arXiv subjects

Nevin Bilgiçli

Publications and source records attributed to Nevin Bilgiçli.

5 recordsLinked to original sources

arXiv2021

Norms of maximal functions between generalized and classical Lorentz spaces

In this paper we calculate the norm of the generalized maximal operator $M_{ϕ,Λ^α(b)}$, defined with $0 < α< \infty$ and functions $b,\,ϕ: (0,\infty) \rightarrow (0,\infty)$ for all measurable functions $f$ on ${\mathbb R}^n$ by \begin{equation*} M_{ϕ,Λ^α(b)}f(x) : = \sup_{Q \ni x} \frac{\|f χ_Q\|_{Λ^α(b)}}{ϕ(|Q|)}, \qquad x \in {\mathbb R}^n, \end{equation*} from ${\operatorname{GΓ}}(p,m,v)$ into $Λ^q(w)$. Here $Λ^α(b)$ and ${\operatorname{GΓ}}(p,m,w)$ are the classical and generalized Lorentz spaces, defined as a set of all measurable functions $f$ defined on ${\mathbb R}^n$ for which $$ \|f\|_{Λ^α(b)} = \bigg( \int_0^{\infty} [f^*(s)]^α b(s)\,ds \bigg)^{\frac{1}α} < \infty \quad \mbox{and} \quad \|f\|_{\operatorname{GΓ}(p,m,w)} = \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (τ)]^p\,dτ\bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} < \infty, $$ respectively. We reduce the problem to the solution of the inequality \begin{equation*} \bigg( \int_0^{\infty} \big[ T_{u,b}f^* (x)\big]^q \, w(x)\,dx\bigg)^{\frac{1}{q}} \le C \, \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (τ)]^p\,dτ\bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} \end{equation*} where $w$ and $v$ are weight functions on $(0,\infty)$. Here $f^*$ is the non-increasing rearrangement of $f$ defined on ${\mathbb R}^n$ and $T_{u,b}$ is the iterated Hardy-type operator involving suprema, which is defined for a measurable non-negative function $f$ on $(0,\infty)$ by $$ (T_{u,b} g)(t) : = \sup_{τ\in [t,\infty)} \frac{u(τ)}{B(τ)} \int_0^τ g(s)b(s)\,ds,\qquad t \in (0,\infty), $$ where $u$ and $b$ are appropriate weight functions on $(0,\infty)$ and the function $B(t) : = \int_0^t b(s)\,ds$ satisfies $0 < B(t) < \infty$ for every $t \in (0,\infty)$..

math.FA↗

arXiv2021

On some restricted inequalities for the iterated Hardy-type operator involving suprema and their applications

In this paper we characterize the inequality \begin{equation*} \bigg( \int_0^{\infty} \bigg( \int_0^x \big[ T_{u,b}f^* (t)\big]^r\,dt\bigg)^{\frac{q}{r}} w(x)\,dx\bigg)^{\frac{1}{q}} \le C \, \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (τ)]^p\,dτ\bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} \end{equation*} for $1 < m < p \le r < q < \infty$ or $1 < m \le r < \min\{p,q\} < \infty$, where $w$ and $v$ are weight functions on $(0,\infty)$. The inequality is required to hold with some positive constant $C$ for all measurable functions defined on measure space $({\mathbb R}^n,dx)$. Here $f^*$ is the non-increasing rearrangement of a measurable function $f$ defined on ${\mathbb R}^n$ and $T_{u,b}$ is the iterated Hardy-type operator involving suprema, whish is defined for a measurable non-negative function $f$ on $(0,\infty)$ by $$ (T_{u,b} g)(t) : = \sup_{t \le τ< \infty} \frac{u(τ)}{B(τ)} \int_0^τ g(s)b(s)\,ds,\qquad t \in (0,\infty), $$ where $u$ and $b$ are two weight functions on $(0,\infty)$ such that $u$ is continuous on $(0,\infty)$ and the function $B(t) : = \int_0^t b(s)\,ds$ satisfies $0 < B(t) < \infty$ for every $t \in (0,\infty)$. At the end of the paper, as an application of obtained results, we calculate the norm of the generalized maximal operator $M_{ϕ,Λ^α(b)}$, defined with $0 < α< \infty$ and functions $b,\,ϕ: (0,\infty) \rightarrow (0,\infty)$ for all measurable functions $f$ on ${\mathbb R}^n$ by \begin{equation*} M_{ϕ,Λ^α(b)}f(x) : = \sup_{Q \ni x} \frac{\|f χ_Q\|_{Λ^α(b)}}{ϕ(|Q|)}, \qquad x \in {\mathbb R}^n, \end{equation*} from ${\operatorname{GΓ}}(p_1,m_1,v)$ into ${\operatorname{GΓ}}(p_2,m_2,w)$. Here $Λ^α(b)$ and ${\operatorname{GΓ}}(p,m,w)$ are the classical and generalized Lorentz spaces, respectively.

math.FA↗

arXiv2019

Boundedness of weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Cesàro function spaces

In this paper the boundedness of the weighted iterated Hardy-type operators $T_{u,b}$ and $T_{u,b}^*$ involving suprema from weighted Lebesgue space $L_p(v)$ into weighted Cesàro function spaces ${\operatorname{Ces}}_{q}(w,a)$ are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator $R_u$ from $L^p(v)$ into ${\operatorname{Ces}}_{q}(w,a)$ on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator $P_{u,b }$ from $L^p(v)$ into ${\operatorname{Ces}}_{q}(w,a)$ on the cone of monotone non-increasing functions. Under additional condition on $u$ and $b$, we are able to characterize the boundedness of weighted iterated Hardy-type operator $T_{u,b}$ involving suprema from $L^p(v)$ into ${\operatorname{Ces}}_q(w,a)$ on the cone of monotone non-increasing functions. At the end of the paper, as an application of obtained results, we calculate the norm of the fractional maximal function $M_γ$ from $Λ^p(v)$ into $Γ^q(w)$.

math.FA↗

arXiv2018

Multidimensional bilinear Hardy inequalities

Our goal in this paper is to find a characterization of $n$-dimensional bilinear Hardy inequalities \begin{align*} \bigg\| \,\int_{B(0,\cdot)} f \cdot \int_{B(0,\cdot)} g \,\bigg\|_{q,u,(0,\infty)} & \leq C \, \|f\|_{p_1,v_1,{\mathbb R}^n} \, \|g\|_{p_2,v_2,{\mathbb R}^n}, \quad f,\,g \in {\mathfrak M}^+ ({\mathbb R}^n), \end{align*} and \begin{align*} \bigg\| \,\int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)} f \cdot \int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)} g \,\bigg\|_{q,u,(0,\infty)} &\leq C \, \|f\|_{p_1,v_1,{\mathbb R}^n} \, \|g\|_{p_2,v_2,{\mathbb R}^n}, \quad f,\,g \in {\mathfrak M}^+ ({\mathbb R}^n), \end{align*} when $0 < q \le \infty$, $1 \le p_1,\,p_2 \le \infty$ and $u$ and $v_1,\,v_2$ are weight functions on $(0,\infty)$ and ${\mathbb R}^n$, respectively. Since the solution of the first inequality can be obtained from the characterization of the second one by usual change of variables we concentrate our attention on characterization of the latter. The characterization of this inequality is easily obtained for the range of parameters when $p_1 \le q$ using the characterizations of multidimensional weighted Hardy-type inequalites while in the case when $q < p_1$ the problem is reduced to the solution of multidimensional weighted iterated Hardy-type inequality. To achieve the goal, we characterize the validity of multidimensional weighted iterated Hardy-type inequality $$ \left\|\left\|\int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,μ,(0,\infty)}\leq c \|h\|_{θ,v,(0,\infty)},~ h \in \mathfrak{M}^+({\mathbb R}^n) $$ where $0 < p,\,q < +\infty$, $1 \leq θ\le \infty$, $u\in {\mathcal W}(0,\infty)$, $v \in {\mathcal W}({\mathbb R}^n)$ and $μ$ is a non-negative Borel measure on $(0,\infty)$.

math.FA↗

arXiv2015

Generalized fractional maximal functions in Lorentz spaces

In this paper we give the complete characterization of the boundedness of the generalized fractional maximal operator $$ M_{ϕ,Λ^α(b)}f(x) : = \sup_{Q \ni x} \frac{\|f χ_Q\|_{Λ^α(b)}}{ϕ(|Q|)} \qquad (x \in {\mathbb R}^n), $$ between the classical Lorentz spaces $Λ^p (v)$ and $Λ^q(w)$ for appropriate functions $ϕ$, where $0 < p,\,q < \infty$, $0 < α\le r < \infty$, $v,w,\,b$ are weight functions on $(0,\infty)$ such that $0 < B(x): = \int_0^x b < \infty$, $x > 0$, $B \in Δ_2$ and $B(t) / t^{α/ r}$ is quasi-increasing.

math.FA↗