Inequalities of the Kahane-Khinchin type and sections of $L_{p}$-balls

A. Koldobsky; A. Pajor; V. Yaskin

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Displaying similar documents to “Inequalities of the Kahane-Khinchin type and sections of $L_{p}$ -balls”

Moser-Trudinger and logarithmic HLS inequalities for systems

Itai Shafrir, Gershon Wolansky (2005)

Journal of the European Mathematical Society

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We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S^{2}$ , on a bounded domain $Ω \subset ℝ^{2}$ and on all of $ℝ^{2}$ . In some cases we also address the question of existence of minimizers.

Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions

Albert Baernstein II, Robert C. Culverhouse (2002)

Studia Mathematica

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Let $X = \sum_{i = 1}^{k} a_{i} U_{i}$ , $Y = \sum_{i = 1}^{k} b_{i} U_{i}$ , where the $U_{i}$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_{i}, b_{i}$ are real constants. We prove that if $b ²_{i}$ is majorized by $a ²_{i}$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L ² - L^{p}$ Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...

Lower bounds for Schrödinger operators in H¹(ℝ)

Ronan Pouliquen (1999)

Studia Mathematica

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We prove trace inequalities of type $| | u^{'} {| |}_{L^{2}}^{2} + \sum_{j \in ℤ} k_{j} | u (a_{j}) |^{2} {\geq λ | | u | |}_{L^{2}}^{2}$ where $u \in H^{1} (ℝ)$ , under suitable hypotheses on the sequences ${a_{j}}_{j \in ℤ}$ and ${k_{j}}_{j \in ℤ}$ , with the first sequence increasing and the second bounded.

Poincaré Inequalities and Moment Maps

Bo’az Klartag (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We discuss a method for obtaining Poincaré-type inequalities on arbitrary convex bodies in $ℝ^{n}$ . Our technique involves a dual version of Bochner’s formula and a certain moment map, and it also applies to some non-convex sets. In particular, we generalize the central limit theorem for convex bodies to a class of non-convex domains, including the unit balls of $ℓ_{p}$ -spaces in $ℝ^{n}$ for $0 < p < 1$ .

Regularity of stable solutions of $p$ -Laplace equations through geometric Sobolev type inequalities

Daniele Castorina, Manel Sanchón (2015)

Journal of the European Mathematical Society

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We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of $- Δ_{p} u = g (u)$ in a smooth bounded domain $Ω \subset ℝ^{n}$ . In particular, we obtain new $L^{r}$ and $W^{1, r}$ bounds for the extremal solution $u^{☆}$ when the domain is strictly convex. More precisely, we prove that $u^{☆} \in L^{\infty} (Ω)$ if $n \leq p + 2$ and $u^{☆} \in L^{\frac{n p}{n - p - 2}} (Ω) \cap W_{0}^{1, p} (Ω)$ if $n > p + 2$ .

Lieb–Thirring inequalities on the half-line with critical exponent

Tomas Ekholm, Rupert Frank (2008)

Journal of the European Mathematical Society

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We consider the operator $- d^{2} / d r^{2} - V$ in $L_{2} (ℝ_{+})$ with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound $tr {(- d^{2} / d r^{2} - V)}_{-}^{γ} \leq C_{γ, α} \int_{ℝ_{+}} {(V (r) - 1 / (4 r^{2}))}_{+}^{γ + (1 + α) / 2} r^{α} d r$ for any $α \in [0, 1)$ and $γ \geq (1 - α) / 2$ . This includes a Lieb-Thirring inequality in the critical endpoint case.

Geometry and inequalities of geometric mean

Trung Hoa Dinh, Sima Ahsani, Tin-Yau Tam (2016)

Czechoslovak Mathematical Journal

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We study some geometric properties associated with the $t$ -geometric means $A ♯_{t} B : = A^{1 / 2} {(A^{- 1 / 2} B A^{- 1 / 2})}^{t} A^{1 / 2}$ of two $n \times n$ positive definite matrices $A$ and $B$ . Some geodesical convexity results with respect to the Riemannian structure of the $n \times n$ positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding...

Product property for capacities in $ℂ^{N}$

Mirosław Baran, Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

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The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{ν} (E ₁ \times E ₂) = m i n (C_{ν ₁} (E ₁), C_{ν ₂} (E ₂))$ , where $E_{j}$ and $ν_{j}$ are respectively a compact set and a norm in $ℂ^{N_{j}}$ (j = 1,2), and ν is a norm in $ℂ^{N ₁ + N ₂}$ , ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of $ℂ^{N}$ , denote by C(E) the standard L-capacity and by $ω_{E}$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

A characterization of the disc $D^{n}$

Th. Friedrich (1974)

Colloquium Mathematicae

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Геометрическая теория состояний, граница фон Неймана, двойственность $C^{*}$ -алгерб

А.М. Вершик (1972)

Zapiski naucnych seminarov Leningradskogo

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The general methods of finding the sum $S_{n}$ for all kinds of series, II

K. Orlov (1981)

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A characterization of sets in $ℝ^{2}$ with DC distance function

Dušan Pokorný, Luděk Zajíček (2022)

Czechoslovak Mathematical Journal

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We give a complete characterization of closed sets $F \subset ℝ^{2}$ whose distance function $d_{F} : = dist (\cdot, F)$ is DC (i.e., is the difference of two convex functions on $ℝ^{2}$ ). Using this characterization, a number of properties of such sets is proved.

On C $^{*}$ -spaces

P. Srivastava, K. K. Azad (1981)

Matematički Vesnik

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$ℵ_{1}$ -incompactness of Z

Ralph McKenzie (1971)

Colloquium Mathematicae

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On a characterization of $L^{p}$ -norm

Janusz Matkowski (1989)

Annales Polonici Mathematici

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On the $p^{λ}$ problem

Stephan Baier (2004)

Acta Arithmetica

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The existence of $P (α)$ -points of N* for $ℵ_{0} α < 𝔠$

A. Szymański (1977)

Colloquium Mathematicae

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On the proximate order of $f^{(s)} (z) * g^{(s)} (z)$ and ${[f (z) * g (z)]}^{(s)}$

M. K. Sen (1971)

Annales Polonici Mathematici

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On a fixobject property for $l^{c} β$ -semigroupoids

M. Đurić (1973)

Matematički Vesnik

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Localization techniques in $L^{p}$ spaces

A. Pełczyński, H. Rosenthal (1975)

Studia Mathematica

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An application of the method wherein the $p h i$ - and $c h i$ - methods are combined for the determination of the grating constant. [II.]

Swami Jnanananda (1936)

Časopis pro pěstování matematiky a fysiky

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On the equation $ϕ (x + m) = ϕ (x) + m$

A. Makowski (1964)

Matematički Vesnik

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Equivalence of measures of smoothness in $L_{p} (S^{d - 1})$ , 1 < p < ∞

F. Dai, Z. Ditzian, Hongwei Huang (2010)

Studia Mathematica

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Suppose Δ̃ is the Laplace-Beltrami operator on the sphere $S^{d - 1}, Δ_{ρ}^{k} f (x) = Δ_{ρ} Δ_{ρ}^{k - 1} f (x)$ and $Δ_{ρ} f (x) = f (ρ x) - f (x)$ where ρ ∈ SO(d). Then $ω^{m} {(f, t)}_{L_{p} (S^{d - 1})} \equiv s u p ∥ Δ_{ρ}^{m} f ∥_{L_{p} (S^{d - 1})} : ρ \in S O (d), m a x_{x \in S^{d - 1}} ρ x \cdot x \geq c o s t$ and $K ̃ ₘ {(f, t^{m})}_{p} \equiv i n f ∥ f - g ∥_{L_{p} (S^{d - 1})} + t^{m} ∥ {(- Δ ̃)}^{m / 2} g ∥_{L_{p} (S^{d - 1})} : g \in ({(- Δ ̃)}^{m / 2})$ are equivalent for 1 < p < ∞. We note that for even m the relation was recently investigated by the second author. The equivalence yields an extension of the results on sharp Jackson inequalities on the sphere. A new strong converse inequality for $L_{p} (S^{d - 1})$ given in this paper plays a significant role in the proof.

On spirallike functions of order $α$ and type $β$ with fixed second coefficient

M. K. Aouf (1988)

Matematički Vesnik

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Displaying similar documents to “Inequalities of the Kahane-Khinchin type and sections of L p -balls”

Displaying similar documents to “Inequalities of the Kahane-Khinchin type and sections of $L_{p}$ -balls”