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1.
We consider the mixed problem,
in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data,
f
D
, has one derivative in L
p
(D) of the boundary and the Neumann data, f
N
, is in L
p
(N). We find a p
0 > 1 so that for p in an interval (1, p
0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L
p
.
L. Lanzani, L. Capogna and R. M. Brown were supported, in part, by the U.S. National Science Foundation. 相似文献
2.
Ian Wood 《Mathematische Zeitschrift》2007,255(4):855-875
We consider the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains Ω, both with the following
two domains of definition: , or , where B is the boundary operator. We prove that, under certain restrictions on the range of p, these operators generate positive analytic contraction semigroups on L
p
(Ω) which implies maximal regularity for the corresponding Cauchy problems. In particular, if Ω is bounded and convex and
, the Laplacian with domain D
2(Δ) has the maximal regularity property, as in the case of smooth domains. In the last part, we construct an example that
proves that, in general, the Dirichlet–Laplacian with domain D
1(Δ) is not even a closed operator.
The main results of this paper are taken from the author’s Ph.D. thesis, written at the TU Darmstadt under the supervision
of Prof. M. Hieber. The author wishes to thank Prof. Hieber for his guidance, encouragement and support in the last few years.
Many thanks also go to Prof. C. E. Kenig for his hospitality and many ruitful discussions on the subject during a 1-year stay
at the University of Chicago. 相似文献
3.
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
4.
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
5.
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
6.
7.
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
8.
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
9.
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
10.
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
11.
On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-consistency of wavelet estimators 下载免费PDF全文
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
12.
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
13.
We derive interior L
p
-estimates for solutions of linear elliptic systems with oscillatory coefficients. The estimates are independent of ε, the small length scale of the rapid oscillations. So far, such results are based on potential theory and restricted to periodic
coefficients. Our approach relies on BMO-estimates and an interpolation argument, gradients are treated with the help of finite
differences. This allows to treat coefficients that depend on a fast and a slow variable. The estimates imply an L
p
-corrector result for approximate solutions.
相似文献
14.
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
15.
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
16.
Let M be a compact manifold of dimension n, P=P(h) a semiclassical pseudodifferential operator on M, and u=u(h) an L
2 normalized family of functions such that P(h)u(h) is O(h) in L
2(M) as h↓0. Let H⊂M be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L
p
norms, p≥2, of u restricted to H, under the assumption that the u are semiclassically localized and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch
−δ(n,k,p) where k=dim H (except for a logarithmic divergence in the case k=n−2, p=2). When H is a hypersurface, i.e., k=n−1, we have δ(n,n−1, 2)=1/4, which is sharp when M is the round n-sphere and H is an equator. 相似文献
17.
We introduce the concept of Lp-maximal regularity for second order Cauchy problems. We prove Lp-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity
of solutions for nonlinear wave equations.
The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the
travel grants provided by NBMH India and MSF Delhi, India. 相似文献
18.
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
19.
20.
Robert J. Taggart 《Mathematische Zeitschrift》2009,261(4):933-949
Suppose that {T
t
: t ≥ 0} is a symmetric diffusion semigroup on L
2(X) and denote by its tensor product extension to the Bochner space , where belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem
and show that this extends to a maximal theorem for analytic continuations of on . As an application, we show that such continuations exhibit pointwise convergence. 相似文献