Let M be a nonnegative measurable function on (0,$\infty)$ and let $\tilde{M}(x) = \vert x\vert\sp{{n\over p}-{n\over q}-n} M(\vert x\vert), x\in R\sp{n}.$ We can consider a convolution operator: for a suitable f, / (UNFORMATTED TABLE OR EQUATION FOLLOWS) / (a) Suppose $1\le s\le\infty.$ Then $M\in L\sp{t}({dr\over r})$ implies that $T\sb{M}:L\sp{p}(R\sp{n})\to L\sp{q}(R\sp{n})$ is bounded for all $({1\over p},{1\over q})$ in the type-diagram triangle with vertices $(1 - {1\over s},0),\ (1,{1\over s})\ {\rm and}\ (1 - {1\over(n+1)s},{1\over(n+1)s})$ if and only if s = t. / (b) Suppose $1<p<q<\infty.$ Let $s\sb0$ be the smallest value of $s\in\lbrack 1,\infty)$ such that ${1\over q}\ \ge\ {1\over n}({1\over p} - (1 - {1\over s}))$ and ${1\over q}\ \ge\ {n\over p} - n +\ {1\over s}$ Then $T\sb{M}:L\sp{p}(R\sp{n})\to L\sp{q}(R\sp{n})$ for all $M\in L\sp{t}({dr\over r})$ if and only if $s\sb0\le t\le\infty.$ / Results (a) and (b) solve the following problem: If $1<p<q<\infty$ find the ranges of s such that $M\in L\sp{s}({dr\over r})$ implies that $T\sb{M}$ is bounded from $L\sp{p}(R\sp{n})$ to $L\sp{q}(R\sp{n}).$ / Source: Dissertation Abstracts International, Volume: 54-07, Section: B, page: 3657. / Major Professor: Daniel M. Oberlin. / Thesis (Ph.D.)--The Florida State University, 1993.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_76982 |
| Contributors | Rhee, Jungsoo., Florida State University |
| Source Sets | Florida State University |
| Language | English |
| Detected Language | English |
| Type | Text |
| Format | 44 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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