Multidimensional Bilinear Hardy Inequalities

Autor: Bilgiçli, N., Rza Mustafayev, Ünver, T.
Přispěvatelé: Kırıkkale Üniversitesi, KKÜ
Jazyk: angličtina
Rok vydání: 2020
Předmět:
Zdroj: Scopus-Elsevier
Popis: 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 $$ \left\|\left\|\int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,��,(0,\infty)}\leq c \|h\|_{��,v,(0,\infty)},~ h \in \mathfrak{M}^+({\mathbb R}^n) $$ where $0 < p,\,q < +\infty$, $1 \leq ��\le \infty$, $u\in {\mathcal W}(0,\infty)$, $v \in {\mathcal W}({\mathbb R}^n)$ and $��$ is a non-negative Borel measure on $(0,\infty)$.
22 pages. arXiv admin note: text overlap with arXiv:1302.3436
Databáze: OpenAIRE
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