Characterizations are obtained for those pairs of weight functions u and v for which the operators with a and b certain non-negative functions are bounded from to , 0 < p,q < ∞, p≥ 1. Sufficient conditions are given for T to be bounded on the cones of monotone functions. The results are applied to give a weighted inequality comparing differences and derivatives as well as a weight characterization for the Steklov operator.
We analyse mean values of functions with values in the boundary of a convex two-dimensional set. As an application, reverse integral inequalities imply exactly the same inequalities for the monotone rearrangement. Sharp versions of the classical Gehring lemma, the Gurov-Resetnyak theorem and the Muckenhoupt theorem are obtained.
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 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 where and are monotone increasing functions of .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 introduced by F. Qi.
We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere , on a bounded domain and on all of . In some cases we also address the question of existence of minimizers.