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This paper is concerned with the L- L estimates of the solutions for a class of pseudodifferential equations under some suitable degenerate assumptions. As applications, these estimates can be used to show that a generalized Schrödinger operator with some integrable potential generates a fractionally integrated group in L(ℝ). [ABSTRACT FROM AUTHOR]

In this paper, for an arbitrary regular biordered setE, by using biorder-isomorphisms between the ?-ideals ofE, we construct a fundamental regular semigroupWE called NH-semigroup ofE, whose idempotent biordered set is isomorphic toE. We prove further thatWE can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroupSwith the idempotent biordered set isomorphic toE, there exists a homomorphism fromStoWE whose kernel is the greatest idempotent-separating congruence onSand the image is a full symmetric subsemigroup ofWE. Moreover, whenEis a biordered set of a semilatticeE0,WE is isomorphic to the Munn-semigroup; and whenEis the biordered set of a bandB,WE is isomorphic to the Hall-semigroupWB. [ABSTRACT FROM AUTHOR]

It is proved that with at mostO(N11/12+e) exceptions, all positive integersn=Nsatisfying some necessary congruence conditions are the sum of three squares of primes. This improves substantially the previous results in this direction. [ABSTRACT FROM AUTHOR]