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Erschienen in:
Cora Sadosky: Her Mathematics, Mentorship, and Professional Contributions Kapitel lesen Erstes Kapitel lesen

2016 | OriginalPaper | Buchkapitel

BMO: Oscillations, Self-Improvement, Gagliardo Coordinate Spaces, and Reverse Hardy Inequalities

verfasst von : Mario Milman

Erschienen in: Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1)

Verlag: Springer International Publishing

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Abstract

A new approach to classical self improving results for BMO functions is presented. “Coordinate Gagliardo spaces” are introduced and a generalized version of the John-Nirenberg Lemma is proved. Applications are provided.

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Vorheriges Kapitel On the Preservation of Eccentricities of Monge–Ampère Sections
Nächstes Kapitel Besov Spaces, Symbolic Calculus, and Boundedness of Bilinear Pseudodifferential Operators
Fußnoten
1
We refer to section “Interpolation Theory: Some Basic Inequalities” for more details.
 
2
With the usual modification when q = ∞. 
 
3
Paradoxically, except for section “Bilinear Interpolation”, in this paper we do not discuss interpolation theorems per se. For interpolation theorems involving BMO type of spaces there is a large literature. For articles that are related to the developments in this note I refer, for example, to [11, 36, 43, 55, 76, 84].
 
4
Where f ∗ denotes the non-increasing rearrangement of f and \(f^{{\ast}{\ast}}(t) = \frac{1} {t} \int _{0}^{t}f^{{\ast}}(s)ds.\)
 
5
The smallest rearrangement invariant space that contains BMO is e L as was shown by Pustylnik [79].
 
6
Apparently the L(∞, q) spaces for q < ∞ were first introduced and their usefulness shown in [7]. Note that with the usual definition L(∞, ∞) would be L ∞ , and L(∞, q) = { 0}, for q < ∞. The key point here is that the use of the oscillation operator introduces cancellations that make the spaces defined in this fashion nontrivial (cf. section “The Rearrangement Invariant Hull of BMO and Gagliardo Coordinate Spaces”, Example 1).
 
7
Which in turn improves upon the classical exponential integrability result by Trudinger [87].
 
8
Let X be a rearrangement invariant space, Pustylnik [79] has given necessary and sufficient conditions for spaces of functions defined by conditions of the form
$$\displaystyle{\left \Vert \left (f^{{\ast}{\ast}}- f^{{\ast}}\right )t^{-\gamma }\right \Vert _{ X} <\infty }$$
to be linear and normable.
 
9
The improvement is also valid for Besov space inequalities as well (cf. [59]).
 
10
The spaces L(∞, q) allow to interpolate between L ∞  = L(∞, 1) and L(∞, ∞) ⊂ e L . 
 
11
The earlier work of Herz [36] and Holmstedt [38], that precedes [11], should be also mentioned here.
 
12
Interpreting \((\left \Vert D_{1}(t)f\right \Vert _{X_{1}},\left \Vert D_{2}(t)f\right \Vert _{X_{2}})\) as coordinates on the boundary of a Gagliardo diagram (cf. [12]) it follows readily that, for all ɛ > 0, t > 0, we can find nearly optimal decompositions x = x ɛ (t) + y ɛ (t), such that
$$\displaystyle\begin{array}{rcl} (1-\varepsilon )[K(t,f;\vec{ X}) - t \frac{d} {dt}K(t,f;\vec{ X})] \leq \left \Vert x_{\varepsilon }(t)\right \Vert _{X_{1}} \leq (1+\varepsilon )[K(t,f;\vec{ X}) - t \frac{d} {dt}K(t,f;\vec{ X})]& & {}\end{array}$$
(16)
$$\displaystyle\begin{array}{rcl} (1-\varepsilon ) \frac{d} {dt}K(t,f;\vec{ X}) \leq \left \Vert y_{\varepsilon }(t)\right \Vert _{X_{2}} \leq (1+\varepsilon ) \frac{d} {dt}K(t,f;\vec{ X}).& & {}\end{array}$$
(17)
 
13
BMO(R n ) can be normed by \(\left \vert f\right \vert _{BMO}\) if we identify functions that differ by a constant.
 
14
We cannot resist but to offer here our slight twist to the argument
$$\displaystyle\begin{array}{rcl} \left \Vert fg\right \Vert _{L^{p}}& \leq &\left \Vert f\right \Vert _{L^{2p}}\left \Vert g\right \Vert _{L^{2p}} {}\\ & =& \left \Vert f\right \Vert _{[L^{p},BMO]_{1/2,2p}}\left \Vert g\right \Vert _{[L^{p},BMO]_{1/2,2p}} {}\\ & \preceq & \left \Vert f\right \Vert _{L^{p}}^{1/2}\left \Vert f\right \Vert _{ BMO}^{1/2}\left \Vert g\right \Vert _{ L^{p}}^{1/2}\left \Vert g\right \Vert _{ BMO}^{1/2} {}\\ & \preceq & \left \Vert f\right \Vert _{L^{p}}\left \Vert g\right \Vert _{BMO} + \left \Vert g\right \Vert _{L^{p}}\left \Vert f\right \Vert _{BMO}. {}\\ \end{array}$$
 
15
However, keep in mind the epigraph of [68], originally due Douglas Adams, The Restaurant at the End of the Universe, Tor Books, 1988:“For seven and a half million years, Deep Thought computed and calculated, and in the end announced that the answer was in fact Forty-two—and so another, even bigger, computer had to be built to find out what the actual question was.”
 
16
For more background information, we refer to [9] and [83].
 
17
For a recent new approach to the John-Nirenberg Lemma we refer to [31].
 
18
Note that \(f_{Q_{0},1}^{\#} = f_{Q_{0}}^{\#}.\)
 
19
We shall then call \(\vec{X} = (X_{0},X_{1})\) a “Banach pair”. In general, the space V plays an auxiliary role, since once we know that \(\vec{X}\) is a Banach pair we can use \(\Sigma (\vec{X})\) as the ambient space. In particular, the functional \(K(t,f;\vec{ X})\) is in principle only defined on \(\Sigma (\vec{X}).\) On the other hand, the functional \(f \rightarrow \frac{d} {dt}K(t,f;\vec{ X}),\) can make sense for a larger class of elements than \(\Sigma (\vec{X}).\) This occurs for significant examples: For example, on the interval [0, 1], 
$$\displaystyle{L(1,\infty ) =\{ f:\sup _{t}tf^{{\ast}}(t) <\infty \} \not\subseteq L^{1} + L^{\infty } = L^{1}.}$$
 
20
With the usual modification when q = ∞. 
 
21
Here and in what follows we use the convention ∞ 0 = 1. 
 
22
See also [26].
 
23
For more recent developments in extrapolation theory cf. [4].
 
24
For computations of related K-functionals and further references cf. [2, 42].
 
25
Think in terms of a Gagliardo diagram, see, for example, [12, p. 39], [43, 46].
 
26
See also [82], Lesson #3.
 
27
Herz [37] also developed a different technique to extrapolate oscillation rearrangement inequalities for martingale operators.
 
28
(cf. [6, 34, 65, 67]).
 
29
See also [47] for a maximal function approach to oscillation inequalities for the gradient.
 
30
In this example we are not interested on the precise dependence of the constants of equivalence.
 
31
It is an interesting open problem to modify Holmstedt’s method to be able to keep track, in a nearly optimal way, both coordinates in the Gagliardo diagram, when doing reiteration. For more on the computation of Gagliardo coordinate spaces see the forthcoming [74].
 
32
To simplify the computations we model the L p case here. Another simplification is that in this argument we don’t need to be fuzzy about constants.
 
33
In fact, the boundedness of [T, b] for all CZ operators implies that b ∈ BMO (cf. [41])
 
34
I would learn quickly that Mischa’s modesty was legendary.
 
35
From this period I can mention [23‐25].
 
36
Existence here to be taken in a non mathematical sense. Of course at point in time I did NOT *exist* mathematically!
 
37
In Spanish *Corita* means little Cora.
 
38
Many many years later she told me that by then everyone called her Cora, except Mischa and myself and that she would prefer for me to call her Cora. I said, of course, Corita!
 
39
She had a membership at IAS herself that year.
 
40
Harmonic analysts trained in the 1960s had a special place in their hearts for Interpolation theory. It was after all a theory to which the great masters of the Chicago school (e.g., Calderón, Stein, Zygmund) had made fundamental contributions.
 
41
I mean using TeX!
 
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Metadaten
Titel
BMO: Oscillations, Self-Improvement, Gagliardo Coordinate Spaces, and Reverse Hardy Inequalities
verfasst von
Mario Milman
Copyright-Jahr
2016
Buch
Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1)

Print ISBN: 978-3-319-30959-0

Electronic ISBN: 978-3-319-30961-3

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-319-30961-3_13