Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces

Sergei V. Astashkin; Francisco L. Hernández; Evgeni M. Semenov

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Displaying similar documents to “Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces”

Disjoint strict singularity of inclusions between rearrangement invariant spaces

Francisco L. Hernández, Víctor M. Sánchez, Evgueni M. Semenov (2001)

Studia Mathematica

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It is studied when inclusions between rearrangement invariant function spaces on the interval [0,∞) are disjointly strictly singular operators. In particular suitable criteria, in terms of the fundamental function, for the inclusions $L ¹ \cap L^{\infty} ↪ E$ and $E ↪ L ¹ + L^{\infty}$ to be disjointly strictly singular are shown. Applications to the classes of Lorentz and Marcinkiewicz spaces are given.

A singular initial value problem for the equation $u^{(n)} (x) = g (u (x))$

Wojciech Mydlarczyk (1998)

Annales Polonici Mathematici

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We consider the problem of the existence of positive solutions u to the problem $u^{(n)} (x) = g (u (x))$ , $u (0) = u^{'} (0) = . . . = u^{(n - 1)} (0) = 0$ (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition $\int ₀^{δ} 1 / s {[s / g (s)]}^{1 / n} d s < \infty$ is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.

Common zero sets of equivalent singular inner functions

Keiji Izuchi (2004)

Studia Mathematica

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Let μ and λ be bounded positive singular measures on the unit circle such that μ ⊥ λ. It is proved that there exist positive measures μ₀ and λ₀ such that μ₀ ∼ μ, λ₀ ∼ λ, and $| ψ_{μ ₀} | < 1 \cap | ψ_{λ ₀} | < 1 = \emptyset$ , where $ψ_{μ}$ is the associated singular inner function of μ. Let $(μ) = ⋂_{ν; ν \sim μ} Z (ψ_{ν})$ be the common zeros of equivalent singular inner functions of $ψ_{μ}$ . Then (μ) ≠ ∅ and (μ) ∩ (λ) = ∅. It follows that μ ≪ λ if and only if (μ) ⊂ (λ). Hence (μ) is the set in the maximal ideal space of $H^{\infty}$ which relates naturally to the set of measures equivalent...

Commutators on ${(\sum ℓ_{q})}_{p}$

Dongyang Chen, William B. Johnson, Bentuo Zheng (2011)

Studia Mathematica

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Let T be a bounded linear operator on $X = {(\sum ℓ_{q})}_{p}$ with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.

Behaviour of $L^{r}$ -Dini singular integrals in weighted $L^{1}$ spaces

Osvaldo Capri, Carlos Segovia (1989)

Studia Mathematica

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An obstruction to $ℓ^{p}$ -dimension

Nicolas Monod, Henrik Densing Petersen (2014)

Annales de l’institut Fourier

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Let $G$ be any group containing an infinite elementary amenable subgroup and let $2 < p < \infty$ . We construct an exhaustion of $ℓ^{p} G$ by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to $ℓ^{p}$ -dimension and gives an answer to a question of Gaboriau.

Automatic continuity of operators commuting with translations

J. Alaminos, J. Extremera, A. R. Villena (2006)

Studia Mathematica

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Let $τ_{X}$ and $τ_{Y}$ be representations of a topological group G on Banach spaces X and Y, respectively. We investigate the continuity of the linear operators Φ: X → Y with the property that $Φ \circ τ_{X} (t) = τ_{Y} (t) \circ Φ$ for each t ∈ G in terms of the invariant vectors in Y and the automatic continuity of the invariant linear functionals on X.

Boundedness of para-product operators on spaces of homogeneous type

Yayuan Xiao (2017)

Czechoslovak Mathematical Journal

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We obtain the boundedness of Calderón-Zygmund singular integral operators $T$ of non-convolution type on Hardy spaces $H^{p} (𝒳)$ for $1 / (1 + ϵ) < p \leq 1$ , where $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss (1971), and $ϵ$ is the regularity exponent of the kernel of the singular integral operator $T$ . Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast...

Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

Ron Kerman, Luboš Pick (2011)

Studia Mathematica

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We study imbeddings of the Sobolev space $W^{m, ϱ} (Ω)$ : = u: Ω → ℝ with $ϱ (\partial^{α} u / \partial x^{α})$ < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, $τ_{ϱ}$ and $σ_{ϱ}$ , that are optimal with respect to the inclusions $W^{m, ϱ} (Ω) \subset W^{m, τ_{ϱ}} (Ω) \subset L_{σ_{ϱ}} (Ω)$ . General formulas for $τ_{ϱ}$ and $σ_{ϱ}$ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

On invariant, dual invariant and absolute formulas

Andrzej Mostowski

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CONTENTS Introduction..............................................................................................................................................................3 1. Lemmas concerning first order formulas.....................................................................................................5 2. Representability of recursively enumerable sets........................................................................................9 3. Simple theory of types.......................................................................................................................................10...

$L^{p} (ℝ ⁿ)$ boundedness for the commutator of a homogeneous singular integral operator

Guoen Hu (2003)

Studia Mathematica

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The commutator of a singular integral operator with homogeneous kernel Ω(x)/|x|ⁿ is studied, where Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that $Ω \in L {(l o g L)}^{k + 1} (S^{n - 1})$ is a sufficient condition for the kth order commutator to be bounded on $L^{p} (ℝ ⁿ)$ for all 1 < p < ∞. The corresponding maximal operator is also considered.

Structure of Cesàro function spaces: a survey

Sergey V. Astashkin, Lech Maligranda (2014)

Banach Center Publications

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Geometric structure of Cesàro function spaces $C e s_{p} (I)$ , where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that $C e s_{p} [0, 1]$ contains isomorphic and complemented copies of $l_{q}$ -spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces $C e s_{p} [0, 1]$ .

Real method of interpolation on subcouples of codimension one

S. V. Astashkin, P. Sunehag (2008)

Studia Mathematica

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We find necessary and sufficient conditions under which the norms of the interpolation spaces ${(N ₀, N ₁)}_{θ, q}$ and ${(X ₀, X ₁)}_{θ, q}$ are equivalent on N, where N is the kernel of a nonzero functional ψ ∈ (X₀ ∩ X₁)* and $N_{i}$ is the normed space N with the norm inherited from $X_{i}$ (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator $T_{θ} = S - 2^{θ} I$ (S...

Multiplication operators on $L (L_{p})$ and $ℓ_{p}$ -strictly singular operators

William Johnson, Gideon Schechtman (2008)

Journal of the European Mathematical Society

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A classification of weakly compact multiplication operators on $L (L_{p}),$ 1<p< $, i s g i v e n . T h i s a n s w e r s a q u e s t i o n r a i s e d b y S a k s m a n a n d T y l l i i n 1992 . T h e c l a s s i f i c a t i o n i n v o l v e s t h e c o n c e p t o f$ p $- s t r i c t l y s i n g u l a r o p e r a t o r s, a n d w e a l s o i n v e s t i g a t e t h e s t r u c t u r e o f g e n e r a l$ p $- s t r i c t l y s i n g u l a r o p e r a t o r s o n$ Lp $. T h e m a i n r e s u l t i s t h a t i f a n o p e r a t o r$ T $o n$ Lp $,$ 1<p<2 $, i s$ p $- s t r i c t l y s i n g u l a r a n d$ T|X $i s a n i s o m o r p h i s m f o r s o m e s u b s p a c e$ X $o f$ Lp $, t h e n$ X $e m b e d s i n t o$ Lr $f o r a l l$ r<2 $, b u t$ X $n e e d n o t b e i s o m o r p h i c t o a H i l b e r t s p a c e .$ It is also shown that if $T$ is convolution by a biased coin on $L_{p}$ of the Cantor group, $1 \leq p < 2$ , and $T_{| X}$ is an isomorphism for some reflexive subspace $X$ of $L_{p}$ , then $X$ is isomorphic to a Hilbert space. The case $p = 1$ answers a question asked by Rosenthal in 1976.

The Daugavet property and translation-invariant subspaces

Simon Lücking (2014)

Studia Mathematica

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Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{Λ} (G)$ or $L ¹_{Λ} (G)$ and which quotients of the form $C (G) / C_{Λ} (G)$ or $L ¹ (G) / L ¹_{Λ} (G)$ have the Daugavet property. We show that $C_{Λ} (G)$ is a rich subspace of C(G) if and only if $Γ ∖ Λ^{- 1}$ is a semi-Riesz set. If $L ¹_{Λ} (G)$ is a rich subspace of L¹(G), then $C_{Λ} (G)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C (G) / C_{Λ} (G)$ has the Daugavet property if Λ is a Rosenthal set, and that $L ¹_{Λ} (G)$ is a poor subspace of L¹(G)...

Khinchin inequality and Banach-Saks type properties in rearrangement-invariant spaces

F. A. Sukochev, D. Zanin (2009)

Studia Mathematica

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We study the class of all rearrangement-invariant ( = r.i.) function spaces E on [0,1] such that there exists 0 < q < 1 for which $∥ \sum_{k = 1}^{n} ξ_{k} ∥_{E} \leq C n^{q}$ , where ${ξ_{k}}_{k \geq 1} \subset E$ is an arbitrary sequence of independent identically distributed symmetric random variables on [0,1] and C > 0 does not depend on n. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces $e x p (L_{p})$ , p ≥ 1. We further apply our results to the study of Banach-Saks...

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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We study the chemotaxis system with singular sensitivity and logistic-type source: $u_{t} = Δ u - χ \nabla \cdot (u \nabla v / v) + r u - μ u^{k}$ , $0 = Δ v - v + u$ under the non-flux boundary conditions in a smooth bounded domain $Ω \subset ℝ^{n}$ , $χ, r, μ > 0$ , $k > 1$ and $n \geq 1$ . It is shown with $k \in (1, 2)$ that the system possesses a global generalized solution for $n \geq 2$ which is bounded when $χ > 0$ is suitably small related to $r > 0$ and the initial datum is properly small, and a global bounded classical solution for $n = 1$ .

A note on integer translates of a square integrable function on ℝ

Maciej Paluszyński (2010)

Colloquium Mathematicae

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We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${ψ (\cdot - n)}_{n = - \infty}^{\infty}$ , which is equivalent to the weight function $\sum_{n = - \infty}^{\infty} | ψ ̂ (\cdot + n) | ²$ being > 0 a.e.

The Lizorkin-Freitag formula for several weighted $L_{p}$ spaces and vector-valued interpolation

Irina Asekritova, Natan Krugljak, Ludmila Nikolova (2005)

Studia Mathematica

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A complete description of the real interpolation space $L = {(L_{p ₀} (ω ₀), . . ., L_{p ₙ} (ω ₙ))}_{θ ⃗, q}$ is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces $Ω_{i}$ (i ∈ I) such that L is an $l_{q}$ sum of the restrictions of L to $Ω_{i}$ , and L on each $Ω_{i}$ is a result of interpolation of just two weighted $L_{p}$ spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

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We study the existence of positive solutions to ⎧ ${d i v (| \nabla u |}^{p - 2} \nabla u) + q (x) u^{- γ} = 0$ on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is $ℝ^{N}$ or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

Invariant theory and the $𝒲_{1 + \infty}$ algebra with negative integral central charge

Andrew Linshaw (2011)

Journal of the European Mathematical Society

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The vertex algebra $𝒲_{1 + \infty, c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n \geq 1$ , it was conjectured in the physics literature that $𝒲_{1 + \infty, - n}$ should have a minimal strong generating set consisting of $n^{2} + 2 n$ elements. Using a free field realization of $𝒲_{1 + \infty, - n}$ due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation...

Corrigendum to “Commutators on ${(\sum ℓ_{q})}_{p}$ ” (Studia Math. 206 (2011), 175-190)

Dongyang Chen, William B. Johnson, Bentuo Zheng (2014)

Studia Mathematica

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We give a corrected proof of Theorem 2.10 in our paper “Commutators on ${(\sum ℓ_{q})}_{p}$ ” [Studia Math. 206 (2011), 175-190] for the case 1 < q < p < ∞. The case when 1 = q < p < ∞ remains open. As a consequence, the Main Theorem and Corollary 2.17 in that paper are only valid for 1 < p,q < ∞.

Continuity of halo functions associated to homothecy invariant density bases

Oleksandra Beznosova, Paul Hagelstein (2014)

Colloquium Mathematicae

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Let be a collection of bounded open sets in ℝⁿ such that, for any x ∈ ℝⁿ, there exists a set U ∈ of arbitrarily small diameter containing x. The collection is said to be a density basis provided that, given a measurable set A ⊂ ℝⁿ, for a.e. x ∈ ℝⁿ we have $l i m_{k \to \infty} 1 / | R_{k} | \int_{R_{k}} χ_{A} = χ_{A} (x)$ for any sequence $R_{k}$ of sets in containing x whose diameters tend to 0. The geometric maximal operator $M_{}$ associated to is defined on L¹(ℝⁿ) by $M_{} f (x) = s u p_{x \in R \in} 1 / | R | \int_{R} | f |$ . The halo function ϕ of is defined on (1,∞) by $ϕ (u) = s u p 1 / | A | | x \in ℝ ⁿ : M_{} χ_{A} (x) > 1 / u | : 0 < | A | < \infty$ and on [0,1] by ϕ(u) = u. It is shown...

Singular $φ$ -Laplacian third-order BVPs with derivative dependance

Smaïl Djebali, Ouiza Saifi (2016)

Archivum Mathematicum

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This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a $φ$ -Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition...

A counter-example in singular integral theory

Lawrence B. Difiore, Victor L. Shapiro (2012)

Studia Mathematica

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An improvement of a lemma of Calderón and Zygmund involving singular spherical harmonic kernels is obtained and a counter-example is given to show that this result is best possible. In a particular case when the singularity is O(|log r|), let $f \in C ¹ (ℝ^{N} ∖ 0)$ and suppose f vanishes outside of a compact subset of $ℝ^{N}$ , N ≥ 2. Also, let k(x) be a Calderón-Zygmund kernel of spherical harmonic type. Suppose f(x) = O(|log r|) as r → 0 in the $L^{p}$ -sense. Set $F (x) = \int_{ℝ^{N}} k (x - y) f (y) d y \forall x \in ℝ^{N} ∖ 0$ . Then F(x) = O(log²r) as r → 0 in the $L^{p}$ -sense, 1 <...

Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups

Lingli Zeng, Jizhu Nan (2016)

Czechoslovak Mathematical Journal

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Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$ th root of unity and $char K \neq p$ . Suppose that a classical group $G$ acts on the $F$ -vector space $V$ . Then it can induce the actions on the vector space $V \oplus V$ and on the group algebra $K [V \oplus V]$ , respectively. In this paper we determine the structure of $G$ -invariant ideals of the group algebra $K [V \oplus V]$ , and establish the relationship between the invariant ideals of $K [V]$ and the vector invariant ideals of $K [V \oplus V]$ , if $G$ is a unitary group or orthogonal...

On a translation property of positive definite functions

Lars Omlor, Michael Leinert (2010)

Banach Center Publications

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If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant $C_{h} > 0$ such that $\int L_{x} h \cdot g \leq C_{h} \int h g$ for every continuous positive definite g≥0, where $L_{x}$ is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That “for one h” implies “for all h” seemed obvious, but turned out not to be obvious at...