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1.
Much of the recent literature on risk measures is concerned with essentially bounded risks in L
∞. In this paper we investigate in detail continuity and representation properties of convex risk measures on L
p
spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded
random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful
tools for approximations. As particular examples of risk measures on L
p
we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation
problem for L
p
risks. 相似文献
2.
We consider the operator ${\cal A}$ formally defined by ${\cal
A}u(x)=\alpha(x)\Delta u(x)$
for any $x$ in a sufficiently smooth bounded open set
$\Om\subset\R^N$ ($N\ge 1$), where $\alpha\in C(\ov\Omega)$ is a
continuous nonnegative function vanishing only on $\partial\Omega$,
and such that $1/\alpha$ is integrable in $\Omega$.
We prove that the realization $A_p$ of ${\cal A}$, equipped with
suitable nonlinear boundary conditions is an m-dissipative operator in
suitably weighted $L^p(\Omega)$-spaces in the
case where either $(p,N)\in (1,+\infty)\times\{1\}$ or
$(p,N)=\{2\}\times\N$. Moreover, we prove that $A_p$ is a densely
defined closed operator.
We consider nonlinear boundary conditions of the following type: in the one
dimensional case we take $\Omega=(0,1)$ and we assume that
$u(j)=(-1)^j\beta_j(u(j))$ ($j=0,1$), $\beta_0$ and $\beta_1$ being
nondecreasing continuous functions in $\R$ such that
$\beta_0(0)=\beta_1(0)=0$; in the $N$-dimensional setting we
assume that
$(D_{\nu}u)_{|\partial\Omega}=-\beta(u_{|\partial\Omega})$, $\beta$
being a nondecreasing Lipschitz continuous function in $\R$ such that
$\beta(0)=0$. Here $\nu$ denotes the unit outward normal to
$\partial\Om$. 相似文献
3.
V. M. Kaplitskiĭ 《Siberian Mathematical Journal》2010,51(2):255-266
We show that Peetre’s classical interpolation theorem in weighted L
p
-spaces is carried over to some classes of nonlinear operators containing in particular the Lipschitz operators and operators
close to them in the properties satisfying less restrictive conditions than Lipschitz in each of the spaces of a Banach pair. 相似文献
4.
F. Abtahi 《Acta Mathematica Hungarica》2011,133(4):324-331
Let G be a locally compact group, ω a weight function on G, and 1<p<∞. We introduce the Lebesgue weighted L p -space \({\mathcal{L}}_{\omega}^{1,p}(G)= L^{p}(G,\omega)\cap L^{1}(G)\) as a Banach space and introduce its dual. Furthermore, we consider this space as a Banach algebra with respect to the usual convolution and show that \({\mathcal{L}}_{\omega}^{1,p}(G)\) admits a bounded approximate identity if and only if G is discrete. In addition, we prove that amenability of this algebra implies that G is discrete and amenable. Moreover, we discuss the converse of this result. 相似文献
5.
Let Γ ⊂ ℝn, n ≥ 2, be the boundary of a bounded domain. We prove that the translates by elements of Γ of functions which transform according
to a fixed irreducible representation of the orthogonal group form a dense class in L
p
(ℝn) for
. A similar problem for noncompact symmetric spaces of rank one is also considered. We also study the connection of the above
problem with the injectivity sets for weighted spherical mean operators.
The first author was supported in part by a grant from UGC via DSA-SAP Phase IV. 相似文献
6.
V. E. Maiorov 《Ukrainian Mathematical Journal》2010,62(3):452-466
We study the approximation of the classes of functions by the manifold R
n
formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W
r
p
from the set R
n
of ridge functions in the space L
q
(B
d
) satisfies the sharp order n
-r/(d-1). 相似文献
7.
We will discuss about the mapping property of Radon transform on L
p
spaces with power weight. It will be shown that the Pitt’s inequality together with the weighted version of Hardy-Littlewood-Sobolev
lemma imply weighted inequality for the Radon transform. 相似文献
8.
In this article we study the problem of extending Fourier
Multipliers on L
p
(T) to those on L
p
(R)
by taking convolution with a kernel, called a summability
kernel. We characterize the space of such kernels
for the cases p = 1 and p = 2. For other values of p we give a
necessary condition for a function to be a
summability kernel. For the case p = 1, we present
properties of measures which are transferred from M(T) to
M(R) by summability kernels. Furthermore it is
shown that every l
p
sequence can be extended to some
L
q
(R) multipliers for certain values of p and q. 相似文献
9.
Let X be a Banach space and let
A be a closed linear operator on
X. It is shown that the abstract Cauchy problem
enjoys maximal regularity in weighted
L
p
-spaces with weights
, where
,
if and only if it has the property of maximal
L
p
-regularity.
Moreover, it is also shown that the derivation operator
admits an
-calculus in weighted
L
p
-spaces.
Received: 26 February 2003 相似文献
10.
In this paper Lambert multipliers acting between L
p
spaces are characterized by using some properties of conditional expectation operator. Also, Fredholmness of corresponding
bounded operators is investigated. 相似文献
11.
We give a characterization of structurally stable diffeomorphisms by making use of the notion of L
p
-shadowing property. More precisely, we prove that the set of structurally stable diffeomorphisms coincides with the C
1-interior of the set of diffeomorphisms having L
p
-shadowing property. 相似文献
12.
<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates of rough maximal functions along surfaces with applications
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In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(Sn-1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results. 相似文献
13.
14.
In this paper, the Lp(Rn)-boundedness of the commutators generalized by BMO(Rn) function and the singular integral operator T with rough kernel Ω∈ Llog+ L(Sn-1) is proved by using the Bony's formula for the paraproduct of two functions. 相似文献
15.
In order to approximate functions defined on (0, +∞), the authors consider suitable Lagrange polynomials and show their convergence
in weighted L
p
-spaces.
相似文献
16.
A weighted norm inequality for the Marcinkiewicz integral operator
is proved when belongs to
. We also give the weighted
Lp-boundedness for a class of Marcinkiewicz integral operators with rough
kernels
and
related to the Littlewood-Paley
-function and the
area integral S, respectively. 相似文献
17.
Milutin Dostanić 《Journal d'Analyse Mathématique》2008,104(1):13-23
It is shown that the Berezin transform B on L
p
(D), where D is the unit disc, has norm
. Furthermore, the norms of a family of operators (on L
p
(D)) whose kernels are moduli of Bergman type kernels are also calculated.
Partially supported by MNZZS, Grant
ON144010 相似文献
18.
Functions whose translates span L
p
(R) are called L
p-cyclic functions. For a fixed
p \memb [1, \infty], we construct Schwartz-class functions which are L
r
-cyclic for r > p and not L
r
-
cyclic for r \le p. We then construct Schwartz-class functions which are L
r
-cyclic for r \ge p and
not L
r
-cyclic for r < p. The constructions differ for p \memb (1, 2) and p > 2. 相似文献
19.
On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-consistency of wavelet estimators
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This paper deals with the L p -consistency of wavelet estimators for a density function based on size-biased random samples. More precisely, we firstly show the L p -consistency of wavelet estimators for independent and identically distributed random vectors in R d . Then a similar result is obtained for negatively associated samples under the additional assumptions d = 1 and the monotonicity of the weight function. 相似文献
20.
Let G be a locally compact group. For 1 < p < ∞, it is well-known that f * g exists and belongs to Lp(G) for all f, g
Lp
(G) if and only if G is compact. Here, for 2 < p < ∞, we show that f * g exists for all f, g
Lp(G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2.
Received: 23 April 2006 相似文献