BOUNDEDNESS OF THE FRACTIONAL INTEGRAL ON WEIGHTED LEBESGUE AND LIPSCHITZ SPACES
BOUNDEDNESS OF THE FRACTIONAL INTEGRAL ON WEIGHTED LEBESGUE AND LIPSCHITZ SPACES
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DOI：
10.1090/s0002-9947-97-01644-9
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发表时间：
1997
影响因子：
1.3
通讯作者：
E. Harboure;O. Salinas;B. Viviani
中科院分区：
文献类型：
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作者：
E. Harboure;O. Salinas;B. Viviani
给出了分数次积分算子1 I从p > n/a范围内的加权强、弱LP空间有界到适当的加权BMO和Lipschitz空间的充要条件.我们还刻画了1 I可以从加权BMO扩展到O阶加权Lipschitz空间的有界算子的权。最后，在加权的一个附加假设下，得到了I，>在加权Lipschitz空间之间有界的充要条件. 0.引言一个非负函数w定义在tn被称为一个重量，如果它是局部可积的。我们用JEl表示E的勒贝格测度，并且w（E）= fEw（x）dx。给定一个球B，GB将意味着球与B具有相同的中心，并且半径为0倍长。在本文中，字母C将表示一个常数，不一定每次出现时都相同。如果存在一个常数C使得对于每个球B C R '，w（B）（-i 1（B）p-1 RBI IBI，则称权w属于Muckenhoupt类Ap，1定义了类A1，用1 W-1XB 11 ~ w（B）<0 C 1 o IBI代替上述不等式，其中X B是球B的特征函数.如果存在一个常数C使得w（2 B）< Cw（B），我们就说权w满足加倍条件。1991年数学学科分类。第42 B25章
Necessary and sufficient conditions are given for the fractional integral operator 1I to be bounded from weighted strong and weak LP spaces within the range p > n/a into suitable weighted BMO and Lipschitz spaces. We also characterize the weights for which 1I can be extended to a bounded operator from weighted BMO into a weighted Lipschitz space of order o. Finally, under an additional assumption on the weight, we obtain necessary and sufficient conditions for the boundedness of I,> between weighted Lipschitz spaces. 0. INTRODUCTION A nonnegative function w defined on tn is called a weight if it is locally integrable. We denote by JEl the Lebesgue measure of E, and w(E) = fEw(x)dx. Given a ball B, GB will mean the ball with the same center as B and with radius 0 times as long. Throughout this paper, the letter C will denote a constant not necessarily the same at each occurrence. A weight w is said to belong to the Muckenhoupt class Ap, 1 < p < oo, if there exists a constant C such that w(B) ( -i1 (B) p-1 RBI IBI for every ball B C R'. The class A1 is defined replacing the above inequality by 1W-1XB11~ w(B) <0C 1 o IBI where XB is the characteristic function of the ball B. We shall say that a weight w satisfies a doubling condition if there exists a constant C such that w(2B) < Cw(B) Received by the editors June 26, 1995. 1991 Mathematics Subject Classification. Primary 42B25.