Vol. 45, No. 1, 1973

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ISSN: 0030-8730
Two-sided Lp estimates of convolution transforms

Gary Sampson

Vol. 45 (1973), No. 1, 347–356
Abstract

Let f and g be two Lebesgue measurable functions on the real line. Then the equation

           ∫
+∞
(f ∗ g)(x) = −∞ f(t)g(x − t)dt

defines the convolution transform of f and g. In an earlier paper [4] we obtained sharp upper and lower eetimates for the expression

     ∫
sup   |(f1 ∗⋅⋅⋅∗fn)(x)|pd(x)
|E|≦u∗ E
fi∼gi
(A)

where p = 1,2 and 4, with applications to Fourier transform inequalities. This paper contains estimates of (A) for all values of p(p 1) in the case where E = (−∞,+). For example, one of our theorems implies the following:

“If gi is bounded and has compact support for all i, then there exists a constant K,1(p + 1)p K (p+ 1)p(2n1)p, such that

       ∫ ∞
(A) = K    |xn− 1(g∗1∗− g∗1)⋅⋅⋅(g∗n∗− g∗n)|pd(x).′′
0

Here gi are preassigned decreasing functions and the symbol fi gi means

|{x : |fi(x)| > y}| = |{x : g∗i(x ) > y}| for all y > 0.

Mathematical Subject Classification 2000
Primary: 44A35
Milestones
Received: 26 July 1971
Published: 1 March 1973
Authors
Gary Sampson