Convergence of multi-dimensional integral operators and applications

Szarvas, Kristóf; Weisz, Ferenc

doi:10.1007/s10998-016-0157-9

Cited by 10 publications

(3 citation statements)

References 17 publications

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“…This means that those mixed-norm Herz spaces Ė * q (R n ) are the best choice in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. We refer the reader to [31,32,98,99,101,102] for applications of Herz spaces in the summability of both Fourier transforms and series, to [94,103,104,100,87] for applications of Herz spaces in other convergence problems related to integral operators, Lebesgue points, and Gabor expansions and, particularly, to the monograph [105] of Weisz for a detailed study on these subjects. However, to the best of our knowledge, no other properties of these mixed-norm Herz spaces and their applications in related Hardy spaces are known so far.…”

Section: Introductionmentioning

confidence: 99%

Mixed-Norm Herz Spaces and Their Applications in Related Hardy Spaces

Zhao¹,

Yang²,

Zhang³

2022

Preprint

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In this article, the authors introduce a class of mixed-norm Herz spaces, Ė α, p q (R n ), which is a natural generalization of mixed Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz-Thorin interpolation theorem on Ė α, p q (R n ). Applying these Riesz-Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy-Littlewood maximal operator and the Fefferman-Stein vector-valued maximal inequality on Ė α, p q (R n ). As applications, the authors develop various real-variable theory of Hardy spaces associated with Ė α, p q (R n ) by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of Ė α, p q (R n ) and the non-trivial constructions of auxiliary functions in the Riesz-Thorin interpolation theorem.

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Section: Introductionmentioning

confidence: 99%

Mixed-Norm Herz Spaces and Their Applications in Related Hardy Spaces

Zhao¹,

Yang²,

Zhang³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In [19] we investigate the θ-summation of the Fourier transform of functions from the variable Lebesgue spaces over cone-like sets. To this end we need the inequalities with respect to the maximal operator M γ,δ proved in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…To this end we need the inequalities with respect to the maximal operator M γ,δ proved in this paper. More exactly, in [19] we estimate pointwise the maximal operator of the θ-means of the Fourier transforms by the maximal operator M γ,δ . This implies the almost everywhere convergence of the θ-means of f to the function f from the variable Lebesgue spaces.…”

Section: Introductionmentioning

confidence: 99%