Let l and p be primes, let F/Q(p) be a finite extension with absolute Galois group G(F), let F be a finite field of characteristic l, and let (rho) over bar : G(F) -> GL(n) (F) be a continuous representation. Let R-square((rho) over bar) be the universal framed deformation ring for (rho) over bar. If l =3D p, then the Breuil-Mezard conjecture ( as recently formulated by Emerton and Gee) relates the mod l reduction of certain cycles in R-square((rho) over bar) to the mod l reduction of certain representations of GL(n) (O-F) . We state an analogue of the Breuil-Mezard conjecture when l not equal p, and we prove it whenever l > 2 using automorphy lifting theorems. We give a local proof when l is "quasibanal" for F and (rho) over bar is tamely ramified. We also analyze the reduction modulo l of the types sigma(tau) defined by Schneider and Zink.
Let G be an infinite compact abelian group. If its dual group G contains an element of infinite order, then it is known that, for every 4 < p < 8, there exists a function g is an element of L-p(G) whose associated convolution operator C-g : f bar right arrow f * g (on L-p(G)) is absolutely summing but the Fourier series of g fails to be unconditionally convergent to g in L-p(G). It is shown that the restriction on G containing an element of infinite order can be removed and also that the range of p can be extended to arbitrary p is an element of (2, infinity).
In this paper, we explore a general method to derive H(p) -> L(p) boundedness from H(p) -> H(p) boundedness of linear operators, an idea originated in the work of Han and Lu in dealing with the multiparameter flag singular integrals ([19]). These linear operators include many singular integral operators in one parameter and multiparameter settings. In this paper, we will illustrate further that this method will enable us to prove the H(p) -> L(p) boundedness on product spaces of homogeneous type in the sense of Coifman and Weiss ([5]) where maximal function characterization of Hardy spaces is not available. Moreover, we also provide a particularly easy argument in those settings such as one parameter or multiparameter Hardy spaces H(p)(R(n)) and H(p)(R(n) x R(m)) where the maximal function characterization exists. The key idea is to prove parallel to f parallel to(Lp) <= C parallel to f parallel to(Hp) for f is an element of L(q) boolean AND H(p) (1 < q < infinity, 0 < p <= 1). It is surprising that this simple result even in this classical setting has been absent in the literature.
In one of our previous papers we proved that, for an infinite set A and p is an element of [1, infinity), the embedded version of the Lipscomb's space L(A) in l(p)(A), p is an element of [1, infinity), with the metric induced from l(p)(A), denoted by omega(A)(p). is the attractor of an infinite iterated function system comprising affine transformations of l(p) (A). In the present paper we point out that omega(A)(p) = omega(A)(q) for all p, q is an element of [1, infinity) and, by providing a complete description of the convergent sequences from omega(A)(p), we prove that the topological structure of omega(A)(p) is independent of p. (C) 2012 Elsevier B.V. All rights reserved.
In this paper, we study the existence of multiple solutions to the L-p-Minkowski problem. We prove if p < -n, then for any integer N > 0, there exists a smooth positive function f on S-n such that the L-p-Minkowski problem admits at least N different smooth solutions. We also construct nonsmooth, positive function f for which the L-p-Minkowski problem has infinitely many C-1,C-1 solutions.
Let F be a totally real field, l and p distinct odd prime unramified in F and l a prime above l. Let K/F be a p-ordinary CM quadratic extension and lambda an arithmetic Hecke character over K. Hida constructed a measure on the l-anticyclotomic class group of K interpolating the normalised Hecke L-values L-alg,L-l (0, lambda nu), as. varies over the finite order l-power conductor anticyclotomic characters. In this article, we interpolate the measures as lambda varies in a p-adic family. In particular, this gives p-adic deformation of the measures. An analogue holds in the case of self-dual Rankin-Selberg convolution of a Hilbert modular form and a theta series. In the case of root number -1, we describe an upcoming analogous interpolation of the p-adic Abel-Jacobi image of generalised Heegner cycles associated with the convolution.