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Rama Rawat

Publications and source records attributed to Rama Rawat.

9 recordsLinked to original sources
arXiv2024

Fractional Hardy's inequality for half spaces in the Heisenberg group

We establish the following fractional Hardy's inequality $$\int_{\mathbb{H}^n_+}\frac{|f(ξ)|^p}{x_1^{sp}|z|^α}dξ\leq C\int_{\mathbb{H}^n_+}\int_{\mathbb{H}^n_+}\frac{|f(ξ)-f(ξ')|^p}{d(ξ^{-1}\circ ξ')^{Q+sp}|z'-z|^α}dξ'dξ,\ \ \forall\,f\in C_c(\mathbb{H}^n_+)$$ for the half space $\mathbb{H}^n_+:=\{ξ=(z,t)=(x_1,x_2,\ldots, x_n, y_1,y_2,\ldots,y_n,t)\in\mathbb{H}^n:x_1>0\}$ in the Heisenberg group $\mathbb{H}^n$ under the conditions $sp>1$ and $α\geq (2n+sp)/2$. We also provide an alternate proof of a fractional Hardy's inequality in $\mathbb{H}^n$ established in an earlier work.

math.AP↗
arXiv2024

Fractional Caffarelli-Kohn-Nirenberg type inequalities on the Heisenberg group

The aim of this work is to establish some cases of the Caffarelli-Kohn-Nirenberg inequalities on the Heisenberg group for the fractional Sobolev spaces. Here we work with the fractional Sobolev spaces as given by Adimurthi and Mallick in [1]. Our inequalities also give an improvement on the range of indices for the Hardy type inequality established in [1].

math.AP↗
arXiv2023

Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction

The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et.al. "Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity" published in Asymptotic Analysis in 2013, where the linear case p=2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.

math.AP↗
arXiv2023

The Fourier transform on Rearrangement-Invariant Spaces

We study inequalities of the form \begin{equation*} ρ( \lvert \hat{f} \rvert) \leq C σ(f) < \infty, \end{equation*} with $f \in L_{1}(\mathbb{R}^n)$, the Lebesgue-integrable functions on $\mathbb{R}^n$ and \begin{equation*} \hat{f}(ξ) := \int_{\mathbb{R}^n} f(x) \, e^{- 2 πi ξ\cdot x} dx, \ \ \ ξ\in \mathbb{R}^n. \end{equation*} The functionals $ρ$ and $σ$ are so-called rearrangement-invariant (r.i.) norms on $M_{+}(\mathbb{R}^n)$, the nonnegative measurable functions on $\mathbb{R}^n$. Results first proved in the general context of r.i. spaces are then both specialized and expanded on in the special cases of Orlicz spaces and of Lorentz Gamma spaces.

math.CA↗
arXiv2020

Heisenberg uniqueness pairs for the hyperbola

Let $Γ$ be the hyperbola $\{(x,y)\in\mathbb R^2 : xy=1\}$ and $Λ_β$ be the lattice-cross defined by $Λ_β=\left(\mathbb Z\times\{0\}\right)\cup\left(\{0\}\timesβ\mathbb Z\right)$ in $\mathbb R^2,$ where $β$ is a positive real. A result of Hedenmalm and Montes-Rodríguez says that $\left(Γ,Λ_β\right)$ is a Heisenberg uniqueness pair if and only if $β\leq1.$ In this paper, we show that for a rational perturbation of $Λ_β,$ namely \[Λ_β^θ=\left((\mathbb Z+\{θ\})\times\{0\}\right)\cup\left(\{0\}\timesβ\mathbb Z\right),\] where $θ=1/{p},~\text{for some}~{p}\in\mathbb N$ and $β$ is a positive real, the pair $\left(Γ,Λ_β^θ\right)$ is a Heisenberg uniqueness pair if and only if $β\leq{p}.$

math.CA↗
arXiv2017

A sharp form of the Marcinkiewicz Interpolation Theorem for Orlicz spaces

An extension of Marcinkiewicz Interpolation Theorem, allowing intermediate spaces of Orlicz type, is proved. This generalization yields a necessary and sufficient condition so that every quasilinear operator, which maps the set, $S(X,μ)$, of all $μ$-measurable simple functions on $σ$- finite measure space $(X,μ)$ into $M(Y,ν)$, the class of $ν$-measurable functions on $σ$- finite measure space $(Y,ν)$, and satisfies endpoint estimates of type: $1 < p< \infty$, $1 \leq r < \infty$, \begin{equation*} λ\, ν\left( \left\lbrace y \in Y : |(Tf)(y)| > λ\right\rbrace \right)^{\frac{1}{p}} \leq C_{p,r} \left( \int_{\mathbb{R_+}} μ\left( \left\lbrace x \in X : |(f)(x)| > t \right\rbrace \right)^{\frac{r}{p}} t^{r-1}dt \right)^{\frac{1}{r}}, \end{equation*} for all $f \in S(X,μ)$ and $λ\in \mathbb{R_+}$; is bounded from an Orlicz space into another.

math.CA↗
arXiv2017

Dilation-commuting operators on power-weighted Orlicz classes

Let $Φ_1$ and $Φ_2$ be nondecreasing functions from $\mathbb{R_+}=(0,\infty)$ onto itself. For $i=1,2$ and $γ\in \mathbb{R}$, define the Orlicz class $L_{Φ_{i}}(\mathbb{R_+})$ to be the set of Lebesgue-measurable functions $f$ on $\mathbb{R_+}$ such that \begin{equation*} \int_{\mathbb{R_+}} Φ_{i} \left( k|(Tf)(t)| \right) t^γdt < \infty \end{equation*} for some $k>0$. Our goal in this paper is to find conditions on $Φ_1$, $Φ_2$, $γ$ and an operator $T$ so that the assertions \begin{equation} T : L_{Φ_2,t^γ}(\mathbb{R_+}) \rightarrow L_{Φ_1,t^γ}(\mathbb{R_+}), \tag{I} \end{equation} and \begin{equation}\label{modularA} \int_{\mathbb{R_+}} Φ_1 \left( |(Tf)(t)| \right)t^γdt \leq K \int_{\mathbb{R_+}} Φ_2 \left( K|f(s)| \right)s^γds, \tag{M} \end{equation} in which $K>0$ is independent of $f$, say, simple on $\mathbb{R_+}$, are equivalent and to then find necessary and sufficient conditions in order that (\ref{modularA}) holds.

math.FA↗
arXiv2011

Spherical means in annular regions in the $n$-dimensional real hyperbolic spaces

Let $Z_{r,R}$ be the class of all continuous functions $f$ on the annulus $\Ann(r,R)$ in the real hyperbolic space $\mathbb B^n$ with spherical means $M_sf(x)=0$, whenever $s>0$ and $x\in \mathbb B^n$ are such that the sphere $S_s(x)\subset \Ann(r, R) $ and $B_r(o)\subseteq B_s(x).$ In this article, we give a characterization for functions in $Z_{r,R}$. In the case $R=\infty$, this result gives a new proof of Helgason's support theorem for spherical means in the real hyperbolic spaces.

math.FA↗
arXiv2010

Twisted spherical means in annular regions in $C ^n$ and support theorems

Let $Z(Ann(r,R))$ be the class of all continuous functions $f$ on the annulus $Ann(r,R)$ in $\mathbb C^n$ with twisted spherical mean $f \times μ_s(z)=0,$ whenever $z\in \mathbb C^n$ and $s >0$ satisfy the condition that the sphere $S_s(z)\subseteq Ann(r, R) $ and ball $B_r(0)\subseteq B_s(z).$ In this paper, we give a characterization for functions in $Z(Ann(r,R))$ in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in $\mathbb C^n$ which improve some of the earlier results.

math.FA↗