2021/10/26 by Mustafayev, Rza, Bilgiçli, Nevin, Yılmaz, Merve
#42B25 #42B35 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2110.13698
In this paper 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(p,m,v) into Λq(w). Here Λα(b) and GΓ(p,m,w) are the classical and generalized Lorentz spaces, defined as a set of all measurable functions f defined on \mathbb Rn for which ‖f‖Λα(b) = ( ∫0∞ [f^*(s)]α b(s) ds )^\frac1α lt; ∞ and ‖f‖GΓ(p,m,w) = ( ∫0∞ ( ∫0x [f^* (τ)]p dτ)(m)/(p) v(x) dx )(1)/(m) lt; ∞, respectively. We reduce the problem to the solution of the inequality ( ∫0∞ [ Tu,bf^* (x)]q w(x) dx)(1)/(q) ≤ C ( ∫0∞ ( ∫0x [f^* (τ)]p dτ)(m)/(p) v(x) dx )(1)/(m) where w and v are weight functions on (0,∞). Here f^* is the non-increasing rearrangement of f defined on \mathbb Rn and Tu,b is the iterated Hardy-type operator involving suprema, which is defined for a measurable non-negative function f on (0,∞) by (Tu,b g)(t) : = supτ∈ [t,∞) (u(τ))/(B(τ)) ∫0τ g(s)b(s) ds, t ∈ (0,∞), where u and b are appropriate weight functions on (0,∞) and the function B(t) : = ∫0t b(s) ds satisfies 0 < B(t) < ∞ for every t ∈ (0,∞)..