Non-negative Non-increasing Functions Research Articles

On the optimal relationships between [formula omitted]-norms for the Hardy operator and its dual for decreasing functions

We prove sharp inequalities between Lp−norms (1<p<∞) of functions Hf and H∗f, where H is the Hardy operator, H∗ is its dual, and f is a nonnegative nonincreasing function on (0,∞). In particular, we extend one result obtained for integer p≥2 by Boza and Soria (2019), to the whole range of values p≥2.

Construction of a Set of Restricted Inertial Controls for C(1)-Smooth Affine Systems with Multidimensional Control

Affine control systems of the class C(1) with multidimensional control are considered. For those systems which can be reduced to linear systems, the inertial synthesis problem is solved, that is, the problem of finding feedback controls satisfying preassigned constraints on a control and its derivatives up to a given order l. The problem of stabilization by use of inertial controls is also solved. The controllability function method is the basis of the investigation. It is shown that each collection f of r nonnegative non-increasing functions which have no less than n1,?,nr points of decrease ( n1+?+nr=n), respectively, and satisfy certain conditions generates a family of controllability functions {?f,?(x)} and a family of controls {uf,?(x)} which transfer an arbitrary point x0 from a certain neighborhood of the origin to the origin in some finite time Tf,?(x0) and satisfy given constraints if ??2l+1. We estimate the time of motion from below and from above. In the limiting case ?=?, the function ?f(x) is a Lyapunov function and uf(x) solves the stabilization problem and satisfies given constraints.

Norm inequalities for some one-sided operators

We show that the one-sided maximal operators associated with Borel measures are of strong type (p, p) , 1 < p < ∞ , with constant p∗ , and the related one-sided geometric maximal operators are of strong type (p, p) , 0 < p < ∞ , with constant e1/p . We also investigate norm inequalities for integral operators with three measures on the cone of nonnegative nonincreasing functions. Our results show that if we restrict the measures in the inequalities to some particular classes, then a simple characterization for these inequalities to hold can be obtained. Mathematics subject classification (2010): Primary 26D15; Secondary 26D10, 42B25.

Optimal relationships between $$L^p$$ -norms for the Hardy operator and its dual

We obtain sharp two-sided inequalities between $$L^p$$ -norms $$(1<p<\infty )$$ of functions $$\textit{Hf}$$ and $$H^*f$$ , where $$H$$ is the Hardy operator, $$H^*$$ is its dual, and $$f$$ is a nonnegative measurable function on $$(0,\infty ).$$ In an equivalent form, it gives sharp constants in the two-sided relationships between $$L^p$$ -norms of functions $$H\varphi -\varphi $$ and $$\varphi $$ , where $$\varphi $$ is a nonnegative nonincreasing function on $$(0,+\infty )$$ with $$\varphi (+\infty )=0.$$ In particular, it provides an alternative proof of a result obtained by Kruglyak and Setterqvist (Proc Am Math Soc 136:2005–2013, 2008) for $$p=2k \,\,(k\in \mathbb N )$$ and by Boza and Soria (J Funct Anal 260:1020–1028, 2011) for all $$p\ge 2$$ , and gives a sharp version of this result for $$1<p<2$$ .

Boundedness of the anisotropic Riesz potential in anisotropic local Morrey-type spaces

The problem of boundedness of the anisotropic Riesz potential in local Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.

Boundedness of the Riesz Potential in Local Morrey-Type Spaces

The problem of boundedness of the Riesz potential in local Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.

Necessary and Sufficient Conditions for the Boundedness of the Riesz Potential in Local Morrey-type Spaces

The problem of the boundedness of the Riesz potential Iα, 0 < α < n, in local Morrey-type spaces is reduced to the boundedness of the Hardy operator in weighted Lp-spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for the boundedness in local Morrey-type spaces for all admissible values of the parameters. Moreover, for a certain range of the parameters, these sufficient conditions coincide with the necessary ones.

Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces

The problem of the boundedness of the fractional maximal operator M α , 0 < α < n , in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.

Risk Theory with the Generalized Inverse Gaussian Lévy Process

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such model is either a compound Poisson process itself or a strictly increasing Lévy process. Their construction is based on a non-negative non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. We discuss an illustration of their model using a generalized Inverse Gaussian (GIG) Lévy process. This increasing Lévy process has the gamma and the inverse Gaussian process as particular cases. Although mathematically more complex, the GIG Lévy process keeps some of the nice properties of the simpler gamma process.

Risk Theory with the Generalized Inverse Gaussian Lévy Process

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such model is either a compound Poisson process itself or a strictly increasing Lévy process. Their construction is based on a non-negative non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. We discuss an illustration of their model using a generalized Inverse Gaussian (GIG) Lévy process. This increasing Lévy process has the gamma and the inverse Gaussian process as particular cases. Although mathematically more complex, the GIG Lévy process keeps some of the nice properties of the simpler gamma process.

Weighted Generalized Hardy Inequalities for Nonincreasing Functions

AbstractThe nonnegative weight function pairs u, v for which the operator maps the nonnegative nonincreasing functions in LP(v) boundedly into weak Lq(u) are characterized. This result is used, in particular, both to generalize and to provide an alternate proof of certain strong type inequalities recently obtained by Ariño and Muckenhouptfor the Hardy averaging operator restricted to nonnegative nonincreasing functions.