$ is an $α$-dimensional fractal measure for some $0<α<n$. Inspired by the results proved by R. Strichartz in 1990, we discuss the $L^p$-asymptotics of the Fourier transform of $fdμ$ by estimating bounds of $$\underset{L\rightarrow\infty}{\liminf}\ \frac{1}{L^k} \int_{|ξ|\leq L}\ |\widehat{fdμ}(ξ)|^pdξ,$$ for $f\in L^p(dμ)$ and $2<p<2n/α$. In a different direction, we prove a Hardy type inequality, that is, $$\int\frac{|f(x)|^p}{(μ(E_x))^{2-p}}dμ(x)\leq C\ \underset{L\rightarrow\infty}{\liminf} \frac{1}{L^{n-α}} \int_{B_L(0)} |\widehat{fdμ}(ξ)|^pdξ$$ where $1\leq p\leq 2$ and $E_x=E\cap(-\infty,x_1]\times(-\infty,x_2]...(-\infty,x_n]$ for $x=(x_1,...x_n)\in\R^n$ generalizing the one dimensional results proved by Hudson and Leckband in 1992.

math.CA↗
arXiv2015

$L^p$-Asymptotics of Fourier transform of fractal measures

One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\mathbb{R}^n$. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let $f\in C_c^{\infty}(\mathbb{R}^n)$ and $dσ$ be the surface measure on the sphere $S^{n-1}\subset\mathbb{R}^n$. Then $$|\widehat{fdσ}(ξ)|\leq\ C\ (1+|ξ|)^{-\frac{n-1}{2}}.$$ It follows that $\widehat{fdσ}\in L^p(\mathbb{R}^n)$ for all $p>\frac{2n}{n-1}$. This result can be extended to compactly supported measure on $(n-1)$-dimensional manifolds with appropriate assumptions on the curvature. Similar results are known for measures supported in lower dimensional manifolds in $\mathbb{R}^n$ under appropriate curvature conditions. However, the picture for fractal measures is far from complete. This thesis is a contribution to the study of asymptotic properties of the Fourier transform of measures supported in sets of fractal dimension $0<α<n$ for $p\leq 2n/α$. In 2004, Agranovsky and Narayanan proved that if $μ$ is a measure supported in a $C^1$-manifold of dimension $d<n$, then $\widehat{fdμ}\notin L^p(\mathbb{R}^n)$ for $1\leq p\leq \frac{2n}{d}$. We prove that the Fourier transform of a measure $μ_E$ supported in a set $E$ of fractal dimension $α$ does not belong to $L^p(\mathbb{R}^n)$ for $p\leq 2n/α$. We also study $L^p$-asymptotics of the Fourier transform of fractal measures $μ_E$ under appropriate conditions on $E$ and give quantitative versions of the above statement by obtaining lower and upper bounds for the following: $$\underset{L\rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{|ξ|\leq L}|\widehat{fdμ_E}(ξ)|^pdξ.$$

math.CA↗