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1.
In this paper we consider operators acting on a subspace ℳ of the space L
2 (ℝm; ℂm) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace
ℳ is defined as the orthogonal sum of spaces ℳs,k of specific Clifford basis functions of L
2(ℝm; ℂm).
Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic
operators are characterized in terms of commutation relations and they transform a space ℳs,k into a similar space ℳs′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known.
Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is
studied in detail. 相似文献
2.
An interpretation is given to point interactions of the form −Δ+d inL
p
(ℝ
N
), where Δ is the Laplacian operator andd is a pseudopotential related to the ‘Dirac measure at 0', depending on the dimension. They are described as extensions of
−Δ, defined on the space {u∈C
0
∞
(ℝ
N
)|u(0)=0} that are negative generators of analytic semigroups. This is done forN=1,2 and 1<p<∞ and forN=3 and 3/2<p<3. 相似文献
3.
LetT(t) be the translation group onY=C
0(ℝ×K)=C
0(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY
⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y
⊙=L
1(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT
0(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT
0(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ
⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C
0(Ω;X) andL
p
(μ;X).
This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful
to the CWI and the Dutch National Science Foundation NWO for financial support. 相似文献
4.
Bálint Farkas 《Czechoslovak Mathematical Journal》2011,61(2):309-322
For a given bi-continuous semigroup (T(t))
t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space Cb(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures
(endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of
bi-continuous semigroups on Cb(Ω). We also prove that the class of bi-continuous semigroups on Cb(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict
topology. In general, if is not a Polish space this is not the case. 相似文献
5.
In this paper, the boundedness of Toeplitz operator T
b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε
(ℝn) is discussed from L
p(ℝn) to L
q(ℝn),
, and from L
p(ℝn) to Triebel-Lizorkin space
. We also obtain the boundedness of generalized Toeplitz operator Θ
α0
b
from L
p(ℝn) to L
q(ℝn),
. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator
T
b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L
p(ℝn), 1 < p < ∞. 相似文献
6.
Subhash J. Bhatt 《Proceedings Mathematical Sciences》2006,116(2):161-173
Given anm-tempered strongly continuous action α of ℝ by continuous*-automorphisms of a Frechet*-algebraA, it is shown that the enveloping ↡-C
*-algebraE(S(ℝ, A∞, α)) of the smooth Schwartz crossed productS(ℝ,A
∞, α) of the Frechet algebra A∞ of C∞-elements ofA is isomorphic to the Σ-C
*-crossed productC
*(ℝ,E(A), α) of the enveloping Σ-C
*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK
*(S(ℝ, A∞, α)) =K
*(C
*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC
*-algebra defined by densely defined differential seminorms is given. 相似文献
7.
We study the boundedness of the H
∞ functional calculus for differential operators acting in L
p
(R
n
; C
N
). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients,
we extend our recent results for the L
p
theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients
Π
B
as treated in L
2(R
n
; C
N
) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that Π
B
has a bounded H
∞ functional calculus, in terms of randomized boundedness conditions of its resolvent. This allows us to deduce stability under
small perturbations of this functional calculus. 相似文献
8.
J. Hounie Rafael Augusto dos Santos Kapp 《Journal of Fourier Analysis and Applications》2009,15(2):153-178
In this work, we study the continuity of pseudodifferential operators on local Hardy spaces h
p
(ℝ
n
) and generalize the results due to Goldberg and Taylor by showing that operators with symbols in S
1,δ
0(ℝ
n
), 0≤δ<1, and in some subclasses of S
1,10(ℝ
n
) are bounded on h
p
(ℝ
n
) (0<p≤1). As an application, we study the local solvability of the planar vector field L=∂
t
+ib(x,t)∂
x
, b(x,t)≥0, in spaces of mixed norm involving Hardy spaces.
Work supported in part by CNPq, FINEP, and FAPESP. 相似文献
9.
M. Langenbruch 《manuscripta mathematica》2000,103(2):241-263
Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ
n
. Let L(P
m
) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P
m
. Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ
n
×(ℝ\{ 0}) for any Q∈L(P
m
) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P
m
must be locally hyperbolic.
Received: 24 January 2000 相似文献
10.
We investigate the Caucy problem for linear elliptic operators withC
∞-coefficients at a regular domain ℝ ⊂ ℝ, which is a classical example of an ill-posed problem. The Cauchy data are given at
the manifold Γ⊂∂Ω and our goal is to obtain a stability estimate inH
4(Ω). 相似文献
11.
Deguang Han 《Journal of Fourier Analysis and Applications》2009,15(2):201-217
Let
be a full rank time-frequency lattice in ℝ
d
×ℝ
d
. In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L
2(ℝ
d
) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift
invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪
j=1
N
G(g
j
,Λ)) for L
2(ℝ
d
). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L
2(ℝ
d
). Related results for affine systems are also discussed.
Communicated by Chris Heil. 相似文献
12.
Francesco Altomare Sabina Milella Graziana Musceo 《Journal of Evolution Equations》2011,11(4):771-792
Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of continuous functions on
\mathbbRN, N 3 1{{\mathbb{R}}^{N},\; N\geq1} . It is proved that such differential operators, defined on their maximal domains, are pre-generators of positive quasicontractive
C
0-semigroups of operators that fulfill the Feller property. Accordingly, these semigroups are associated with a suitable probability
transition function and hence with a Markov process on
\mathbbRN{{\mathbb{R}}^{N}} . An approximation formula for these semigroups is also stated in terms of iterates of integral operators that generalize
the classical Gauss-Weierstrass operators. Some applications of such approximation formula are finally shown concerning both
the semigroups and the associated Markov processes. 相似文献
13.
B. J. C. Baxter 《Foundations of Computational Mathematics》2008,8(3):395-407
A radial basis function (RBF) has the general form
where the coefficients a
1,…,a
n
are real numbers, the points, or centres, b
1,…,b
n
lie in ℝ
d
, and φ:ℝ
d
→ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance
(Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline
spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported
when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy
certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities,
with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions.
Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore,
we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite
functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we
are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore,
the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric
integration.
Dedicated to Arieh Iserles on the occasion of his 60th birthday. 相似文献
14.
In this paper we consider positive semigroups on Lp(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups
as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L∞(Ω) is real and satisfies b ≥ δ for some δ > 0.
In memoriam Helmut H. Schaefer 相似文献
15.
Shu-Yu Hsu 《Mathematische Annalen》2003,325(4):665-693
We prove that the solution u of the equation u
t
=Δlog u, u>0, in (Ω\{x
0})×(0,T), Ω⊂ℝ2, has removable singularities at {x
0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C
1, C
2>0, such that C
1
|x−x
0|α≤u(x,t)≤C
2|x−x
0|−α holds for all 0<|x−x
0|≤ρ0 and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u
0∉L
1
(ℝ2) is radially symmetric and u
0L
loc
1(ℝ2).
Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003
Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65 相似文献
16.
Boundedness of Multilinear Operators in Herz-type Hardy Space 总被引:1,自引:0,他引:1
Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝn are bounded from HK
α1,p1
q1
(ℝn) ×···×HK
αk,pk
qk
(ℝn) into HK
α,p
q
(ℝn) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments
satisfied by the multilinear operators are also necessary when αj≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals
of any orders.
Received September 6, 1999, Revised November 17, 1999, Accepted December 9, 1999 相似文献
17.
The purpose of this paper is to study the L
2 boundedness of operators of the form f ↦ ψ(x) ∫ f (γ
t
(x))K(t)dt, where γ
t
(x) is a C
∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ
N
× ℝ
n
, satisfying γ
0(x) ≡ x, ψ is a C
∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ
n
, and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ
N
. The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L
2. The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case
when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later
two of which are joint with E. M. Stein. The second paper deals with the related question of L
p
boundedness, while the third paper deals with the special case when γ is real analytic. 相似文献
18.
Kernel regression estimation for continuous spatial processes 总被引:1,自引:0,他引:1
We investigate here a kernel estimate of the spatial regression function r(x) = E(Y
u | X
u = x), x ∈ ℝd, of a stationary multidimensional spatial process { Z
u = (X
u, Y
u), u ∈ ℝ
N
}. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the
bandwidth, when the process is observed over a rectangular domain of ℝN. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal
rate is preserved by using a suitable spatial sampling scheme.
相似文献
19.
Markus Poppenberg 《manuscripta mathematica》1991,72(1):257-274
In a previous paper, the quotient spaces of (s) in the tame category of nuclear Fréchet spaces have been characterized by property (ΩDZ) corresponding to the topological
condition (Ω) of D. Vogt and M. J. Wagner. In addition, a splitting theorem has been proved which provides the existence of
a tame linear right inverse of a tame linear map on the assumption that the kernel of the given map has property (ΩDZ) and
that certain tameness conditions hold. In this paper it is proved that property (Ω) in standard form (i.e., the dual norms
‖ ‖
n
*
are logarithmically convex) implies the tame splitting condition (ΩDZ) for any tamely nuclear Fréchet space equipped with
a grading defined by sermiscalar products. As an application, property (ΩDZ) is verified for the kernels of any hypoelliptic
system of linear partial differential operators with constant coefficients on ℝN or on a bounded convex region in ℝN. 相似文献
20.
Strongly elliptic differential operators with (possibly) unbounded lower order coefficients are shown to generate analytic
semigroups of linear operators onL
p(R
n
), 1≦p≦∞. An explicit characterization of the domain is given for 1<p<∞. An application to parabolic problems is also included.
This work has been partially supported by the Research Funds of the Ministero della Pubblica Istruzione.
The authors are members of GNAFA (Consiglio Nazionale delle Ricerche). 相似文献