Abstract
For a function its modified strong dyadic integral and the modified strong dyadic derivative are defined. A criterion for the existence of a modified strong dyadic integral for an integrable function is proved, and the equalities and are established under the assumption that . A countable system of eigenfunctions of the operators and is found. The linear span of this set is shown to be dense in the dyadic Hardy space , and the linear operator , , is proved to be bounded. Hence this operator can be uniquely continuously extended to and the resulting linear operator is bounded.
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