[Paper Review] Multidimensional bilinear Hardy inequalities

Nevin Bilgiçli, Rza Mustafayev+1 more|arXiv (Cornell University)|May 17, 2018

Advanced Harmonic Analysis Research16 references3 citations

TL;DR

This paper characterizes multidimensional bilinear Hardy inequalities involving integrals over balls and their complements in $\mathbb{R}^n$, establishing necessary and sufficient conditions for the boundedness of $n$-dimensional bilinear Hardy operators from weighted Lebesgue spaces into another weighted Lebesgue space. The key contribution is a complete characterization for all $0 < q \leq \infty$, $1 \leq p_1, p_2 \leq \infty$, using duality, iterated Hardy-type inequalities, and conditions on weights and measures, with explicit constants derived via discretization and duality techniques.

ABSTRACT

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)} &amp; \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)} &amp;\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 $0 &lt; q \le \infty$, $1 \le p_1,\,p_2 \le \infty$ 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 $q &lt; p_1$ 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 $$ \left\|\left\|\int_{\,^{^{\mathsf{c}}}\! B(0,\cdot)}h(z)dz ight\|_{p,u,(0,t)} ight\|_{q,μ,(0,\infty)}\leq c \|h\|_{θ,v,(0,\infty)},~ h \in \mathfrak{M}^+({\mathbb R}^n) $$ where $0 &lt; p,\,q &lt; +\infty$, $1 \leq θ\le \infty$, $u\in {\mathcal W}(0,\infty)$, $v \in {\mathcal W}({\mathbb R}^n)$ and $μ$ is a non-negative Borel measure on $(0,\infty)$.

Motivation & Objective

• To characterize the boundedness of $n$-dimensional bilinear Hardy operators defined via integrals over balls and their complements in $\mathbb{R}^n$.
• To determine necessary and sufficient conditions on weight functions $u$, $v_1$, $v_2$ and measure $\mu$ for the inequality $\| \int_{B(0,\cdot)}f \cdot \int_{B(0,\cdot)}g \|_{q,u,(0,\infty)} \leq C \|f\|_{p_1,v_1} \|g\|_{p_2,v_2}$ to hold.
• To extend the analysis to the case involving complements of balls, $\int_{^cB(0,\cdot)}f \cdot \int_{^cB(0,\cdot)}g$, and reduce it to iterated Hardy-type inequalities.
• To provide a complete characterization for all $0 < q \leq \infty$, $1 \leq p_1, p_2 \leq \infty$, including the challenging case $q < p_1$.
• To derive explicit expressions for the best constant $C$ in terms of norms of weight functions and measures.

Proposed method

• Use of duality techniques to reduce the bilinear inequality to a dual form involving essential suprema and weighted $L^p$ norms.
• Reduction of the main inequality to a multidimensional iterated Hardy-type inequality of the form $\| \| \int_{^cB(0,s)} h(z) dz \|_{p,u,(0,t)} \|_{q,\mu,(0,\infty)} \leq c \|h\|_{\theta,v,(0,\infty)}$.
• Application of known characterizations of weighted Hardy-type inequalities, particularly under the condition that the measure $\mu$ is non-degenerate with respect to $U^{q/p}$.
• Use of discretization and iteration methods to handle the case $q < p_1$, which is more complex than $p_1 \leq q$.
• Leveraging known results from [mu.emb] and [Krep] to derive sharp estimates via duality and norm comparisons.
• Interchange of suprema and use of weak-type estimates to derive equivalent conditions on the weights and measures.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on the weights $u$, $v_1$, $v_2$ and measure $\mu$ for the $n$-dimensional bilinear Hardy inequality involving the complement of balls to hold?
• RQ2 How does the boundedness of the bilinear Hardy operator change when $q < p_1$, compared to the case $p_1 \leq q$?
• RQ3 Can the characterization of the bilinear inequality be reduced to a known class of iterated Hardy-type inequalities?
• RQ4 What is the precise expression for the best constant $C$ in the bilinear Hardy inequality in terms of the given weights and measures?
• RQ5 Under what conditions on the measure $\mu$ does the characterization of the iterated Hardy-type inequality remain valid?

Key findings

• For $p_1 < \infty$, the best constant $C$ in the inequality is equivalent to $\sup_{t \in (0,\infty)} u(t) \|v_1^{-1/p_1}\|_{p_1', \,^cB(0,t)} \|v_2^{-1/p_2}\|_{p_2', \,^cB(0,t)}$ when $p_2 < \infty$.
• When $p_2 = \infty$, the best constant satisfies $C \approx \sup_{t \in (0,\infty)} u(t) \|v_1^{-1/p_1}\|_{p_1', \,^cB(0,t)} \|v_2^{-1}\|_{1, \,^cB(0,t)}$.
• In the case $p_1 = p_2 = \infty$, the best constant satisfies $C \approx \sup_{t \in (0,\infty)} u(t) \|v_1^{-1}\|_{1, \,^cB(0,t)} \|v_2^{-1}\|_{1, \,^cB(0,t)}$.
• The characterization of the bilinear Hardy inequality is fully determined by the behavior of the weights on the complements of balls and the measure $\mu$.
• The problem reduces to characterizing a multidimensional iterated Hardy-type inequality, which is solved under the non-degeneracy condition on $\mu$ with respect to $U^{q/p}$.
• The solution for $p_1 \leq q$ follows from known multidimensional Hardy-type inequalities, while the case $q < p_1$ requires deeper analysis via duality and iterated inequalities.