Two-sided estimates for the approximation numbers of Hardy-type operators in $L^{∞}$ and L¹

W. Evans; D. Harris; J. Lang

Displaying similar documents to “Two-sided estimates for the approximation numbers of Hardy-type operators in $L^{\infty}$ and L¹”

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

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

Studia Mathematica

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We consider the Volterra integral operator $T : L^{p} (ℝ^{+}) \to L^{p} (ℝ^{+})$ 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 $l i m_{n \to \infty} n a_{n} (T) = π^{- 1} ʃ_{0}^{\infty} | u (t) v (t) | d t$ . We also provide upper and lower estimates for the $ℓ^{α}$ and weak $ℓ^{α}$ norms of (an(T)) when 1 < α < ∞.

$(H_{p}, L_{p})$ -type inequalities for the two-dimensional dyadic derivative

Ferenc Weisz (1996)

Studia Mathematica

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It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space $H_{p, q}$ to $L_{p, q}$ (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type $(L_{1}, L_{1})$ . As a consequence we show that the dyadic integral of a ∞ function $f \in L_{1}$ is dyadically differentiable and its derivative is f a.e.

The local versions of $H^{p} (ℝ^{n})$ spaces at the origin

Shan Lu, Da Yang (1995)

Studia Mathematica

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Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces $H K ̇_{q}^{α, p} (ℝ^{n})$ which are the local versions of $H^{p} (ℝ^{n})$ spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth’s sense. We also prove an interpolation theorem for operators on $H K ̇_{q}^{α, p} (ℝ^{n})$ and discuss the $H K ̇_{q}^{α, p} (ℝ^{n})$ -boundedness of Calderón-Zygmund operators. Similar results can also be obtained...

Integrability theorems for trigonometric series

Bruce Aubertin, John Fournier (1993)

Studia Mathematica

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We show that, if the coefficients (an) in a series $a_{0} / 2 + \sum_{n = 1}^{\infty} a_{n} c o s (n t)$ tend to 0 as n → ∞ and satisfy the regularity condition that $\sum_{m = 0}^{\infty} {\sum_{j = 1}^{\infty} [\sum_{n = j 2^{m}}^{(j + 1) 2^{m} - 1} | a_{n} - a_{n + 1} |] ²}^{1 / 2} < \infty$ , then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series $\sum_{n = 1}^{\infty} b_{n} s i n (n t)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if $\sum_{n = 1}^{\infty} | b_{n} | / n < \infty$ . These conclusions were previously known to hold under stronger restrictions on the sizes...

Isomorphism of certain weak $L^{p}$ spaces

Denny Leung (1993)

Studia Mathematica

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It is shown that the weak $L^{p}$ spaces $ℓ^{p, \infty}, L^{p, \infty} [0, 1]$ , and $L^{p, \infty} [0, \infty)$ are isomorphic as Banach spaces.

Functions with derivatives in spaces of Morrey type and elliptic equations in unbounded domains

Anna Canale, Patrizia Di Gironimo, Antonio Vitolo (1998)

Studia Mathematica

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We introduce a sort of "local" Morrey spaces and show an existence and uniqueness theorem for the Dirichlet problem in unbounded domains for linear second order elliptic partial differential equations with principal coefficients "close" to functions having derivatives in such spaces.

On some singular integral operatorsclose to the Hilbert transform

T. Godoy, L. Saal, M. Urciuolo (1997)

Colloquium Mathematicae

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Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^{p} (ℝ)$ -boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_{m} f (x) = p . v . \int k (x - y) m (x + y) f (y) d y$ where $k (x) = \sum_{j \in ℤ} 2^{j} φ_{j} (2^{j} x)$ for a family of functions ${φ_{j}}_{j \in ℤ}$ satisfying conditions (1.1)-(1.3) given below.

Nonconvolution transforms with oscillating kernels that map $Ḃ_{1}^{0, 1}$ into itself

G. Sampson (1993)

Studia Mathematica

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We consider operators of the form $(Ω f) (y) = ʃ_{- \infty}^{\infty} Ω (y, u) f (u) d u$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h \in L^{\infty}$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space $Ḃ_{1}^{0, 1}$ (= B) into itself. In particular, all operators with $h (y) = e^{{i | y |}^{a}}$ , a > 0, a ≠ 1, map B into itself.

Subspaces of the Bourgain-Delbaen space

Richard Haydon (2000)

Studia Mathematica

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It is shown that every infinite-dimensional closed subspace of the Bourgain-Delbaen space $X_{a, b}$ has a subspace isomorphic to some $ℓ^{p}$ .

On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system

G. Gát (1998)

Studia Mathematica

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Let G be the Walsh group. For $f \in L^{1} (G)$ we prove the a. e. convergence σf → f(n → ∞), where $σ_{n}$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator $σ * f ≔ s u p_{n} | σ_{n} f | .$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, $∥ σ * f ∥_{1} \leq c ∥ | f | ∥_{H}$ , where H is the Hardy space on the Walsh group.

Two-parameter Hardy-Littlewood inequality and its variants

Chang-Pao Chen, Dah-Chin Luor (2000)

Studia Mathematica

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Let s* denote the maximal function associated with the rectangular partial sums $s_{m n} (x, y)$ of a given double function series with coefficients $c_{j k}$ . The following generalized Hardy-Littlewood inequality is investigated: ${| | s * | |}_{p, μ} \leq C_{p, α, β} {Σ_{j = 0}^{\infty} Σ_{k = 0}^{\infty} {(j ̅)}^{p - α - 2} {(k ̅)}^{p - β - 2} {| c_{j k} |}^{p}}^{1 / p}$ , where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_{j k}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property...

On integrability in F-spaces

Mikhail Popov (1994)

Studia Mathematica

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Some usual and unusual properties of the Riemann integral for functions x : [a,b] → X where X is an F-space are investigated. In particular, a continuous integrable $l_{p}$ -valued function (0 < p < 1) with non-differentiable integral function is constructed. For some class of quasi-Banach spaces X it is proved that the set of all X-valued functions with zero derivative is dense in the space of all continuous functions, and for any two continuous functions x and y there is a sequence...

On the complemented subspaces of the Schreier spaces

I. Gasparis, D. Leung (2000)

Studia Mathematica

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It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^{ξ}$ generated by subsequences $(e_{l_{n}}^{ξ})$ and $(e_{m_{n}}^{ξ})$ , respectively, of the natural Schauder basis $(e_{n}^{ξ})$ of $X^{ξ}$ are isomorphic if and only if $(e_{l_{n}}^{ξ})$ and $(e_{m_{n}}^{ξ})$ are equivalent. Further, $X^{ξ}$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $(e_{n}^{ξ})$ . It is also shown that there exists a complemented subspace spanned by a block basis of $(e_{n}^{ξ})$ , which is not isomorphic to a subspace generated by a subsequence of $(e_{n}^{ζ})$ ,...

A rigid space admitting compact operators

Paul Sisson (1995)

Studia Mathematica

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A rigid space is a topological vector space whose endomorphisms are all simply scalar multiples of the identity map. The first complete rigid space was published in 1981 in [2]. Clearly a rigid space is a trivial-dual space, and admits no compact endomorphisms. In this paper a modification of the original construction results in a rigid space which is, however, the domain space of a compact operator, answering a question that was first raised soon after the existence of complete rigid...

Displaying similar documents to “Two-sided estimates for the approximation numbers of Hardy-type operators in L ∞ and L¹”

Displaying similar documents to “Two-sided estimates for the approximation numbers of Hardy-type operators in $L^{\infty}$ and L¹”