Abstract
We introduce the concept of Gaussian integral isoperimetric transfer and show how it can be applied to obtain a new class of sharp Sobolev-Poincaré inequalities with constants independent of the dimension. In the special case of \(L^{q}\) spaces on the unit \(n\)-dimensional cube our results extend the recent inequalities that were obtained in Fiorenza et al. (2012) using extrapolation.
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Notes
Dealing with a large number of variables (high dimensionality) gives rise to a set of specific problems and issues in analysis, e.g. in approximation theory, numerical analysis, optimization. (cf. [14]).
As the associated space of the ‘grand’ Lebesgue spaces, introduced by Iwaniec and Sbordone [15].
The basic fractional inequalities that underlie our analysis were obtained in [20].
See Sect. 2 below; in particular we assume that \(I(t)\) is concave and symmetric about \(1/2.\)
Precise definitions and properties of rearrangements and related topics coming into play in this section are contained in Sect. 2.
as far as the condition on the left hand side (or target space).
which of course is still weaker than the optimal inequality (1.6).
Note that this notation is somewhat unconventional. In the literature it is common to denote the decreasing rearrangement of \(\left| u\right| \) by \(u_{\mu }^{*},\) while here it is denoted by \(\left| u\right| _{\mu }^{*}\) since we need to distinguish between the rearrangements of \(u\) and \(\left| u\right| .\) In particular, the rearrangement of \(u\) can be negative. We refer the reader to [27] and the references quoted therein for a complete treatment.
In fact one can define \(\left| \nabla f\right| \) for functions \(f\) that are Lipschitz on every ball in \((\Omega ,d)\) (cf. [6, pp. 2, 3] for more details).
This means that if \(f_{n}\ge 0,\) and \(f_{n}\uparrow f,\) then \(\left\| f_{n}\right\| _{X}\uparrow \left\| f\right\| _{X}\) (i.e. Fatou’s Lemma holds in the \(X\) norm).
where if \(a=0\) we simply let \(Q:=\) \(Q_{0}\).
Introduced by D. W. Boyd in [7].
For a discussion of the extrapolation properties of a more general class of spaces we refer to [2].
Let \(f\ \)be a measurable function, a real number \(med(f)\) will be called a median of \(f\) if
$$\begin{aligned} \mu \left\{ f\ge med(f)\right\} \ge 1/2 \, \text {and }\mu \left\{ f\le med(f)\right\} \ge 1/2. \end{aligned}$$note that \(u_{\mu }^{*}(s)=u^{*}(\gamma _{n}s)\).
Note that \(u_{\frac{dx_{n}}{\omega _{n}}}^{*}(s)=u_{dx_{n}}^{*}(\omega _{n}s)\).
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We are very grateful to the referees for their comments and suggestions to improve the quality of the paper.
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J. Martín partially supported in part by Grants MTM2010-14946, MTM-2010-16232. M. Milman work was partially supported by a Grant from the Simons Foundation (#207929 to Mario Milman).
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Martín, J., Milman, M. Integral isoperimetric transference and dimensionless Sobolev inequalities. Rev Mat Complut 28, 359–392 (2015). https://doi.org/10.1007/s13163-014-0153-7
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DOI: https://doi.org/10.1007/s13163-014-0153-7