If $P$ is the Hardy averaging operator - or some of its generalizations, then weighted modular inequalities of the form u (Pf) Cv (f) are established for a general class of functions $\phi$. Modular inequalities for the two- and higher dimensional Hardy averaging operator are also given.

Relations between moduli of smoothness of the derivatives of a function and those of the function itself are investigated. The results are for $L_p\left(T\right)$ and $L_p[-1,1]$ for 0 < p < ∞ using the moduli of smoothness ${\omega }^r{(f,t)}_p$ and ${\omega }_{\phi }^r{(f,t)}_p$ respectively.

This paper is an informal presentation of material from [28]–[34]. The monotone envelopes of a function, including the level function, are introduced and their properties are studied. Applications to norm inequalities are given. The down space of a Banach function space is defined and connections are made between monotone envelopes and the norms of the down space and its dual. The connection is shown to be particularly close in the case of universally rearrangement invariant spaces. Next, two equivalent...

We show here that a wide class of integral inequalities concerning functions on $[0,1]$ can be obtained by purely combinatorial methods. More precisely, we obtain modulus of continuity or other high order norm estimates for functions satisfying conditions of the type ${\int }_0^1{\int }_0^1\Psi \left(\frac{f\left(x\right)-f\left(y\right)}{p(x-y)}\right)dxdy\&lt;\infty$ where $\Psi \left(u\right)$ and $p\left(u\right)$ are monotone increasing functions of $\left|u\right|$.Several applications are also derived. In particular these methods are shown to yield a new condition for path continuity of general stochastic processes

The author gives a new simple proof of monotonicity of the generalized extended mean values $M(r,s)=\le {(\left(\int f^{s}d\mu \right)/\left(\int f^{r}d\mu \right))}^{1/(s-r)}$ introduced by F. Qi.

Let 1 < p < ∞, q = p/(p-1) and for $f\in L^p(0,\infty )$ define $F\left(x\right)=(1/x){ʃ}_0^{x}f\left(t\right)dt$, x > 0. Moser’s Inequality states that there is a constant $C_p$ such that $su{p}_{a\le 1}su{p}_{f\in B_p}{ʃ}_0^{\infty }exp[a{x}^q{\left|F\left(x\right)\right|}^q-x]dx=C_p$ where $B_p$ is the unit ball of $L^p$. Moreover, the value a = 1 is sharp. We observe that $F=K_1$ f where the integral operator $K_1$ has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for the analogue...

We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S^2$, on a bounded domain $\Omega \subset {ℝ}^2$ and on all of ${ℝ}^2$. In some cases we also address the question of existence of minimizers.