The Bochner-Riesz means of order delta greater-than-or-equal-to 0 for suitable test functions on R(N) are defined via the Fourier transform by (S(R)(delta)f)(xi) = (1 - \xi\2/R2)+(delta)f(xi). We show that the means of the critical index delta = N/P - N + 1/2, 1 < p < 2N/N + 1, do not map L(p,infinity)(R(N)) into L(p,infinity) (R(N)), but they map radial functions of L(p,infinity) (R(N)) into L(p,infinity) (R(N)). Moreover, if f is radial and in the L(p,infinity) (R(N)) closure of test functions, S(R)(delta)f (x) converges, as R --> + infinity, to f(x) in norm and for almost every x in R(N). We also observe that the means of the function absolute value of x-N/p, which belongs to L(p,infinity) (R(N)) but not to the closure of test functions, converge for no x
Colzani, L., Travaglini, G., Vignati, M. (1993). Bochner-Riesz means of functions in weak-$L\sp p$. MONATSHEFTE FÜR MATHEMATIK, 115(1-2), 35-45 [10.1007/BF01311209].
Bochner-Riesz means of functions in weak-$L\sp p$
COLZANI, LEONARDO;TRAVAGLINI, GIANCARLO;
1993
Abstract
The Bochner-Riesz means of order delta greater-than-or-equal-to 0 for suitable test functions on R(N) are defined via the Fourier transform by (S(R)(delta)f)(xi) = (1 - \xi\2/R2)+(delta)f(xi). We show that the means of the critical index delta = N/P - N + 1/2, 1 < p < 2N/N + 1, do not map L(p,infinity)(R(N)) into L(p,infinity) (R(N)), but they map radial functions of L(p,infinity) (R(N)) into L(p,infinity) (R(N)). Moreover, if f is radial and in the L(p,infinity) (R(N)) closure of test functions, S(R)(delta)f (x) converges, as R --> + infinity, to f(x) in norm and for almost every x in R(N). We also observe that the means of the function absolute value of x-N/p, which belongs to L(p,infinity) (R(N)) but not to the closure of test functions, converge for no x
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