A weighted Plancherel formula II. The case of the ball

Genkai Zhang

Displaying similar documents to “A weighted Plancherel formula II. The case of the ball”

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.

On invariant measures for power bounded positive operators

Ryotaro Sato (1996)

Studia Mathematica

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We give a counterexample showing that $\bar{(I - T *) L_{\infty}} \cap L_{\infty}^{+} = 0$ does not imply the existence of a strictly positive function u in $L_{1}$ with Tu = u, where T is a power bounded positive linear operator on $L_{1}$ of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.

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.

Sums of an entire function in certain weighted L-spaces.

Bruno Brive (2003)

Publicacions Matemàtiques

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We consider the functional equation f(z+σ) - f(z) = g(z) where σ is a complex number, f and g are entire functions of a complex variable z, with growth conditions. We prove the existence of certain types of solutions of this equation by an a priori estimate method in certain weighted L-spaces.

Two problems related to the non-vanishing of $L (1, χ)$

Paolo Codecà, Roberto Dvornicich, Umberto Zannier (1998)

Journal de théorie des nombres de Bordeaux

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We study two rather different problems, one arising from Diophantine geometry and one arising from Fourier analysis, which lead to very similar questions, namely to the study of the ranks of matrices with entries either zero or $((x y / q)), (0 \leq x, y < q)$ , where $((u)) = u - [u] - 1 / 2$ denotes the “centered” fractional part of $x$ . These ranks, in turn, are closely connected with the non-vanishing of the Dirichlet $L$ -functions at $s = 1$ .

On weighted James' spaces.

M. Angeles Miñarro (1996)

Collectanea Mathematica

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In this note we study the topological structure of weighted James spaces J(h). In particular we prove that J(h) is isomorphic to J if and only if the weight h is bounded. We also provide a description of J(h) if the weight is a non-decreasing sequence.

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.

On the extension of Lipschitz-Hölder maps on $L^{P}$ spaces

Lynn Williams, J. Wells, T. Hayden (1971)

Studia Mathematica

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On subharmonicity inequalities involving solutions of generalized Cauchy-Riemann equations

R. Coifman, Guido Weiss (1970)

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...

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...

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...

Several characterizations for the special atom spaces with applications.

Geraldo Soares de Souza, Richard O&#039;Neil, Gary Sampson (1986)

Revista Matemática Iberoamericana

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The theory of functions plays an important role in harmonic analysis. Because of this, it turns out that some spaces of analytic functions have been studied extensively, such as H-spaces, Bergman spaces, etc. One of the major insights that has developed in the study of H-spaces is what we call the real atomic characterization of these spaces.