Monotonicity in Banach function spaces

Sinnamon, Gord

  • Nonlinear Analysis, Function Spaces and Applications, Publisher: Institute of Mathematics of the Academy of Sciences of the Czech Republic(Praha), page 205-240

Abstract

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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 norms are given for the down spaces and these are applied to advance a factorization theory for Hardy inequalities and to characterize embeddings of the classes of generalized quasiconcave functions between Lebesgue spaces. This embedding theory is, in turn, applied to find an expression for the dual space of Lorentz Γ -space and to find necessary and sufficient conditions for the boundedness of the Fourier transform, acting as a map between Lorentz spaces. A new Lorentz space, the Θ -space, is introduced and shown to be the key to extending the characterization of Fourier inequalities to a greater range of Lorentz spaces. Finally, the scale of down spaces of universally rearrangement invariant spaces is completely characterized by means of interpolation theory, when it is shown that the down spaces of L 1 and L ∞ (with general measures) form a Calderón couple.

How to cite

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Sinnamon, Gord. "Monotonicity in Banach function spaces." Nonlinear Analysis, Function Spaces and Applications. Praha: Institute of Mathematics of the Academy of Sciences of the Czech Republic, 2007. 205-240. <http://eudml.org/doc/220888>.

@inProceedings{Sinnamon2007,
abstract = {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 norms are given for the down spaces and these are applied to advance a factorization theory for Hardy inequalities and to characterize embeddings of the classes of generalized quasiconcave functions between Lebesgue spaces. This embedding theory is, in turn, applied to find an expression for the dual space of Lorentz $\Gamma $-space and to find necessary and sufficient conditions for the boundedness of the Fourier transform, acting as a map between Lorentz spaces. A new Lorentz space, the $\Theta $-space, is introduced and shown to be the key to extending the characterization of Fourier inequalities to a greater range of Lorentz spaces. Finally, the scale of down spaces of universally rearrangement invariant spaces is completely characterized by means of interpolation theory, when it is shown that the down spaces of $L^1$ and $L^\infty $ (with general measures) form a Calderón couple.},
author = {Sinnamon, Gord},
booktitle = {Nonlinear Analysis, Function Spaces and Applications},
keywords = {Monotone envelope; level function; pushing mass; down space; Hardy inequality; Lorentz pace; rearrangement invariant space; quasiconcave function; Fourier inequality; interpolation; Calderón couple},
location = {Praha},
pages = {205-240},
publisher = {Institute of Mathematics of the Academy of Sciences of the Czech Republic},
title = {Monotonicity in Banach function spaces},
url = {http://eudml.org/doc/220888},
year = {2007},
}

TY - CLSWK
AU - Sinnamon, Gord
TI - Monotonicity in Banach function spaces
T2 - Nonlinear Analysis, Function Spaces and Applications
PY - 2007
CY - Praha
PB - Institute of Mathematics of the Academy of Sciences of the Czech Republic
SP - 205
EP - 240
AB - 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 norms are given for the down spaces and these are applied to advance a factorization theory for Hardy inequalities and to characterize embeddings of the classes of generalized quasiconcave functions between Lebesgue spaces. This embedding theory is, in turn, applied to find an expression for the dual space of Lorentz $\Gamma $-space and to find necessary and sufficient conditions for the boundedness of the Fourier transform, acting as a map between Lorentz spaces. A new Lorentz space, the $\Theta $-space, is introduced and shown to be the key to extending the characterization of Fourier inequalities to a greater range of Lorentz spaces. Finally, the scale of down spaces of universally rearrangement invariant spaces is completely characterized by means of interpolation theory, when it is shown that the down spaces of $L^1$ and $L^\infty $ (with general measures) form a Calderón couple.
KW - Monotone envelope; level function; pushing mass; down space; Hardy inequality; Lorentz pace; rearrangement invariant space; quasiconcave function; Fourier inequality; interpolation; Calderón couple
UR - http://eudml.org/doc/220888
ER -

References

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