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Weighted norm inequalities for multipliers
| Content Provider | Semantic Scholar |
|---|---|
| Author | Gutiérrez, Cristian E. |
| Copyright Year | 1988 |
| Abstract | We consider the two-weight function problem for a class of multiplier operators that include the Riesz and Bessel potentials. 1. Let o(t) be a nonnegative function on (0, oo) such that m(g~) = | e0 (tl/2)e-te dt is finite for every g > 0. We define the operator T by (1.1) (Tf)A(x) = m(7rlx12)f(x), where f (x) = fRn f (y)e 2xY dy. In this note we prove weighted norm inequalities for the class of multiplier operators defined by (1.1). This class includes the Riesz potentials and the Bessel potentials by taking p(t) = t' and p(t) = t&e-t, a > 0, respectively. The case of the Riesz potentials has been studied in [5] and [3]. A weight function u is said to belong to D,, ,u > 1, if u(Bt,(x)) ? cttn/u(B,(x)) for every t > 1, s > 0 and x E R , where BE,(x) denotes the ball with center x and radius s and u(B,(x)) its u-measure. We write Doo = Ul>? D1. Analogously, u E RD,, v > 0, if u(B,t(x)) > ctnvu(B8(x)) for every t > 1, s > 0 and x E Rn. It is not hard to see that if u E Doo then u E RDv for some v > O. LP is the class of functions g such that 119I1P,U = (fRn lg(x)lPu(x)dx)1/P is finite. So,o will denote the class of Schwartz functions whose Fourier transforms have compact support not including the origin. The space HuP, 0 o IF(x,t)I (see [8]). We prove the following THEOREM 1. Let 0 ,tp. If (1.2) p(t)u(Bt(x))111 0 and every x E Rn then IITf llq,u < CllIf IIHP for every f E So,o. THEOREM 2. Let O < q < 1 and u, v E Dm . If u(Q)j (/s p(t) ds < cv(Q) Received by the editors August 23, 1986 and, in revised form, October 30, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 42B15, 42B30, 42B25. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page |
| Starting Page | 290 |
| Ending Page | 294 |
| Page Count | 5 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1988-0920988-4 |
| Volume Number | 102 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1988-102-02/S0002-9939-1988-0920988-4/S0002-9939-1988-0920988-4.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1988-0920988-4 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
