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Displaying similar documents to “A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces”

A Whitney extension theorem in L p and Besov spaces

Alf Jonsson, Hans Wallin (1978)

Annales de l'institut Fourier

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The classical Whitney extension theorem states that every function in Lip ( β , F ) , F R n , F closed, k < β k + 1 , k a non-negative integer, can be extended to a function in Lip ( β , R n ) . Her Lip ( β , F ) stands for the class of functions which on F have continuous partial derivatives up to order k satisfying certain Lipschitz conditions in the supremum norm. We formulate and prove a similar theorem in the L p -norm. The restrictions to R d , d < n , of the Bessel potential spaces in R n and the Besov or generalized Lipschitz...

Two-sided estimates for the approximation numbers of Hardy-type operators in L and L¹

W. Evans, D. Harris, J. Lang (1998)

Studia Mathematica

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In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator T : L p ( + ) L p ( + ) defined by ( T f ) ( x ) v ( x ) ʃ 0 u ( t ) f ( t ) d t when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].

Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

Andreas Defant, Marius Junge (1997)

Studia Mathematica

Similarity:

We determine the set of all triples 1 ≤ p,q,r ≤ ∞ for which the so-called Marcinkiewicz-Zygmund inequality is satisfied: There exists a constant c≥ 0 such that for each bounded linear operator T : L q ( μ ) L p ( ν ) , each n ∈ ℕ and functions f 1 , . . . , f n L q ( μ ) , ( ʃ ( k = 1 n | T f k | r ) p / r d ν ) 1 / p c T ( ʃ ( k = 1 n | f k | r ) q / r d μ ) 1 / q . This type of inequality includes as special cases well-known inequalities of Paley, Marcinkiewicz, Zygmund, Grothendieck, and Kwapień. If such a Marcinkiewicz-Zygmund inequality holds for a given triple (p,q,r), then we calculate the best constant c ≥ 0 (with the...

Purely non-atomic weak L p spaces

Denny Leung (1997)

Studia Mathematica

Similarity:

Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If L p , ( Ω , , μ ) is isomorphic, as a Banach space, to L p , ( Ω ' , ' , μ ' ) for some purely atomic measure space (Ω’,∑’,μ’), then there is a measurable partition Ω = Ω 1 Ω 2 such that ( Ω 1 , Σ Ω 1 , μ | Σ Ω 1 ) is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable σ Ω 2 . In particular, L p , ( Ω , , μ ) is isomorphic to p , .

Two-sided estimates of the approximation numbers of certain Volterra integral operators

D. Edmunds, W. Evans, D. Harris (1997)

Studia Mathematica

Similarity:

We consider the Volterra integral operator T : L p ( + ) L p ( + ) defined by ( T f ) ( x ) = v ( x ) ʃ 0 x u ( t ) f ( t ) d t . Under suitable conditions on u and v, upper and lower estimates for the approximation numbers a n ( T ) of T are established when 1 < p < ∞. When p = 2 these yield l i m n n a n ( T ) = π - 1 ʃ 0 | u ( t ) v ( t ) | d t . We also provide upper and lower estimates for the α and weak α norms of (an(T)) when 1 < α < ∞.

Fourier analysis, Schur multipliers on S p and non-commutative Λ(p)-sets

Asma Harcharras (1999)

Studia Mathematica

Similarity:

This work deals with various questions concerning Fourier multipliers on L p , Schur multipliers on the Schatten class S p as well as their completely bounded versions when L p and S p are viewed as operator spaces. For this purpose we use subsets of ℤ enjoying the non-commutative Λ(p)-property which is a new analytic property much stronger than the classical Λ(p)-property. We start by studying the notion of non-commutative Λ(p)-sets in the general case of an arbitrary discrete group before turning...

The converse of the Hölder inequality and its generalizations

Janusz Matkowski (1994)

Studia Mathematica

Similarity:

Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if ʃ Ω x y d μ ϕ - 1 ( ʃ Ω ϕ x d μ ) ψ - 1 ( ʃ Ω ψ x d μ ) for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then...

Multiplier transformations on H p spaces

Daning Chen, Dashan Fan (1998)

Studia Mathematica

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The authors obtain some multiplier theorems on H p spaces analogous to the classical L p multiplier theorems of de Leeuw. The main result is that a multiplier operator ( T f ) ( x ) = λ ( x ) f ̂ ( x ) ( λ C ( n ) ) is bounded on H p ( n ) if and only if the restriction λ ( ε m ) m Λ is an H p ( T n ) bounded multiplier uniformly for ε>0, where Λ is the integer lattice in n .

Estimates of Fourier transforms in Sobolev spaces

V. Kolyada (1997)

Studia Mathematica

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We investigate the Fourier transforms of functions in the Sobolev spaces W 1 r 1 , . . . , r n . It is proved that for any function f W 1 r 1 , . . . , r n the Fourier transform f̂ belongs to the Lorentz space L n / r , 1 , where r = n ( j = 1 n 1 / r j ) - 1 n . Furthermore, we derive from this result that for any mixed derivative D s f ( f C 0 , s = ( s 1 , . . . , s n ) ) the weighted norm ( D s f ) L 1 ( w ) ( w ( ξ ) = | ξ | - n ) can be estimated by the sum of L 1 -norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin

Paweł Głowacki (1998)

Studia Mathematica

Similarity:

Let A be a pseudodifferential operator on N whose Weyl symbol a is a strictly positive smooth function on W = N × N such that | α a | C α a 1 - ϱ for some ϱ>0 and all |α|>0, α a is bounded for large |α|, and l i m w a ( w ) = . Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin. ...

A characterization of probability measures by f-moments

K. Urbanik (1996)

Studia Mathematica

Similarity:

Given a real-valued continuous function ƒ on the half-line [0,∞) we denote by P*(ƒ) the set of all probability measures μ on [0,∞) with finite ƒ-moments ʃ 0 ƒ ( x ) μ * n ( d x ) (n = 1,2...). A function ƒ is said to have the identification propertyif probability measures from P*(ƒ) are uniquely determined by their ƒ-moments. A function ƒ is said to be a Bernstein function if it is infinitely differentiable on the open half-line (0,∞) and ( - 1 ) n ƒ ( n + 1 ) ( x ) is completely monotone for some nonnegative integer n. The purpose...