arXiv subjects

A. Turan Gürkanlı

Publications and source records attributed to A. Turan Gürkanlı.

3 recordsLinked to original sources

arXiv2024

Ditkin sets for some functional spaces and applications to grand Lebesgue spaces

Let $G$ be a locally compact Abelian group with dual group $\widehat G $ and Haar measures $dμ$ and $\hat dμ$ respectively. In this work we have proved that if $X$ is an essential Banach ideal in Beurling algebra $ L^1_ω(G),$ then a closed subset $E\subset \widehat G$ is a Ditkin set for $X$ if and only if $E$ is a Ditkin set for $ L^1_ω(G).$ Next, as an application we have investigated the Ditkin sets for grand Lebesgue space $L^{p),θ}(G)$ and Ditkin sets for $[L^p(G)]_{L^{p),θ}}$, where $[L^p(G)]_{L^{p),θ}}$ is the closure of the set $C_c(G)$ in $L^{p),θ}(G)$.

math.FA↗

arXiv2024

The Generalized Grand Wiener Amalgam Spaces and the boundedness of Hardy-Littlewood maximal operators

In \cite{g5}, we defined and investigated the grand Wiener amalgam space $W(L^{p),θ_1}(Ω), L^{q),θ_2}(Ω))$ , where $1 0, θ_2>0$, $Ω\subset\mathbb R^{n} $ and the Lebesgue measure of $Ω$ is finite. In the present paper we generalize this space and define the generalized grand Wiener amalgam space $W(L_{a}^{p)}(\mathbb R^{n}), L_{b}^{q)}(\mathbb R^{n})),$ where $L_{a}^{p)}(\mathbb R^{n})$ and $L_{b}^{q)}(\mathbb R^{n}),$ are the generalized grand Lebesgue spaces, (see \cite{u}, \cite{su3}). Later we investigate some basic properties. Next we study embeddings for these spaces and we discuss boundedness and unboundedness of the Hardy-Littlewood maximal operator between some generalized grand Wiener amalgam spaces.

math.FA↗

arXiv2020

Bilinear Multipliers of Small Lebesgue spaces

Let $G$ be a locally compact abelian metric group with Haar measure $λ$ and $\hat{G}$ its dual with Haar measure $μ,$ and $λ( G) $ is finite. Assume that$~1<p_{i}<\infty $, $p_{i}^{\prime }=\frac{ p_{i}}{p_{i}-1}$, $( i=1,2,3) $ and $θ\geq 0$. Let $ L^{(p_{i}^{\prime },θ}( G) ,$ $( i=1,2,3) $ be small Lebesgue spaces. A bounded measurable function $m( ξ,η) $ defined on $\hat{G}\times \hat{G}$ is said to be a bilinear multiplier on $G$ of type $[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] _{θ}$ if the bilinear operator $B_{m}$ associated with the symbol $m$, \begin{equation} B_{m}(f,g) ( x) =\sum_{s\in \hat{G} }\sum_{t\in \hat{G}}\hat{f}(s) \hat{g}(t) m(s,t) \langle s+t,x\rangle \end{equation} defines a bounded bilinear operator from $L^{(p_{1}^{\prime },θ}( G) \times L^{(p_{2}^{\prime },θ}( G) $ into $ L^{(p_{3}^{\prime },θ}(G) $. We denote by $BM_{θ} [ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] $ the space of all bilinear multipliers of type $[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] _{θ}$. In this paper, we discuss some basic properties of the space $BM_{θ}[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] $ and give examples of bilinear multipliers.

math.FA↗