2021/09/14 by Mustafayev, Rza, Bilgiçli, Nevin, Yılmaz, Merve
#42B25 #42B35 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2109.06745
In this paper we characterize the inequality ( ∫0∞ ( ∫0x [ Tu,bf^* (t)]r dt)(q)/(r) w(x) dx)(1)/(q) ≤ C ( ∫0∞ ( ∫0x [f^* (τ)]p dτ)(m)/(p) v(x) dx )(1)/(m) for 1 < m < p ≤ r < q < ∞ or 1 < m ≤ r < min\p,q\ < ∞, where w and v are weight functions on (0,∞). The inequality is required to hold with some positive constant C for all measurable functions defined on measure space (\mathbb Rn,dx). Here f^* is the non-increasing rearrangement of a measurable function f defined on \mathbb Rn and Tu,b is the iterated Hardy-type operator involving suprema, whish is defined for a measurable non-negative function f on (0,∞) by (Tu,b g)(t) : = supt ≤ τlt; ∞ (u(τ))/(B(τ)) ∫0τ g(s)b(s) ds, t ∈ (0,∞), where u and b are two weight functions on (0,∞) such that u is continuous on (0,∞) and the function B(t) : = ∫0t b(s) ds satisfies 0 < B(t) < ∞ for every t ∈ (0,∞). At the end of the paper, as an application of obtained results, we calculate the norm of the generalized maximal operator Mϕ,Λα(b), defined with 0 < α< ∞ and functions b, ϕ: (0,∞) → (0,∞) for all measurable functions f on \mathbb Rn by Mϕ,Λα(b)f(x) : = supQ \ni x \frac‖f χQ‖Λα(b)ϕ(|Q|), x ∈ \mathbb Rn, from GΓ(p1,m1,v) into GΓ(p2,m2,w). Here Λα(b) and GΓ(p,m,w) are the classical and generalized Lorentz spaces, respectively.