Weighted multidimensional inequalities for monotone functions

summary:We discuss the characterization of the inequality \biggl(\int_{{\Bbb R}^N_+} f^q u\biggr)^{1/q} \leq C \biggl(\int_{{\Bbb R}^N_+} f^p v \biggr)^{1/p},\quad0<q, p <\infty, for monotone functions $f\geq0$ and nonnegative weights $u$ and $v$ and $N\geq1$. We prove a new multidimensional integral modular inequality for monotone functions. This inequality generalizes and unifies some recent results in one and several dimensions

Hardy's inequalities for monotone functions on partially ordered measure spaces

We characterize the weighted Hardy's inequalities for monotone functions in ${\mathbb R^n_+}.$ In dimension $n=1$, this recovers the classical theory of $B_p$ weights. For $n>1$, the result was only known for the case $p=1$. In fact, our main theorem is proved in the more general setting of partially ordered measure spaces.Comment: 14 page

End-point Norm Estimates for Ces\`aro and Copson Operators

For a large class of operators acting between weighted $\ell^\infty$ spaces, exact formulas are given for their norms and the norms of their restrictions to the cones of nonnegative sequences; nonnegative, nonincreasing sequences; and nonnegative, nondecreasing sequences. The weights involved are arbitrary nonnegative sequences and may differ in the domain and codomain spaces. The results are applied to the Ces\`aro and Copson operators, giving their norms and their distances to the identity operator on the whole space and on the cones. Simplifications of these formulas are derived in the case of these operators acting on power-weighted $\ell^\infty$. As an application, best constants are given for inequalities relating the weighted $\ell^\infty$ norms of the Ces\`aro and Copson operators both for general weights and for power weights.Comment: 27 page

Weighted multidimensional integral inequalities and applications

In this thesis some new weighted integral inequalities for monotone functions in higher dimensions are proved. These results extend previous results in one dimension, and also those in higher dimensions for particular choices of the weights (power weights, etc.). All inequalities are sharp. A new duality principle (of Sawyer type) over the cone of multidimensional monotone functions is proved and applied. Weighted Chebyshev type inequalities for monotone functions and modular inequalities in higher dimensions are proved. A new type of weighted function spaces are introduced. In particular these spaces generalize the classical Lebesgue spaces. The weights such that they become quasi-Banach spaces are completely characterized. A multidimensional multiplicative inequality (of Carlson type) for weighted Lebesgue spaces with homogeneous weights is proved and applied. The inequality is sharp and all cases of equality are pointed out.Godkänd; 1999; 20061117 (haneit

Strong and weak-type weighted norm inequalities for the geometric fractional maximal operator

We characterize the strong and weak-type boundedness of the geometric fractional maximal operator between weighted Lebesgue spaces in the case $

Sharp constants between equivalent norms in weighted Lorentz spaces

For an increasing weight $w$ in $B_p$ (or equivalently in $A_p$), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space $\Lambda^p(w)$ and the dual norm. doi:10.1017/S144678870900046

Some new sharp limit Hardy-type inequalities via convexity

Some new limit cases of Hardy-type inequalities are proved, discussed and compared. In particular, some new refinements of Bennett’s inequalities are proved. Each of these refined inequalities contain two constants, and both constants are in fact sharp. The technique in the proofs is new and based on some convexity arguments of independent interest