ACTA issues

On the uniform and the absolute convergence of Dirichlet integrals of functions in Besov spaces

Dang Vu Giang, Ferenc Móricz

Acta Sci. Math. (Szeged) 59:1-2(1994), 257-265

Abstract. We show that if a function $f$ defined on the real line ${\bf R}$ belongs to the Besov space $B^{1/p}_{p,1},$ where $1\le p< \infty,$ then its Dirichlet integral $s_T(f,x)$ converges uniformly as $T\to\infty.$ In the case where $1\le p\le2$, we even prove that $s_T(f,x)$ converges absolutely. It follows easily that any function $f$ in $B^{1/p}_{p,1}, 1\le p\le2,$ is a Fourier transform on $L^1({\bf R}).$ The counterparts of our results were proved by A. M. Garsia (1976) for functions defined on the unit circle ${\bf T}.$ Our proofs are basically different from those given there.

AMS Subject Classification (1991): 42A38, 46E35, 41A17

Keyword(s): Besov space, modulus of continuity, Fourier transform, Dirichlet integral, Riesz mean, Hardy-Littlewood inequality, L^1., multiplier of Fourier transform on

Received September 15, 1993. (Registered under 5586/2009.)