2023/06/14 by José Ángel Peláez, Peláez, José Ángel, Elena de la Rosa +3 · 1 citation
Mathematics · #Holomorphic and Operator Theory #Algebraic and Geometric Analysis #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2306.08255
Let ω and ν be radial weights on the unit disc of the complex plane, and denote σ=ωp'ν^-\fracp'p and ωx =∫01 sx ω(s) ds for all 1≤ x<∞. Consider the one-weight inequality \beginequation ‖Pω(f)‖Lpν≤ C‖f‖Lpν, 11. In addition, an analogous result for the one weight inequality ‖Pω(f)‖_Dpν,k≤ C‖f‖Lpν, where \Vert f \Vert_Dpν, kp =∑j=0k-1| f(j)(0)|p+ ∫_\mathbbD \vert f(k)(z)\vertp (1-|z| )kpν(z) dA(z)lt;∞, k∈ ℕ, is established. The inequality \eqrefab1 is further studied by using the necessary condition Dp(ω,ν)<∞ in the case of the exponential type weights ν(r)=exp (-\fracα(1-rl)β ) and ω(r)= exp (-\frac\widetildeα(1-r^\widetildel)\widetildeβ ), where 0<α, \widetildeα, l, \widetildel<∞ and 0<β, \widetildeβ≤ 1.