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Monotonicity in Banach function spaces

Sinnamon, Gord — 2007

Nonlinear Analysis, Function Spaces and Applications

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...

Spaces defined by the level function and their duals

Gord Sinnamon — 1994

Studia Mathematica

The classical level function construction of Halperin and Lorentz is extended to Lebesgue spaces with general measures. The construction is also carried farther. In particular, the level function is considered as a monotone map on its natural domain, a superspace of L p . These domains are shown to be Banach spaces which, although closely tied to L p spaces, are not reflexive. A related construction is given which characterizes their dual spaces.

The level function in rearrangement invariant spaces.

Gord Sinnamon — 2001

Publicacions Matemàtiques

An exact expression for the down norm is given in terms of the level function on all rearrangement invariant spaces and a useful approximate expression is given for the down norm on all rearrangement invariant spaces whose upper Boyd index is not one.

Embeddings of concave functions and duals of Lorentz spaces.

Gord Sinnamon — 2002

Publicacions Matemàtiques

A simple expression is presented that is equivalent to the norm of the Lp v → Lq u embedding of the cone of quasi-concave functions in the case 0 < q < p < ∞. The result is extended to more general cones and the case q = 1 is used to prove a reduction principle which shows that questions of boundedness of operators on these cones may be reduced to the boundedness of...

Transferring monotonicity in weighted norm inequalities.

Gord Sinnamon — 2003

Collectanea Mathematica

Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions...

From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces

María CarroLeonardo ColzaniGord Sinnamon — 2007

Studia Mathematica

Let X be a quasi-Banach rearrangement invariant space and let T be an (ε,δ)-atomic operator for which a restricted type estimate of the form T χ E X D ( | E | ) for some positive function D and every measurable set E is known. We show that this estimate can be extended to the set of all positive functions f ∈ L¹ such that | | f | | 1 , in the sense that T f X D ( | | f | | ) . This inequality allows us to obtain strong type estimates for T on several classes of spaces as soon as some information about the galb of the space X is known. In this paper...

A new algorithm for approximating the least concave majorant

Martin FrancůRon KermanGord Sinnamon — 2017

Czechoslovak Mathematical Journal

The least concave majorant, F ^ , of a continuous function F on a closed interval, I , is defined by F ^ ( x ) = inf { G ( x ) : G F , G concave } , x 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 𝒞 4 ( I ) , it can be well-approximated on I by a clamped cubic spline S . We show that S ^ is then a good approximation to F ^ . We give two examples, one to illustrate, the other to apply our algorithm.

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