共查询到20条相似文献,搜索用时 31 毫秒
1.
Violeta Petkova 《Archiv der Mathematik》2009,93(4):357-368
We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces
Lw2(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with
f ? Cc(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces
L2w(\mathbb Rk){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form. 相似文献
2.
Françoise Lust-Piquard 《Potential Analysis》2006,24(1):47-62
Let L=?Δ+|ξ|2 be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|2 be the harmonic oscillator on
\mathbbRn\mathbb{R}^{n}
, with the associated Riesz transforms R2j−1=(∂/∂ξj)L−1/2,R2j=ξjL−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant Cp,
||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}. 相似文献
3.
Erik Talvila 《Journal of Fourier Analysis and Applications》2012,18(1):27-44
Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional
derivative of a continuous function. This space of distributions is denoted
Ac(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}) and is a Banach space under the Alexiewicz norm,
||f||\mathbbT=sup|I| £ 2p|òI f|\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|, the supremum being taken over intervals of length not exceeding 2π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of
L
1 Fourier series continue to hold for this larger space, with the L
1 norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form [^(f)](n)=o(n)\hat{f}(n)=o(n) as |n|→∞. The convolution is defined for
f ? Ac(\mathbbT)f\in{\mathcal{A}}_{c}(\mathbb{T}) and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative.
There is the estimate
||f*g||¥ £ ||f||\mathbbT ||g||BV\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}. For
g ? L1(\mathbbT)g\in L^{1}(\mathbb{T}),
||f*g||\mathbbT £ ||f||\mathbb T ||g||1\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}. As well, [^(f*g)](n)=[^(f)](n) [^(g)](n)\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n). There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The
trigonometric polynomials are dense in
Ac(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}). The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D
n
be the Dirichlet kernel and let
f ? L1(\mathbbT)f\in L^{1}(\mathbb{T}). Then
||Dn*f-f||\mathbbT?0\|D_{n}\ast f-f\|_{\mathbb{T}}\to0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem. 相似文献
4.
We consider the weighted Bergman spaces
HL2(\mathbb Bd, ml){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dml(z) = cl(1-|z|2)l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators
on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which
the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized
Bergman spaces. 相似文献
5.
A Toeplitz operator TfT_\phi with symbol f\phi in
L¥(\mathbbD)L^{\infty}({\mathbb{D}}) on the Bergman space
A2(\mathbbD)A^{2}({\mathbb{D}}), where
\mathbbD\mathbb{D} denotes the open unit disc, is radial if f(z) = f(|z|)\phi(z) = \phi(|z|) a.e. on
\mathbbD\mathbb{D}. In this paper, we consider the numerical ranges of such operators. It is shown that all finite line segments, convex hulls
of analytic images of
\mathbbD\mathbb{D} and closed convex polygonal regions in the plane are the numerical ranges of radial Toeplitz operators. On the other hand,
Toeplitz operators TfT_\phi with f\phi harmonic on
\mathbbD\mathbb{D} and continuous on
[`(\mathbbD)]{\overline{\mathbb{D}}} and radial Toeplitz operators are convexoid, but certain compact quasinilpotent Toeplitz operators are not. 相似文献
6.
We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operators on a commutative algebra in such a way that derivations of the
commutative algebra are replaced by
\mathbbDer(A){\mathbb{D}{\rm er}(A)}, the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A){\mathcal{F}(A)}, a certain ‘Fock space’ associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)){\mathcal{D}(\mathcal{F}(A))} of differential operators is filtered and gr D(F(A)){\mathcal{D}(\mathcal{F}(A))}, the associated graded algebra, is commutative in some ‘wheeled’ sense. The resulting ‘wheeled’ Poisson structure on gr D(F(A)){\mathcal{D}(\mathcal{F}(A))} is closely related to the double Poisson structure on
TA \mathbbDer(A){T_{A} \mathbb{D}{\rm er}(A)} introduced by Van den Bergh. Specifically, we prove that gr
D(F(A)) @ F(TA(\mathbbDer(A)),{\mathcal{D}(\mathcal{F}(A))\cong\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)),} provided the algebra A is smooth. Our construction is based on replacing vector spaces by the new symmetric monoidal category of wheelspaces. The Fock space F(A){\mathcal{F}(A)} is a commutative algebra in this category (a “commutative wheelgebra”) which is a structure closely related to the notion of wheeled PROP. Similarly, we have Lie, Poisson, etc., wheelgebras.
In this language, D(F(A)){\mathcal{D}(\mathcal{F}(A))} becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper,
we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule
connection on
\mathbbDer(A){\mathbb{D}{\rm er}(A)} gives rise to a second-order (wheeled) differential operator, a noncommutative analogue of the Batalin-Vilkovisky (BV) operator,
that makes
F(TA(\mathbbDer(A))){\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)))} a BV wheelgebra. In the final section, we explain how the wheeled differential operators D(F(A)){\mathcal{D}(\mathcal{F}(A))} produce ordinary differential operators on the varieties of n-dimensional representations of A for all n ≥ 1. 相似文献
7.
Dennis Courtney Paul S. Muhly Samuel W. Schmidt 《Complex Analysis and Operator Theory》2012,6(1):163-188
If b is an inner function, then composition with b induces an endomorphism, β, of
L¥(\mathbbT){L^\infty({\mathbb{T}})} that leaves
H¥(\mathbbT){H^\infty({\mathbb{T}})} invariant. We investigate the structure of the endomorphisms of
B(L2(\mathbbT)){B(L^2({\mathbb{T}}))} and
B(H2(\mathbbT)){B(H^2({\mathbb{T}}))} that implement β through the representations of
L¥(\mathbbT){L^\infty({\mathbb{T}})} and
H¥(\mathbbT){H^\infty({\mathbb{T}})} in terms of multiplication operators on
L2(\mathbbT){L^2({\mathbb{T}})} and
H2(\mathbbT){H^2({\mathbb{T}})} . Our analysis, which is based on work of Rochberg and McDonald, will wind its way through the theory of composition operators
on spaces of analytic functions to recent work on Cuntz families of isometries and Hilbert C*-modules. 相似文献
8.
Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t ?1/ω ?1(t ?1) for ${t\in (0,\infty).}
9.
Harald Woracek 《Monatshefte für Mathematik》2012,33(3):105-149
A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ¥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L2(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D\mathfrakmDx{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as
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