Wyniki wyszukiwania

1

Comparison of Metric Spectral Gaps

100%

Naor A.

Analysis and Geometry in Metric Spaces

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2014

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tom 2

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nr 1

EN

Let A = (aij) ∊ Mn(ℝ) be an n by n symmetric stochastic matrix. For p ∊ [1, ∞) and a metric space (X, dX), let γ(A, dpx) be the infimum over those γ ∊ (0,∞] for which every x1, . . . , xn ∊ X satisfy [...] Thus γ (A, dpx) measures the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel dpX : X × X →[0,∞). We study pairs of metric spaces (X, dX) and (Y, dY ) for which there exists Ψ: (0,∞)→(0,∞) such that γ (A, dpX) ≤Ψ (A, dpY ) for every symmetric stochastic A ∊ Mn(ℝ) with (A, dpY ) < ∞. When Ψ is linear a complete geometric characterization is obtained. Our estimates on nonlinear spectral gaps yield new embeddability results as well as new nonembeddability results. For example, it is shown that if n ∊ ℕ and p ∊ (2,∞) then for every f1, . . . , fn ∊ Lp there exist x1, . . . , xn ∊ L2 such that [...] and [...] This statement is impossible for p ∊ [1, 2), and the asymptotic dependence on p in (0.1) is sharp. We also obtain the best known lower bound on the Lp distortion of Ramanujan graphs, improving over the work of Matoušek. Links to Bourgain-Milman-Wolfson type and a conjectural nonlinear Maurey-Pisier theorem are studied.

2

Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

63%

Mendel M., Naor A.

Analysis and Geometry in Metric Spaces

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2013

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tom 1

163-199

EN

The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.

3

Pisier's inequality revisited

63%

Hytönen T., Naor A.

Studia Mathematica

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2013

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tom 215

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nr 3

221-235

EN

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant 𝔓 ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: {-1,1}ⁿ → X satisfies $∫_{{-1,1}ⁿ} ||∑_{j=1}^{n} ∂_{j}f_{j}(ε)||^{p} dμ(ε) ≤ 𝔓^{p} ∫_{{-1,1}ⁿ} ∫_{{-1,1}ⁿ} ||∑_{j=1}^{n} δ_{j} Δf_{j}(ε)||^{p} dμ(ε)dμ(δ)$, where μ is the uniform probability measure on the discrete hypercube {-1,1}ⁿ, and ${∂_j}_{j=1}^{n}$ and $Δ = ∑_{j=1}^{n}∂_{j}$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by $𝔓ⁿ_{p}(X)$, we show that $𝔓ⁿ_{p}(X) ≤ ∑_{k=1}^{n} 1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special case $f_{j} = Δ^{-1}∂_{j} f$ for some f: {-1,1}ⁿ → X. We show that $sup_{n∈ ℕ }𝔓ⁿ_{p}(X) < ∞$ if either the dual X* is a UMD⁺ Banach space, or for some θ ∈ (0,1) we have $X = [H,Y]_{θ}$, where H is a Hilbert space and Y is an arbitrary Banach space. It follows that $sup_{n∈ ℕ}𝔓ⁿ_{p}(X) < ∞$ if X is a Banach lattice of nontrivial type.

4

Lipschitz sums of convex functions

63%

Csörnyei M., Naor A.

Studia Mathematica

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2003

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tom 158

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nr 3

269-286

EN

We give a geometric characterization of the convex subsets of a Banach space with the property that for any two convex continuous functions on this set, if their sum is Lipschitz, then the functions must be Lipschitz. We apply this result to the theory of Δ-convex functions.

Ograniczanie wyników

2 Analysis and Geometry in Metric Spaces

2 Studia Mathematica

4 Naor A.

1 Csörnyei M.

1 Hytönen T.

1 Mendel M.

1 2014

2 2013

1 2003

Comparison of Metric Spectral Gaps

Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

Pisier's inequality revisited

Lipschitz sums of convex functions