Sur Les Fonctions Conjuguées À Plusieurs Variables

Horváth, J. I.

doi:10.1016/s1385-7258(53)50004-6

Cited by 33 publications

(17 citation statements)

References 2 publications

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“…(16) The reader is referred to [4] where the material in the preceding paragraphs is discussed. The identities (5.3) and (5.4) may be thought of as generalizations of the CauchyRiemann equations.…”

Section: Ri(f) In L(«)mentioning

confidence: 99%

On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz

Stein¹

1958

Trans. Amer. Math. Soc.

324

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Section: Ri(f) In L(«)mentioning

confidence: 99%

On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz

Stein¹

1958

Trans. Amer. Math. Soc.

324

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“…, n}, Q (j) t is the conjugant Poisson kernel as in (2.39). From f ∈ L 2 (R n ) and [16] (see also [31, p. 78, 4.4]), we deduce that F satisfies the generalized Cauchy-Riemann equation (2.16). Thus, we know that, for q ∈ [ n−1 n , i(ϕ) q(ϕ) ), |F | q is subharmonic (see, for example, [35, p. 234, Theorem 4.14]).…”

Section: T)mentioning

confidence: 93%

Riesz Transform Characterizations of Musielak-Orlicz Hardy Spaces

Cao

Chang

Yang

et al. 2017

Lecture Notes in Mathematics

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Abstract. Let ϕ be a Musielak-Orlicz function satisfying that, for any (x, t) ∈ R n × (0, ∞), ϕ(·, t) belongs to the Muckenhoupt weight class A ∞ (R n ) with the critical weight exponent q(ϕ) ∈ [1, ∞) and ϕ(x, ·) is an Orlicz function with 0 < i(ϕ) ≤ I(ϕ) ≤ 1 which are, respectively, its critical lower type and upper type. In this article, the authors establish the Riesz transform characterizations of the Musielak-Orlicz-Hardy spaces H ϕ (R n ) which are generalizations of weighted Hardy spaces and Orlicz-Hardy spaces. Precisely, the authors characterize H ϕ (R n ) via all the first order Riesz transforms whenn , and via all the Riesz transforms with the order not more than m ∈ N when i(ϕ) q(ϕ) > n−1 n+m−1 . Moreover, the authors also establish the Riesz transform characterizations of H ϕ (R n ), respectively, by means of the higher order Riesz transforms defined via the homogenous harmonic polynomials or the odd order Riesz transforms. Even if when ϕ(x, t) := tw(x) for all x ∈ R n and t ∈ [0, ∞), these results also widen the range of weights in the known Riesz characterization of the classical weighted Hardy space H 1 w (R n ) obtained by R. L. Wheeden from w ∈ A 1 (R n ) into w ∈ A ∞ (R n ) with the sharp range q(w) ∈ [1, n n−1 ), where q(w) denotes the critical index of the weight w.

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“…It was Horváth who, already in his 1953 paper [36], introduced the Clifford vector valued Hilbert operator…”

Section: Hilbert Transforms In Euclidean Spacementioning

confidence: 99%

Hilbert Transforms in Clifford Analysis

Brackx

Knock

Schepper

2010

Geometric Algebra Computing

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The Hilbert transform on the real line has applications in many fields. In particular in one-dimensional signal processing, the Hilbert operator is used to extract global as well as instantaneous characteristics, such as frequency, amplitude and phase, from real signals. The multidimensional approach to the Hilbert transform usually is a tensorial one, considering the so-called Riesz transforms in each of the cartesian variables separately. In this paper we give an overview of generalized Hilbert transforms in Euclidean space, developed within the framework of Clifford analysis. Roughly speaking, this is a function theory of higher dimensional holomorphic functions, which is particularly suited for a treatment of multidimensional phenomena since all dimensions are encompassed at once as an intrinsic feature.

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Sur Les Fonctions Conjuguées À Plusieurs Variables

Cited by 33 publications

References 2 publications

On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz

On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz

Riesz Transform Characterizations of Musielak-Orlicz Hardy Spaces

Hilbert Transforms in Clifford Analysis

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