The boundedness of hausdorff operators on function spaces

For a fixed kernel function phi, the one dimensional Hausdorff operator is defined in the integral form by hFf x=0x Ft tf&parl0;xt&parr0; dt. By the Minkowski inequality, it is easy to check that the Hausdorff operator is bounded on the Lebesgue spaces Lp when p ≥ 1, with some size condition assumed on the kernel functions phi. However, people discovered that the above boundedness property is quite different on the Hardy space Hp when 0 < p < 1. To establish the boundedness on the Hardy space for 0 < p < 1, some smoothness must be assumed on the kernel functions phi.;In this thesis, we first study the boundedness h phi on the Hardy space H1, and on the local Hardy space h1( R ). Our work shows that for phi(t) ≥ 0, the Hausdorff operator hphi is bounded on the Hardy space H1 if and only if phi is a Lebesgue integrable function; and hphi is bounded on the local Hardy space h1( R ) if and only if the functions phi(t)&khgr; (1,infinity)(t) and phi(t)&khgr; (0,1)(t) log 1t are Lebesgue integrable. These results solve an open question posed by the Israeli mathematician Liflyand. We also establish an H 1( R ) → H1,infinity( R ) boundedness theorem for hphi. As applications, we obtain many decent properties for the Hardy operator and the k th order Hardy operators. For instance, we know that the Hardy operator H is bounded from H1( R ) → H1,infinity( R ), bounded on the atomic space H1A&parl0;R+ &parr0; +), but it is not bounded on both H 1( R ) and the local Hardy space h1( R ).;We also extend part of these results to the high dimensional Hausdoff operators. Here, we study two high dimensional extentions on the Hausdorff operatorhphi: H&d5;F,b fx =Rn Fy &vbm0;y&vbm0;n-bf&parl0; xy&parr0; dy,n⩾b⩾0, and HF,bf x= RnF&parl0; xy &parr0;&vbm0;y&vbm0;n-bf &parl0;y&parr0;dy,n⩾b ⩾0, where phi is a local integrable function.;For 0 < p < 1, we obtain a sufficient condition for the Hp boundedness for the Hausdorff operator in the one dimensional case. This theorem needs less smoothness on the kernel phi than any other theorems in the literature. Since there is no result involving the boundedness on Hp( Rn ) in the literature for the high dimensional Hausdorff operators, if 0 < p < 1 and n ≥ 2, it is interesting to study such problems in the high dimensional spaces. We establish several sufficient conditions by using a duality argument.;Additionally, we study boundedness of Hausdorff operators on some Herz type spaces, and some bilinear Hausdorff operators operators and fractional Hausdorff operators.