arXiv subjects

R. Kerman

Publications and source records attributed to R. Kerman.

2 recordsLinked to original sources

arXiv2026

Boundedness of Positive Integral Operators on Lorentz-Gamma Spaces

We characterize the boundedness of a positive integral operator $T_K$, with kernel $K\in M_+(\R^{2n})$, between Lorentz-Gamma spaces $\Gamma_{p,\phi_2}(\R^n)$ and $\Gamma_{q,\phi_1}(\R^n)$, $1<p\le q<\infty$. The key step reduces the $n$-dimensional problem to a one-dimensional weighted norm inequality for the composed operator $T_LS$, where $L=(K^{*_2})^{*_1}$ is the iterated rearrangement of $K$ introduced by Blozinski~\cite{B} and $S$ is the Stieltjes transform. Explicit Muckenhoupt-type conditions are obtained for the case $L(t,s)=(t+s)^{-1}$, corresponding to the iterated Stieltjes operator $S^2$.

math.FA↗

arXiv2012

A New Proof Of The Asymptotic Limit Of The $Lp$ Norm Of The Sinc Function

We improve on the inequality $\displaystyle{\frac{1}{\pi}\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt\leq \frac{1}{\sqrt p}, {0.2 cm}p\geq 1,}$ showing that $\displaystyle{\frac{1}{\pi}\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt\leq C(p) \frac{\sqrt{3/\pi}}{\sqrt p},}$ with $\displaystyle{\lim_{p\longrightarrow \infty} C(p)=1,}$ and indeed that {align*} \displaystyle{\lim_{p\longrightarrow \infty}\frac{1}{\pi}\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt/ \frac{\sqrt{3/\pi}}{\sqrt p}=1.} {align*}

math.FA↗