Using a generalized spherical mean operator, we obtain a generalization of Titchmarsh's theorem for the Dunkl transform for functions satisfying the ('; p)-Dunkl Lipschitz condition in the space Lp(Rd;wl(x)dx), 1 < p 6 2, where wl is a weight function invariant under the action of an associated re ection group.

We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative Lp space for some p > 1. This is a noncommutative version of Rosenthal’s result for commutative Lp spaces. Similarly for 1 ≤ q < 2, an infinite dimensional subspace X of a noncommutative Lq space either contains lq or embeds in Lp for some q < p < 2. The novelty in the noncommutative setting i...

A classification of weakly compact multiplication operators on L(Lp), 1 < p < ∞, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of `p-strictly singular operators, and we also investigate the structure of general `p-strictly singular operators on Lp. The main result is that if an operator T on Lp, 1 < p < 2, is `p-strictly singular ...

A classification of weakly compact multiplication operators on L(Lp), 1 < p < ∞, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of `p-strictly singular operators, and we also investigate the structure of general `p-strictly singular operators on Lp . The main result is that if an operator T on Lp , 1 < p < 2, is `p-strictly singula...

Let ? be a weight function defined on locally compact group G, 1?p<+?, S?G and let us assume that for any s?S, the left translation operator Ts is continuous from weighted Lp-space Lp(G,?) into itself. For given set ??C, vector f?Lp(G,?) said to (?,S)-dense if {?Tsf:???,s?S} dense in Lp(G,?). In this paper, we characterize existence of vectors terms ?.

The space of linear programs (LP) can be partitioned into a finite number of sets, each corresponding to a basis. This partition is thus called the basis partition. The closed-form solution on the space of LP can be determined with the basis partition if we can characterize the basis partition. A differential equation on the Grassmann manifold which represents the space of LP provides a powerfu...

Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. We begin with the classical definition of Sobolev spaces. Definition 1. Let k be a nonnegative integer and let 1 < p < ∞ ...

Let M be the Doob maximal operator on a filtered measure space and let v an Ap weight with 1&lt;p&lt;+∞. We try proving that ∥Mf∥Lp(v)≤p′[v]Ap1p−1∥f∥Lp(v), where 1/p+1/p′=1. Although we do not find approach which gives constant p′, obtain ∥Mf∥Lp(v)≤p1p−1p′[v]Ap1p−1∥f∥Lp(v), limp→+∞p1p−1=1.

There are several characterizations of coarse embeddability of a discrete metric space into a Hilbert space. In this note we give such characterizations for general metric spaces. By applying these results to the spaces Lp(μ), we get their coarse embeddability into a Hilbert space for 0 < p < 2. This together with a theorem by Banach and Mazur yields that coarse embeddability into l2 and into L...

For each p^ 1, Lp is a linear metric complete separable space. Let E be a set in Lp. The linear manifold M(E) determined by E is the set of all linear combinations (finite) of elements of £ . The span SP(E) of E in Lp is the closure in Lp of M (E) ; an element cj> of Lp belongs to SP(E) if and only if to each e > 0 corresponds an element f€ of M(E) such that | | 0 /« | | <e . L e t / G L p . Fo...