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The harmonic Cesáro and Copson operators on the spaces $L^{p}(ℝ)$, 1 ≤ p ≤ 2

100%

Móricz F.

nr 3

267-279

The harmonic Cesàro operator 𝓒 is defined for a function f in $L^{p}(ℝ)$ for some 1 ≤ p < ∞ by setting $𝓒(f)(x): = ∫^{∞}_{x} (f(u)/u)du$ for x > 0 and $𝓒(f)(x): = -∫_{-∞}^{x} (f(u)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L¹_{loc}(ℝ)$ by setting $𝓒*(f)(x): = (1/x) ∫^{x₀} f(u)du$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f ∈ L^{p}(ℝ)$ for some 1 ≤ p ≤ 2, then $(𝓒(f))^{∧}(t) = 𝓒*(f̂)(t)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f ∈ L^{p}(ℝ)$ for some 1 < p ≤ 2, then $(𝓒*(f))^{∧}(t) = 𝓒(f̂)(t)$ a.e. As a by-product of our proofs, we obtain representations of $(𝓒(f))^{∧}(t)$ and $(𝓒*(f))^{∧}(t)$ in terms of Lebesgue integrals in case f belongs to $L^{p}(ℝ)$ for some 1 < p ≤ 2. These representations are valid for almost every t and may be useful in other contexts.

Multiple conjugate functions and multiplicative Lipschitz classes

100%

Móricz F.

2009

tom 115

nr 1

21-32

We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class $Lip(α₁,..., α_N)$ for some $0 < α₁,...,α_N < 1$ and its marginal functions satisfy $f(·,x₂,...,x_N) ∈ Lip β₁,...,f(x₁,...,x_{N-1},·) ∈ Lip β_N$ for some $0 < β₁,...,β_N < 1$ uniformly in the indicated variables $x_{l}$, 1 ≤ l ≤ N, then $f̃^{(η₁, ..., η_N)} ∈ Lip(α₁, ..., α_N)$ for each choice of $(η₁,...,η_N)$ with $η_{l} = 0$ or 1 for 1 ≤ l ≤ N.

Addendum to the paper "Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅ ²₊" (Studia Math. 201 (2010), 287-304)

63%

Kórus P. , Móricz F.

2012

tom 212

nr 3

285-286

Bundle Convergence in a von Neumann Algebra and in a von Neumann Subalgebra

63%

Le Gac B. , Móricz F.

nr 3

283-295

Let H be a separable complex Hilbert space, 𝓐 a von Neumann algebra in 𝓛(H), ϕ a faithful, normal state on 𝓐, and 𝓑 a commutative von Neumann subalgebra of 𝓐. Given a sequence (Xₙ: n ≥ 1) of operators in 𝓑, we examine the relations between bundle convergence in 𝓑 and bundle convergence in 𝓐.

On the integrability and L¹-convergence of double trigonometric series

38%

Móricz F.

nr 3

203-225

On the integrability and L¹-convergence of sine series

32%

Móricz F.

nr 2

187-200