Vol. 38, No. 1, 1971

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Sharp estimates of convolution transforms in terms of decreasing functions

Gary Sampson

Vol. 38 (1971), No. 1, 213–231
Abstract

Let f g denote the convolution transform of two Lebesgue measurable functions on the real line defined by the equation

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

We get best possible upper and lower estimates for the expression

          ∫
fi ∼ g∗i sup   |(f1 ∗∖⋅⋅⋅∗fn)(x)|pd(x)
|E |≦u  E

where p = 1 and 2, with applications to Fourier transform inequalities. Here gλ are preassigned decreasing functions and the symbol fi gi means

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

Mathematical Subject Classification 2000
Primary: 42A96
Secondary: 44A35
Milestones
Received: 18 November 1970
Published: 1 July 1971
Authors
Gary Sampson