共查询到20条相似文献,搜索用时 31 毫秒
1.
Michael Langenbruch 《Results in Mathematics》1999,36(3-4):281-296
Let F be a closed proper subset of ?n and let ?* be a class of ultradifferentiable functions. We give a new proof for the following result of Schmets and Valdivia on analytic modification of smooth functions: for every function ? ∈ ?* (?n) there is ${\widetilde f} \in {\cal E}_{*}(\rm R ^{n})$ which is real analytic on ?nF and such that ?a ? ¦ F = ?a ? ¦ F for any a ∈ ?0 n. For bounded ultradifferentiable functions ? we can obtain ${\widetilde f}$ by means of a continuous linear operator. 相似文献
2.
Nazih Faour 《Rendiconti del Circolo Matematico di Palermo》1989,38(1):121-129
Let ? be a non-constant function inL ∞(D) such thatφ=φ 1+φ 2, whereφ 1 is an element in the Bergman spaceL a 2 (D), and \(\phi _2 \in \overline {L_a^2 (D)} \) , the space of all complex conjugates of functions inL a 2 (D). In this paper, it is shown that if 1 is an element in the closure of the range of the self-commutator ofT ?, \(T_{\bar \phi } T_\phi - T_\phi T\phi \) , then the Toeplitz operatorT ? defined ofL a 2 (D) is not quasinormal. Moreover, if \(\phi = \psi + \lambda \bar \psi \) , whereψ∈ H ∞(D), and λεC, it is proved that ifT ? is quasinormal, thenT ? is normal. Also, the spectrum of a class of pure hyponormal Toeplitz operators is determined. 相似文献
3.
Björn Gustafsson Jacqueline Mossino Colette Picard 《Annali di Matematica Pura ed Applicata》1992,162(1):87-104
Let \(\Omega = \Omega _0 \backslash \bar \Omega _1\) be a regular annulus inR N and \(\phi :\bar \Omega \to R\) be a regular function such that φ=0 on ?Ω0, φ=1 on ?Ω1 and ▽φ ≠ 0. Let Kn be the subset of functions v ε W1,p (Ω) such that v=0 on ?Ω0, v=1 on ?Ω1, v=(unprescribed) constant on n given level surfaces of φ. We study the convergence of sequences of minimization problems of the type $$Inf\left\{ {\int\limits_\Omega {\frac{1}{{a_n \circ \phi }}G(x,(a_n \circ \phi )\nabla v)dx;v \in K_n } } \right\},$$ where an ε L∞ (0,1) and G: (x, ζ) ε Ω × RN → G(x, ζ εR is convex with respect to ξ and verifies some standard growth conditions. 相似文献
4.
L. A. Bordag 《Journal of Mathematical Sciences》1983,23(1):1875-1877
One considers the one-dimensional Dirac operator with a slowly oscillating potential (1) $$H = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right)\frac{d}{{dx}} + q\left( {\begin{array}{*{20}c} {\cos z(x)} & {\sin z(x)} \\ {\sin z(x)} & { - \cos z(x)} \\ \end{array} } \right)_, x \in ( - \infty ,\infty ),q - const,$$ where . The following statement holds. The double absolutely continuous spectrum of the operator (1) fills the intervals (?∞,?¦q¦), (¦q¦, ∞). The interval (?¦q¦, ¦q¦) is free from spectrum. The operator has a simple eigenvalue only for singn C+=sign C?, situated either at the point (under the condition C+>0) or at the point λ=?¦q¦ (under the condition). The proof is based on the investigation of the coordinate asytnptotics of the corresponding equation. 相似文献
5.
G. V. Rozenblyum 《Mathematical Notes》1977,21(3):222-227
The Schrödinger operator Hu = -Δu + V(x)u, where V(x) → 0 as ¦x¦ → ∞, is considered in L2(Rm) for m?3. The asymptotic formula $$N(\lambda ,V) \sim \Upsilon _m \int {(\lambda - V(x))_ + ^{{m \mathord{\left/ {\vphantom {m {2_{dx} }}} \right. \kern-\nulldelimiterspace} {2_{dx} }}} ,} \lambda \to ---0,$$ is established for the number N(λ, V) of the characteristic values of the operator H which are less than λ. It is assumed about the potential V that V = Vo + V1; Vo < 0, ¦Vo =o (¦Vo¦3/2) as ¦x¦ → ∞; σ (t/2, Vo) ?cσ (t. Vo) and V1∈Lm/2,loc, σ(t, V1) =o (σ (t, Vo)), where σ (t,f)= mes {x:¦f (x) ¦ > t). 相似文献
6.
This study concerns the class A K D of functions x analytic in a domain D of an open Riemann surface and satisfying there the inequality ¦x¦?1 with metric defined by the norm of the space C(K) of functions continuous on the compact subsetK ? D. The asymptotic formula $$\mathop {\lim }\limits_{n \to \infty } [d_n (A_K^D )]^{{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}} = e^{ - 1/\tau } $$ is established, where D is a finitely connected domain of Carathéodory type,K ? D is a regular compact subset such thatd?k is connected, and τ = τ (D, K) is the flux of harmonic measure of the set ?D relative to the domaind?k through any rectifiable contour separating ?D and K. 相似文献
7.
A. Khatamov 《Mathematical Notes》1977,21(3):198-207
The following inequalities are shown to hold for the least uniform rational deviations Rn(f) of a function f(x), continuous and convex in the interval [a, b]: $$R_n (f) \leqslant C(v)\Omega (f)n^{ - 1} \overbrace {\ln \ldots \ln }^{vtimes}n$$ (ν is an integer, C(ν) depends only on ν, and Ω(f) is the total oscillation of f); $$R_n (f) \leqslant C_1 n^{ - 1} \overbrace {\ln \ldots \ln }^{vtimes}n\mathop {\inf }\limits_{(b - a)\chi _n \leqslant \lambda< b - a} \left\{ {\omega (\lambda ,f) + M(f)n^{ - 1} \ln \frac{{b - a}}{\lambda }} \right\}$$ (ν is an integer, C1(ν) depends only on ν, xn = exp (-n/(500 In2n)), ω (δ,f) is the modulus of continuity of f, and M(f) = max¦f(x) ¦. 相似文献
8.
Let $\mathcal{X}$ be a metric space with doubling measure and L a nonnegative self-adjoint operator in $L^{2}(\mathcal{X})$ satisfying the Davies–Gaffney estimates. Let $\varphi:\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that φ(x,?) is an Orlicz function, $\varphi(\cdot,t)\in\mathbb{A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ)∈(0,1], and it satisfies the uniformly reverse Hölder inequality of order 2/[2?I(φ)]. In this paper, the authors introduce a Musielak–Orlicz–Hardy space $H_{\varphi,L}(\mathcal{X})$ , by the Lusin area function associated with the heat semigroup generated by L, and a Musielak–Orlicz BMO-type space $\mathrm{BMO}_{\varphi,L}(\mathcal{X})$ , which is further proved to be the dual space of $H_{\varphi,L}(\mathcal{X})$ and hence whose φ-Carleson measure characterization is deduced. Characterizations of $H_{\varphi,L}(\mathcal{X})$ , including the atom, the molecule, and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize $H_{\varphi,L}(\mathcal{X})$ in terms of the Littlewood–Paley $g^{\ast}_{\lambda}$ -function $g^{\ast}_{\lambda,L}$ and establish a Hörmander-type spectral multiplier theorem for L on $H_{\varphi,L}(\mathcal{X})$ . Moreover, for the Musielak–Orlicz–Hardy space H φ,L (? n ) associated with the Schrödinger operator L:=?Δ+V, where $0\le V\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ , the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom, and the molecule; finally, the authors show that the Riesz transform ?L ?1/2 is bounded from H φ,L (? n ) to the Musielak–Orlicz space L φ (? n ) when i(φ)∈(0,1], and from H φ,L (? n ) to the Musielak–Orlicz–Hardy space H φ (? n ) when $i(\varphi)\in(\frac{n}{n+1},1]$ , where i(φ) denotes the uniformly critical lower type index of φ. 相似文献
9.
A. Kroó 《Analysis Mathematica》1981,7(2):121-130
ПустьC 2π — пространств о 2π-периодических вещественных непрер ывных функций, W{rLip α={f∈C 2π r : ω(f (r), δ)≦δα}, Y?[?π,π] — некоторое дискр етное множество точе к на периоде, плотность ко торого задается соот ношением ?(Y)= max min ¦x-у¦. Дляf∈C2π x∈[?π,π] y∈Y обозначим через pk(f) pk(f)y т ригонометрические полиномы степени не в ышеk наилучшего чебышевского прибли жения функцииf на все м периоде и на дискретном множес тве Y соответственно. Тогда величина $$\Omega _{k,r + \alpha } (d) = \mathop {\sup }\limits_{f \in W_r Lip\alpha } \mathop {\sup }\limits_{\mathop {Y \subset [ - \pi ,\pi ]}\limits_{\rho (Y) \leqq d} } \left\| {p_k (f) - p_k (f)_Y } \right\| (d > 0)$$ xарактеризует отклон ение наилучших равно мерных и дискретных чебышевс ких приближений равномерно на классе функций WrLip а. В работе да ются точные оценки для ?k,r+α(d) пр и всехk, r и 0-?1. 相似文献
10.
A. A. Kon'kov 《Mathematical Notes》1996,60(1):22-28
Let Ω be an arbitrary, possibly unbounded, open subset of ? n , and letL be an elliptic operator of the form $$L = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial x_i }}\left( {a_{ij} (x)\frac{\partial }{{\partial x_j }}} \right)} $$ . The behavior at infinity of the solutions of the equationLu=?(¦u¦)signu in Ω is studied, where? is a measurable function. In particular, given certain conditions at infinity, the uniqueness theorem for the solution of the first boundary value problem is proved. 相似文献
11.
For b?∈?BMO(? n ) and 0?<?α?≤?1/2, the commutator of the fractional integral operator T Ω,α with rough variable kernel is defined by $$ [b, T_{\Omega, \alpha}]f(x)= \int_{\mathbb{R}^n} \frac{\Omega(x,x-y)}{|x-y|^{n-\alpha}}(b(x)-b(y))f(y)dy. $$ In this paper the authors prove that the commutator [b, T Ω,α ] is a bounded operator from $L^{\frac{2n}{n+2\alpha}}(\mathbb{R}^n)$ to L 2(? n ). The result obtained in this paper is substantial improvement and extension of some known results. 相似文献
12.
LetQ(x,y,z) be an indefinite ternary quadratic form of type (2,1) and determinantD(<0). Let 0≤t≤1/3 and \(f(t) = \frac{4}{{(1 + t)^2 (1 + 5t)}}\) . Then given any real numbersx 0,y 0,z 0 there exist integersx,y,z satisfying $$ - t(f(t)|D|)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}< Q (x + x_0 ,y + y_0 ,z + z_0 ) \leqslant (f(t)|D|)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} $$ All the cases when equality holds are also obtained. 相似文献
13.
LetL be the space of rapidly decreasing smooth functions on ? andL * its dual space. Let (L 2)+ and (L 2)? be the spaces of test Brownian functionals and generalized Brownian functionals, respectively, on the white noise spaceL * with standard Gaussian measure. The Donsker delta functionδ(B(t)?x) is in (L 2)? and admits the series representation $$\delta (B(t) - x) = (2\pi t)^{ - 1/2} \exp ( - x^2 /2t)\sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} H_n (x/\sqrt {2t} )} \times H_n (B(t)/\sqrt {2t} )$$ , whereH n is the Hermite polynomial of degreen. It is shown that forφ in (L 2)+,g t,φ(x)≡〈δ(B(t)?x), φ〉 is inL and the linear map takingφ intog t,φ is continuous from (L 2)+ intoL. This implies that forf inL * is a generalized Brownian functional and admits the series representation $$f(B(t)) = (2\pi t)^{ - 1/2} \sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} \langle f,\xi _{n, t} \rangle } H_n (B(t)/\sqrt {2t} )$$ , whereξ n,t is the Hermite function of degreen with parametert. This series representation is used to prove the Ito lemma forf inL *, $$f(B(t)) = f(B(u)) + \int_u^t {\partial _s^ * } f'(B(s)) ds + (1/2)\int_u^t {f''} (B(s)) ds$$ , where? s * is the adjoint of \(\dot B(s)\) -differentiation operator? s . 相似文献
14.
In this paper we give Lp-bound edness for the operator Tu defined bywhere P(x,y) is a real nontrivial polynomial on Rn×Rn,Ωis homogeneous of degree zero,Ω∈Lq(Sn-1),q>1/(1-μ) and b(r)∈BV(R+),The result can be regarded as an improvement of F.Ricci and E.M.Stein's result for fractional oscillatory integral operator with smoothness kernel. 相似文献
15.
Letf(x) ∈L p[0,1], 1?p? ∞. We shall say that functionf(x)∈Δk (integerk?1) if for anyh ∈ [0, 1/k] andx ∈ [0,1?kh], we have Δ h k f(x)?0. Denote by ∏ n the space of algebraic polynomials of degree not exceedingn and define $$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$ We prove that for any positive integerk, iff(x) ∈ Δ k ∩ L p[0, 1], 1?p?∞, then we have $$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$ whereC is a constant only depending onk. 相似文献
16.
V. S. Panferov 《Mathematical Notes》1973,14(5):936-942
Let ∥·∥ be a norm in R2 and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R2 rational if (x1 ? y1)/(x2 ? y2) or (x2 ? y2)/(x1 ? y1) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T2 of positive planar measure implies ∥T v ∥L4 (T2) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T v } and a set E ? T2, ¦E¦ > 0, such that T v (x) → 0(v → ∞) on E; however, ¦cn¦ ? 0 for ∥n∥ → ∞. 相似文献
17.
Zhu Xuexian 《分析论及其应用》2001,17(4):65-76
Let Mg be the maximal operator defined by $$M_g f\left( x \right) = \sup \frac{{\int_a^b {f\left( y \right)g\left( y \right){\text{d}}y} }}{{\int_a^b {g\left( y \right){\text{d}}y} }}$$ , where g is a positive locally integrable function on R and the supremum is taken over all intervals [a,b] such that 0≤a≤x≤b/η(b?a), here η is a non-increasing function such that η (0) = 1 and $\mathop {{\text{lim}}}\limits_{t \to {\text{ + }}\infty } \eta \left( t \right) = 0$ η (t) = 0. This maximal function was introduced by H. Aimar and L. L. Forzani [AF]. Let Φ be an N - function such that Φ and its complementary N - function satisfy Δ2. It gives an A′Φ(g) type characterization for the pairs of weights (u,v) such that the weak type inequality $$u\left( {\left\{ {x \in {\text{R}}\left| {M_g f\left( x \right) >\lambda } \right.} \right\}} \right) \leqslant \frac{C}{{\Phi \left( \lambda \right)}}\int_{\text{R}} {\Phi \left( {\left| f \right|v} \right)} $$ holds for every f in the Orlicz space LΦ(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′Φ(g). 相似文献
18.
We deal with several classes of integral transformations of the form $$f(x) \to D\int_{\mathbb{R}_ + ^2 } {\frac{1} {u}} \left( {e^{ - u\cosh (x + v)} + e^{ - u\cosh (x - v)} } \right)h(u)f(v)dudv,$$ , where D is an operator. In case D is the identity operator, we obtain several operator properties on L p (?+) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L 2(?+) and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type. 相似文献
19.
Yu Liu 《Monatshefte für Mathematik》2012,165(1):41-56
Let ${\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)}$ with the non-negative potential V belonging to reverse H?lder class with respect to the measure ??(x)dx, where ??(x) satisfies the A 2 condition of Muckenhoupt and a i,j (x) is a real symmetric matrix satisfying ${\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. }$ We obtain some estimates for ${V^{\alpha}\mathcal{L}^{-\alpha}}$ on the weighted L p spaces and we study the weighted L p boundedness of the commutator ${[b, V^{\alpha} \mathcal{L}^{-\alpha}]}$ when ${b\in BMO_\omega}$ and 0?<??? ?? 1. 相似文献
20.
For Pm ∈ ?[z1, …, zn], homogeneous of degree m we investigate when the graph of Pm in ?n+1 satisfies the Phragmén-Lindelöf condition PL(?n+1, log), or equivalently, when the operator $i{\partial \over \partial_{x_{n+1}}}+P_{m}(D)$ admits a continuous solution operator on C∞(?n+1). This is shown to happen if the varieties V+- ? {z ∈ ?n: Pm(z) = ±1} satisfy the following Phragmén-Lindelöf condition (SPL): There exists A ≥ 1 such that each plurisubharmonic function u on V+- satisfying u(z) ≤ ¦z¦+ o(¦z¦) on V+- and u(x) ≤ 0 on V+- ∩ ?n also satisfies u(z) A¦ Im z¦ on V+-. Necessary as well as sufficient conditions for V+- to satisfy (SPL) are derived and several examples are given. 相似文献