Weighted inequalities for iterated convolutions

Abstract

Given a fixed exponent p p , 1 ≤ p > ∞ 1\le p>\infty , and suitable nonnegative weight functions v j v_j , j = 1 , … , m j=1,\dots ,m , an optimal associated weight function ω m \omega _m is constructed for which the iterated convolution product satisfies \[ ∫ 0 ∞ | [ ∏ j = 1 m ∗ F j ] ( x ) | p d x ω m ( x ) ≤ ∏ j = 1 m ∫ 0 ∞ | F j ( t ) | p d t v j ( t ) \int _0^{\infty }\bigg |\bigg [\prod _{j=1}^m*F_j\bigg ](x)\bigg |^p\, \dfrac {dx}{\omega _m(x)}\le \prod _{j=1}^m\int _0^{\infty }|F_j(t)|^p\, \dfrac {dt}{v_j(t)} \] for all complex valued measurable functions F j F_j with ∫ 0 ∞ | F j ( t ) | p d t / v j ( t ) > ∞ \int _0^{\infty }|F_j(t)|^p\,dt/v_j(t)>\infty . Here [ ∏ j = 1 2 ∗ F j ] ( x ) = [ F 1 ∗ F 2 ] ( x ) = ∫ 0 x F 1 ( t ) F 2 ( x − t ) d t [\prod _{j=1}^2*F_j](x)=[F_1*F_2](x)= \int _0^xF_1(t)F_2(x-t)\,dt and for each m > 2 m>2 , ∏ j = 1 m ∗ F j = [ ∏ j = 1 m − 1 ∗ F j ] ∗ F m \prod _{j=1}^m*F_j=\bigg [\prod _{j=1}^{m-1}*F_j \bigg ]*F_m . Analogous results are given when R + = ( 0 , ∞ ) R^+=(0,\infty ) is replaced by R n R^n and also when the convolution F 1 ∗ F 2 F_1*F_2 on R + R^+ is taken instead to be ∫ 0 ∞ F ( t ) G ( x / t ) d t / t \int _0^{\infty }F(t)G(x/t)\,dt/t . The extremal functions are also discussed.