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Convolution algebras with weighted rearrangement-invariant norm

R. KermanE. Sawyer — 1994

Studia Mathematica

Let X be a rearrangement-invariant space of Lebesgue-measurable functions on n , such as the classical Lebesgue, Lorentz or Orlicz spaces. Given a nonnegative, measurable (weight) function on n , define X ( w ) = F : n : > F X ( w ) : = F w X . We investigate conditions on such a weight w that guarantee X(w) is an algebra under the convolution product F∗G defined at x n by ( F G ) ( x ) = ʃ n F ( x - y ) G ( y ) d y ; more precisely, when F G X ( w ) F X ( w ) G X ( w ) for all F,G ∈ X(w).

The trace inequality and eigenvalue estimates for Schrödinger operators

R. KermanEric T. Sawyer — 1986

Annales de l'institut Fourier

Suppose Φ is a nonnegative, locally integrable, radial function on R n , which is nonincreasing in | x | . Set ( T f ) ( x ) = R n Φ ( x - y ) f ( y ) d y when f 0 and x R n . Given 1 < p < and v 0 , we show there exists C > 0 so that R n ( T f ) ( x ) p v ( x ) d x C R n f ( x ) p d x for all f 0 , if and only if C ' > 0 exists with Q T ( x Q v ) ( x ) p ' d x C ' Q v ( x ) d x < for all dyadic cubes Q, where p ' = p / ( p - 1 ) . This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

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