Orlicz-Lorentz Gauge Functional Inequalities for Positive Integral Operators. Revised Version

Let $f \in M_+(\R_+)$, the class of nonnegative, Lebesgure-measurable functions on $\R_+=(0, \infty)$. We deal with integral operators of the form $$(T_Kf)(x)=\int_{\R_+}K(x,y)f(y)\, dy, \quad x \in \R_+,$$ with $K \in M_+(\R_+^2)$. We are interested in inequalities $$\rho_{1}((T_Kf)^*)\leq C\rho_2(f^*),$$ in which $\rho_1$ and $\rho_2$ are functionals on functions $h \in M_+(\R_+)$, and $$h^*(t)=\mu_h^{-1}(t), \quad t \in \R_+,$$ where $$\mu_h(\lambda)=|\{x \in \R_+: \, h(x)> \lambda\}|, \lambda \in \R_+.$$ Specifically, $\rho_1$ and $\rho_2$ are so-called Orlicz-Lorentz gauge functionals of the type $$\rho(h)=\rho_{\Phi, u}(h)=\inf\left\{\lambda>0:\, \int_{\R_+}\Phi\left(\frac{h(x)}{\lambda}\right)u(x)\, dx \leq 1\right\}, \quad h \in M_+(\R_+);$$ here $\Phi(x)=\int_0^x\phi(y)\, dy$, $\phi$ an increasing function mapping $\R_+$ onto itself and $u\in M_+(\R_+)$.Comment: arXiv admin note: substantial text overlap with arXiv:2102.1143

Marcinkiewicz interpolation theorems for Orlicz and Lorentz gamma spaces

This research was supported in part by NSERC grant A4021, an USRA grant from NSERC, grant MSM 0021620839 of the Czech Ministry of Education, grants 201/07/0388 and 201/08/0383 of the Grant Agency of the Czech Republic, NATO grant PST.CLG.978798, Leverhulme Trust Grant n.F/00407/E and by the Necas Center for Mathematical Modelling project no. LC06052 financed by the Czech Ministry of Education

A new algorithm for approximating the least concave majorant

summary:The least concave majorant, $\hat F$, of a continuous function $F$ on a closed interval, $I$, is defined by $$\hat F (x) = \inf \{ G(x)\colon G \geq F,\ G \text { concave}\},\quad x \in I.$$ We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on $I$. Given any function $F \in \mathcal {C}^4(I)$, it can be well-approximated on $I$ by a clamped cubic spline $S$. We show that $\hat S$ is then a good approximation to $\hat F$. \endgraf We give two examples, one to illustrate, the other to apply our algorithm

Weighted LΦL_{Φ} integral inequalities for operators of Hardy type

Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for $Φ_2^{-1} (ʃΦ_2(w(x)|Tf(x)|)t(x)dx) ≤ Φ_{1}^{-1}(ʃΦ_{1}(Cu(x)|f(x)|)v(x)dx)$ to hold when $Φ_1$ and $Φ_2$ are N-functions with $Φ_2∘Φ_{1}^{-1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored

On the Brudny˘ı-Krugljak Duality Theory of Spaces Formed by the K-Method of Interpolation

The Brudny˘ı-Krugljak duality theory for the K-method is elaborated for a class of parameters derived from rearrangement-invariant spaces. As examples, concrete expressions are given for the norms dual to certain interpolation spaces between two rearrangement-invariant spaces. These interpolation spaces are formed by the K-method using parameters related to classical Lorentz spaces or Orlicz spaces