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

/pdf/best-constants-for-two-nonconvolution-inequalities-124u1h0qel.pdf

Best constants for two nonconvolution inequalities

Random martingales and localization of maximal inequalities

BYRODRIGO BANUELOSAbstract. The operators obtained by taking conditional expectation of continuoustime martingale transforms are studied, both on the circle T and on R". Using aBurkholder-Gundy inequality for vector-valued martingales, it is shown that thevector formed by any number of these operators is bounded on LP(R"), 1 < p < oo,with constants that depend only on p and the norms of the matrices involved. As acorollary we obtain a recent result of Stein on the boundedness of the Riesztransforms on LP(K'), 1 < p < oo, with constants independent of n.

/pdf/martingale-transforms-and-related-singular-integrals-30z2dwexln.pdf

Martingale transforms and related singular integrals

Let Md be the centered Hardy-Littlewood maximal function associated to cubes in R d with Lebesgue measure, and let cd denote the lowest constant appearing in the weak type (1,1) inequality satised by Md. We show that cd ! 1 as d ! 1, thus answering, for the case of cubes, a longstanding

http://annals.math.princeton.edu/wp-content/uploads/annals-v173-n2-p10-p.pdf

The weak type (1,1) bounds for the maximal function associated to cubes grow to infinity with the dimension

In this paper, the sharp bound for the weak-type (1, 1) inequality for the n-dimensional Hardy operator is obtained. Moreover, the precise norms of generalized Hardy operators on the type of Campanato spaces are worked out. As applications, the corresponding norms of the Riemann-Liouville integral operator and the n-dimensional Hardy operator are deduced. It is also proved that the n-dimensional Hardy operator maps from the Hardy space into the Lebesgue space. The endpoint estimate for the commutator generated by the Hardy operator and the (central) BMO function is also discussed.

Endpoint estimates for n-dimensional Hardy operators and their commutators

What is noteworthy here is that any of the usual covering arguments lead only to a weak-type (1,1) bound which grows exponentially in n, and thus by interpolation one obtains by this method (1.1) with Ap replaced by a bound which increases exponentially in n. Thus the foUowing further questions now present themselves: (1) Does M (") have a weak-type (1, 1) bound independent of n? (2) What can be said when the usual balls are replaced by dilates of more general sets?, We give here some partial answers to these questions:

/pdf/behavior-of-maximal-functios-in-r-n-for-large-n-z2e2w1mcmc.pdf

Behavior of maximal functios in R n for large n

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Behavior of maximal functios in R n for large n