In this note we improve the known L p-bounds for Bochner-Riesz operators and their maximal operators.

Abstract. The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure μ, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces L(μ;X) of functions with val...

Introduction. S. Bochner has shown that an additive homogeneous functional defined over a sufficiently large class C of functions can be realized as a Riemann integral with respect to a finitely additive measure V in the space X over which the functions are defined. His proof makes use of the fact that the constant function belongs to C, as a result, V(X) is finite. I t is the purpose of this n...

We provide some new results on the weak convergence of sequences or nets lying in L,((T, & r), X) = L,(p, X), 1 <p < co, i.e., the space of equivalence classes of X-valued (X is a Banach space) Bochner integrable functions on the finite measure space ( r, z, I”). Our theorems generalize in several directions recent resuls on weak sequential convergence in L,(p, X) obtained by M. A. Khan and M. ...

Let (T,r,p) be a finite measure space, X be a Banach space, P be a metric space and let L,(y,X) denote the space of equivalence classes of X-valued Bochner integrable functions on (T, T, p). We show that if $I: T x P-2x is a set-valued function such that for each fixed p E P, 4(. , p) has a measurable graph and for each fixed TV T, 4(t;) is either upper or lower semicontinuous then the Aumann i...

Abstract. The paper gives a Banach space -valued extension of the “Tb theorem” of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure μ, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces L(μ;X) of functions with v...

where the line integral is taken in a positive sense around the curve C. In [l] Bochner investigated under which conditions (1) holds. There it was shown that if A and B have certain regularity properties and if the integrand on the right of (1) behaves well, then (1) does hold. Here we shall prove (1) under what may be regarded as the weakest possible hypotheses. This question was also treated...

The notion of a <7-continuous matrix-valued positive definite distribution on 5/v(2a) x RM x G is introduced, where G is an abelian separable locally compact group and where Sp/(2a) is an open ball around zero in RA' with radius la > 0 . This notion generalizes that one of strongly continuous positive definite operator-valued functions. For these objects, a Bochner-type theorem gives a suitable...