Best constants for two nonconvolution inequalities

摘要: The norm of the operator which averages If I in LP (Rn) over balls radius (51xl centered at either 0 or x is obtained as a function n, p and 65. Both inequalities proved are n-dimensional analogues classical inequality Hardy R1 . Finally, lower bound for Hardy-Littlewood maximal on given. 0. INTRODUCTION A result [HLP] states that if f LP(R1) > 1, then (0.1) (jt (1-J )P)ldt / d 1 (f ' f(t)nPdt) constant p/(p 1) best possible. By considering two-sided instead one-sided, can be equivalently formulated as: (0.2) ( 21l Jl f(t)ldt) dx) < If(t)IPdt) We denote by D(a, R) ball R Rn a. Let (Tf)(x) average Ifl E LP(Rn) D(O, lxl). analogue inequality: (0.3) IITfJIIP Cp(n)JJfJJIP some Cp(n) depends priori n. Our first satisfies all p' = 1), same dimension one. Another version Hardy's with possible found [F]. Next we consider similar problem. An equivalent formulation (0.4) 00 x+1X1 \p l/p { /p (L (2 Jx lf(t)Idt) dx 21/P(p1) l f(t)I dt Received editors May 5, 1993 and, revised form, September 3, 1993. 1991 Mathematics Subject Classification. Primary 42B25. authors' research was partially supported National Science Foundation. ? 1995 American Mathematical Society 0002-9939/95 $1.00 + $.25 per page