The d-dimensional classical Hardy spaces H_p (T^d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from H_p(T^d)to L_p(T^2) (d/(d+1)<p≤∞) and is of weak t...The d-dimensional classical Hardy spaces H_p (T^d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from H_p(T^d)to L_p(T^2) (d/(d+1)<p≤∞) and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. The same is proved for the conjugate Riemann sums. As a consequence we obtain that every function f∈L_1(T^d)is a.e. Riemann summable to f, provided again that the limit is taken over a positive cone.展开更多
Let q∈(0,∞]andϕbe a Musielak-Orlicz function with uniformly lower type p_(ϕ)^(−)∈(0,∞)and uniformly upper type p_(ϕ)^(−)∈(0,∞).In this article,the authors establish various realvariable characterizations of the ...Let q∈(0,∞]andϕbe a Musielak-Orlicz function with uniformly lower type p_(ϕ)^(−)∈(0,∞)and uniformly upper type p_(ϕ)^(−)∈(0,∞).In this article,the authors establish various realvariable characterizations of the Musielak-Orlicz-Lorentz Hardy space H^(ϕ,q)(R^(n)),respectively,in terms of various maximal functions,finite atoms,and various Little wood-Paley functions.As applications,the authors obtain the dual space of Hϕ,q(Rn)and the summability of Fourier transforms from Hϕ,q(Rn)to the Musielak-Orlicz-Lorentz space L^(ϕ,q)(R^(n))when q∈(0,∞)or from the Musielak-Orlicz Hardy space Hϕ(Rn)to Lϕ,∞(Rn)in the critical case.These results are new when q∈(0,∞)and also essentially improve the existing corresponding results(if any)in the case q=∞via removing the original assumption thatϕis concave.To overcome the essential obstacles caused by both thatϕmay not be concave and that the boundedness of the powered Hardy-Littlewood maximal operator on associated spaces of Musielak-Orlicz spaces is still unknown,the authors make full use of the obtained atomic characterization of H^(ϕ,q)(R^(n)),the corresponding results related to weighted Lebesgue spaces,and the subtle relation between Musielak-Orlicz spaces and weighted Lebesgue spaces.展开更多
A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone ...A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.展开更多
基金This research was partly supported by the Hungarian Scientific Research Funds (OTKA) No F019633.
文摘The d-dimensional classical Hardy spaces H_p (T^d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from H_p(T^d)to L_p(T^2) (d/(d+1)<p≤∞) and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. The same is proved for the conjugate Riemann sums. As a consequence we obtain that every function f∈L_1(T^d)is a.e. Riemann summable to f, provided again that the limit is taken over a positive cone.
文摘Let q∈(0,∞]andϕbe a Musielak-Orlicz function with uniformly lower type p_(ϕ)^(−)∈(0,∞)and uniformly upper type p_(ϕ)^(−)∈(0,∞).In this article,the authors establish various realvariable characterizations of the Musielak-Orlicz-Lorentz Hardy space H^(ϕ,q)(R^(n)),respectively,in terms of various maximal functions,finite atoms,and various Little wood-Paley functions.As applications,the authors obtain the dual space of Hϕ,q(Rn)and the summability of Fourier transforms from Hϕ,q(Rn)to the Musielak-Orlicz-Lorentz space L^(ϕ,q)(R^(n))when q∈(0,∞)or from the Musielak-Orlicz Hardy space Hϕ(Rn)to Lϕ,∞(Rn)in the critical case.These results are new when q∈(0,∞)and also essentially improve the existing corresponding results(if any)in the case q=∞via removing the original assumption thatϕis concave.To overcome the essential obstacles caused by both thatϕmay not be concave and that the boundedness of the powered Hardy-Littlewood maximal operator on associated spaces of Musielak-Orlicz spaces is still unknown,the authors make full use of the obtained atomic characterization of H^(ϕ,q)(R^(n)),the corresponding results related to weighted Lebesgue spaces,and the subtle relation between Musielak-Orlicz spaces and weighted Lebesgue spaces.
基金Supported by the Hungarian Scientific Research Funds (OTKA) No. K67642
文摘A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.
