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Hardy-type operators with general kernels and characterizations of dynamic weighted inequalities

Volume 126 / 2021

S. H. Saker, M. M. Osman, D. O’Regan, R. P. Agarwal Annales Polonici Mathematici 126 (2021), 55-78 MSC: 26A15, 26D10, 26D15, 39A13, 34A40. DOI: 10.4064/ap191222-23-6 Published online: 11 December 2020

Abstract

In this paper, we prove some new characterizations of the weighted functions such that norm dynamic inequalities of mixed type involving operators of Hardy’s type with general kernels, of the form$$ \Vert \mathcal {H}_{\mathcal {K}}f \Vert _{\mathbb {L}_{u}^{q}([a,\infty )_{\mathbb {T}})}\leq A \Vert f \Vert _{\mathbb {L}_{\upsilon }^{p}([a,\infty )_{\mathbb {T}})}, $$ hold for $1 \lt p\leq q \lt \infty $ and $1 \lt q \lt p \lt \infty ,$ where $\mathcal {H}_{\mathcal {K}}f (x ) :=\int _{a}^{\sigma (x ) }\mathcal {K} (\sigma (x ) ,y ) f(y)\,\varDelta y$ (here $u$ and $\upsilon $ are the weight functions). Corresponding results are also obtained for the adjoint operator $\mathcal {H}_{\mathcal {K}}^{\ast }f ( x ) :=\int _{x}^{\infty }\mathcal {K} (x,\sigma (y) ) f(y)\,\varDelta y,$ where $\sigma (x)$ is the forward jump operator on time scales. Our results include some well known inequalities in the literature.

Authors

  • S. H. SakerDepartment of Mathematics
    Faculty of Science
    Mansoura University
    Mansoura, Egypt
    e-mail
  • M. M. OsmanDepartment of Mathematics
    Faculty of Science
    Mansoura University
    Mansoura, Egypt
    e-mail
  • D. O’ReganSchool of Mathematics, Statistics
    and Applied Mathematics
    National University of Ireland
    Galway, Ireland
    e-mail
  • R. P. AgarwalDepartment of Mathematics
    Texas A \& M University
    Kingsville, TX, 78363, U.S.A.
    e-mail

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