arXiv subjects
Publications and source records attributed to Merve Yılmaz.
In this paper, new equivalence theorems for the boundedness of the composition of a quasilinear operator $T$ with the Hardy and Copson operators in weighted Lebesgue spaces are proved. The usefulness of the obtained results is illustrated in the case of weighted Hardy-type and weighted iterated Hardy-type inequalities.
In this paper, we present a solution to the inequality $$ \bigg( \int_0^{\infty} \bigg( \int_x^{\infty} \bigg( \int_0^t h \bigg)^q w(t)\,dt \bigg)^{r / q} u(x)\,ds \bigg)^{1/r}\leq C \, \bigg( \int_0^{\infty} h^p v \bigg)^{1 / p}, \quad h \in {\mathfrak M}^+(0,\infty), $$ using a combination of reduction techniques and discretization. Here $1 \le p < \infty$, $0 < q ,\, r < \infty$ and $u,\,v,\,w$ are weight functions on $(0,\infty)$.
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 $t \in (0,\infty)$..
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.