Intrinsic characterizations of distribution spaces on domains

V. Rychkov

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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 the boundedness of multipliers, commutators and the second derivatives of Green’s operators on $H^{1}$ and $B M O$

Der-Chen Chang, Song-Ying Li (1999)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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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...

$(H_{p}, L_{p})$ -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.

Homogeneous Besov spaces on locally compact Vilenkin groups

C. Onneweer, Su Weiyi (1989)

Studia Mathematica

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On the characterization of Hardy-Besov spaces on the dyadic group and its applications

Jun Tateoka (1994)

Studia Mathematica

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C. Watari [12] obtained a simple characterization of Lipschitz classes $L i p^{(p)} α (W) (1 \geq p \geq \infty, α > 0)$ on the dyadic group using the $L^{p}$ -modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces $B_{p, q}^{α}$ on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces $B_{p, q}^{α}$ by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality...

A characterization of $H^{p} (R^{n})$ in terms of atoms

Robert Latter (1978)

Studia Mathematica

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BMO and smooth truncation in Sobolev spaces

David Adams, Michael Frazier (1988)

Studia Mathematica

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The dual of Besov spaces on fractals

Alf Jonsson, Hans Wallin (1995)

Studia Mathematica

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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...

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.