论文标题
涉及Hardy和Copson运营商的加权不平等
Weighted inequalities involving Hardy and Copson operators
论文作者
论文摘要
我们表征了涉及Hardy操作员和Copson操作员的四重不平等现象。更准确地说，给定$ p_1，p_2，q_1，q_2 \ in（0，\ infty）$，我们在非负可测量函数上找到必要且充分的条件，$ u_1，u_2，v_1，v_1，v_2，v_2 $ on $（0，\ infty）$，存在正常常数$ c $ acign $ c $ quinign* ＆\ bigG（\ int_0^{\ infty} \ big（\ int_0^t f（s）^{p_2} v_2（s）^{p_2} ds \ bigG）^{\ frac {q_2} \ bigG）^{\ frac {1} {q_2}}} \ notag \\＆\ hspace {3cm} \ leq c \ bigG（\ int_0^{\ int_0^{\ infty} \ bigG（\ int_t_t^{ \ bigG）^{\ frac {q_1} {p_1}} u_1（t）^{q_1} dt \ bigG）^{\ frac {1} {1} {q_1}}} \ end End {align {align {align {align*}都可以容纳每个非尼的测量函数，该证明是基于离散和抗化技术的。主要的创新是开发一种新方法，该方法仔细避免了二元技术，因此使我们能够在以前不可用的情况下获得表征，从而解决了一个长期的开放问题。 We then apply the characterization of the inequality to the establishing of criteria for embeddings between weighted Copson spaces $\operatorname{Cop}_{p_1,q_1} (u_1, v_1)$ and weighted Cesàro spaces $\operatorname{Ces}_{p_2, q_2} (u_2, v_2)$, and also between spaces $ s^q（w）$配备了标准$ \ | | _ {s^q（w）} = \ bigG（\ int_0^\ infty [f^{**}（t）（t）-f^*（t）-f^*（t）]^q W（t）]^q W（t）
We characterize a four-weight inequality involving the Hardy operator and the Copson operator. More precisely, given $p_1, p_2, q_1, q_2 \in (0, \infty)$, we find necessary and sufficient conditions on nonnegative measurable functions $u_1, u_2, v_1, v_2$ on $(0,\infty)$ for which there exists a positive constant $c$ such that the inequality \begin{align*} &\bigg(\int_0^{\infty} \bigg(\int_0^t f(s)^{p_2} v_2(s)^{p_2} ds \bigg)^{\frac{q_2}{p_2}} u_2(t)^{q_2} dt \bigg)^{\frac{1}{q_2}} \notag \\ & \hspace{3cm} \leq c \bigg(\int_0^{\infty} \bigg(\int_t^{\infty} f(s)^{p_1} v_1(s)^{p_1} ds \bigg)^{\frac{q_1}{p_1}} u_1(t)^{q_1} dt \bigg)^{\frac{1}{q_1}} \end{align*} holds for every non-negative measurable function $f$ on $(0, \infty)$. The proof is based on discretizing and antidiscretizing techniques. The principal innovation consists in development of a new method which carefully avoids duality techniques and therefore enables us to obtain the characterization in previously unavailable situations, solving thereby a long-standing open problem. We then apply the characterization of the inequality to the establishing of criteria for embeddings between weighted Copson spaces $\operatorname{Cop}_{p_1,q_1} (u_1, v_1)$ and weighted Cesàro spaces $\operatorname{Ces}_{p_2, q_2} (u_2, v_2)$, and also between spaces $S^q(w)$ equipped with the norm $\|f\|_{S^q(w)}= \bigg(\int_0^\infty [f^{**}(t)-f^*(t)]^q w(t)\,dt\bigg)^{{1}/{q}}$ and classical Lorentz spaces of type $Λ$.
