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Keywords:
Monotone envelope; level function; pushing mass; down space; Hardy inequality; Lorentz pace; rearrangement invariant space; quasiconcave function; Fourier inequality; interpolation; Calderón couple
Monotone envelope; level function; pushing mass; down space; Hardy inequality; Lorentz pace; rearrangement invariant space; quasiconcave function; Fourier inequality; interpolation; Calderón couple
Summary:
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.
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