We prove norm inequalities between Lorentz and Besov-Lipschitz spaces of fractional smoothness.

In this paper, characterizations of the embeddings between weighted Copson function spaces ${\mathrm{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\mathrm{Ces}}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $$\begin{array}{cc}\hfill d({\int }_0^{\infty }& {\left({\int }_0^{t}f{\left(\tau \right)}^{p_2}{v}_2\left(\tau \right)\mathrm{d}\tau \right)}^{q_2/p_2}{u}_2\left(t\right)\mathrm{d}t{)}^{1/q_2}\hfill \\ & \le c{\left({\int }_0^{\infty }{\left({\int }_t^{\infty }f{\left(\tau \right)}^{p_1}{v}_1\left(\tau \right)\mathrm{d}\tau \right)}^{q_1/p_1}{u}_1\left(t\right)\mathrm{d}t\right)}^{1/q_1},\hfill \end{array}d$$ where $p_1,p_2,q_1,q_2\in (0,\infty )$, $p_2\le q_2$ and $u_1$, $u_2$, $v_1$, $v_2$ are weights on $(0,\infty )$, are obtained. The most innovative part consists of the fact that possibly different parameters $p_1$ and $p_2$ and possibly different inner weights $v_1$ and $v_2$ are allowed. The proof is based on the combination of duality techniques with estimates...

A simple expression is presented that is equivalent to the norm of the Lpv → Lqu embedding of the cone of quasi-concave functions in the case 0 < q < p < ∞. The result is extended to more general cones and the case q = 1 is used to prove a reduction principle which shows that questions of boundedness of operators on these cones may be reduced to the boundedness of related operators on whole spaces. An equivalent norm for the dual of the Lorentz spaceΓp(v) = { f: ( ∫0∞ (f**)pv...

The aim of this article is to give new refinements and sharpenings of Shafer's inequality involving the arctangent function. These are obtained by means of a change of variables, which makes the computations much easier than the classical approach.