Weighted Hardy Spaces

Strömberg, Jan-Olov; Torchinsky, Alberto

doi:10.1007/bfb0091154

“…By θ ∈ C ∞ (S n−1 ), we deduce, from [37,p. 176,Theorem 14], that, for all p 1 ∈ (0, 1] and w ∈ A ∞ (R n ), K is bounded on the weighted Hardy space H p 1 w (R n ) and that, for all s ∈ (1, ∞), w ∈ A s (R n ) and p 2 ∈ (2s, ∞), K is bounded on L p 2 w (R n ), which, together with Propositions 2.2 and 2.21, and an argument similar to that used in the proofs of Corollaries 2.22 and 2.23, implies that K is bounded on H ϕ (R n ).…”

Section: Proposition 33 ([6])mentioning

confidence: 91%

Riesz Transform Characterizations of Musielak-Orlicz Hardy Spaces

Cao

¹

,

Chang

²

,

Yang

³

et al. 2017

Lecture Notes in Mathematics

0

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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.

show abstract

“…We point out that, if ϕ(x, t) := w(x)t p , with p ∈ (0, 1] and w ∈ A ∞ (R n ), for all (x, t) ∈ R n × [0, ∞), the Musielak-Orlicz-Hardy space H ϕ (R n ) coincides with the weighted Hardy space H p w (R n ) studied in [11,37]; if ϕ(x, t) := Φ(t), with Φ an Orlicz function whose upper type is 1 and lower type p ∈ (0, 1], for all (x, t) ∈ R n × [0 , ∞), H ϕ (R n ) coincides with the Orlicz-Hardy space H Φ (R n ) introduced in [19,36]. Also, the Musielak-Orlicz-Hardy space H ϕ (R n ) has proved useful in the study of other analysis problems when we take various different Musielak-Orlicz functions ϕ (see, for example, [3,22,23]).…”

Section: Definition 12 ([23]mentioning

confidence: 98%

“…It is known that the space H ϕ (R n ) is a generalization of the Orlicz-Hardy space introduced by Strömberg [36] and Janson [19], and the weighted Hardy space H p w (R n ) for w ∈ A ∞ (R n ) and p ∈ (0, 1], introduced by García-Cuerva [11] and Strömberg-Torchinsky [37]. Here, A q (R n ) with q ∈ [1, ∞] denotes the class of Muckenhoupt weights (see, for example, [11,12,13] for their definitions and properties).…”

Section: Theorem 11 ([9]mentioning

confidence: 99%

See 1 more Smart Citation

Riesz transform characterizations of Musielak-Orlicz-Hardy spaces

Cao

¹

,

Chang

²

,

Yang

³

et al. 2016

Trans. Amer. Math. Soc.

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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.

show abstract

“…Sufficient conditions, in order that a space (X,δ,µ) of homogeneous type admits a quasidistance d that is equivalent to δ and such that (X,d,µ) is normal, are given in [14,Lemma 22].…”

Section: Introduction Bramanti and Ceruttimentioning

confidence: 99%