共查询到20条相似文献,搜索用时 31 毫秒
1.
Let ${s,\,\tau\in\mathbb{R}}Let
s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,¥]{q\in(0,\infty]} . We introduce Besov-type spaces
[(B)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ¥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces
[(F)\dot]s, tp, q(\mathbbRn) for p ? (0, ¥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of
[(B)\dot]s, tp, q(\mathbbRn) and [(F)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and
molecular decomposition characterizations of
[(B)\dot]s, tp, q(\mathbbRn) and [(F)\dot]s, tp, q(\mathbbRn){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For
s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ¥), q ? [1, ¥){p\in(1,\,\infty), q\in[1,\,\infty)} and
t ? [0, \frac1(max{p, q})¢]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces
B[(H)\dot]s, tp, q(\mathbbRn){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of
B[(H)\dot]s, tp, q(\mathbbRn){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just
[(B)\dot]-s, tp¢, q¢(\mathbbRn){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,¥){t\in (1,\infty)} . 相似文献
2.
In this paper we study density properties of the Hankel translations of certain positive definite (in the Hankel sense) functions on (0,¥)(0,\infty ). The density is understood in a Fréchet space of smooth functions on (0,¥)(0,\infty ) whose topology is defined by seminorms involving the Bessel operator x-2m-1Dx2m+1Dx^{-2\mu -1}Dx^{2\mu +1}D. 相似文献
3.
Emmanuel Preissmann 《Monatshefte für Mathematik》2007,45(1):233-239
Let X
0 be the germ at 0 of a complex variety and let
f: X0? \Bbb Cn0f:\ X_0\rightarrow {\Bbb C}^n_0
be a holomorphic germ. We say that f is pseudoimmersive if for any
g: \Bbb R0? X0g:\ {\Bbb R}_0\rightarrow X_0
such that
f °g ? C¥ f \circ g \in C^{\infty}
, we have
g ? C¥g\in C^{\infty}
. We prove that f is pseudoimmersive if and only if it is injective. Some results about the real case are also considered. 相似文献
4.
Let H be the symmetric second-order differential operator on L 2(R) with domain ${C_c^\infty({\bf R})}Let H be the symmetric second-order differential operator on L
2(R) with domain Cc¥(R){C_c^\infty({\bf R})} and action Hj = -(c j¢)¢{H\varphi=-(c\,\varphi^{\prime})^{\prime}} where c ? W1,2loc(R){ c\in W^{1,2}_{\rm loc}({\bf R})} is a real function that is strictly positive on R\{0}{{\bf R}\backslash\{0\}} but with c(0) = 0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of H. In particular if n = n+ún-{\nu=\nu_+\vee\nu_-} where n±(x)=±ò±1±x c-1{\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}} then H has a unique self-adjoint extension if and only if n ? L2(0,1){\nu\not\in L_2(0,1)} and a unique submarkovian extension if and only if n ? L¥(0,1){\nu\not\in L_\infty(0,1)}. In both cases, the corresponding semigroup leaves L
2(0,∞) and L
2(−∞,0) invariant. In addition, we prove that for a general non-negative c ? W1,¥loc(R){ c\in W^{1,\infty}_{\rm loc}({\bf R})} the corresponding operator H has a unique submarkovian extension. 相似文献
5.
Vladimir A. Borovikov Francisco Javier Mendoza 《Journal of Fourier Analysis and Applications》2002,8(4):399-406
We study the pointwise convergence problem for the inverse Fourier transform of piecewise smooth functions, i.e., whether SrD f (\bx) ? f (\bx)S_{\rho D} f (\bx) \to f (\bx) as r? ¥\rho \to \infty . r? ¥\rho \to \infty . Here for \bx,\bxi ? \Rn\bx,\bxi \in \Rn SrDf(\bmx)=\dsf1(2p)n/2\intlirD [^(f)](\bxi) e\dst iá\bmx,\bxi? d\bxi . S_{\rho D}f(\bm{x})=\dsf1{(2\pi)^{n/2}}\intli_{\rho D} \widehat{f}(\bxi) e^{\dst i\langle\bm{x},\bxi\rangle} d\bxi~. is the partial sum operator using a convex and open set DD containing the origin, and rD={ r\bxi:\bxi ? D }\rho D=\left\{ \rho \bxi:\bxi\in D \right\}. 相似文献
6.
In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if
div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to
L\frac21-r( 0,T;[(X)\dot]r( \mathbbR3) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique. 相似文献
7.
T.-A. Tanaka 《Archiv der Mathematik》2002,78(3):202-209
Transcendence of the number ?k=0¥ ark \sum_{k=0}^\infty \alpha^{r_k} , where a \alpha is an algebraic number with 0 < | a | \mid\alpha\mid > 1 and {rk}k\geqq0 \{r_k\}_{k\geqq0} is a sequence of positive integers such that limk?¥ rk+1/rk = d ? \mathbbN \{1} \lim_{k\to\infty}\, r_{k+1}/r_k = d \in \mathbb{N}\, \backslash \{1\} , is proved by Mahler's method. This result implies the transcendence of the number ?k=0¥ akdk \sum_{k=0}^\infty \alpha^{kd^k} . 相似文献
8.
Silviu Radu 《The Ramanujan Journal》2009,20(2):215-251
In this paper we present an algorithm that takes as input a generating function of the form $\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n}In this paper we present an algorithm that takes as input a generating function of the form ?d|M?n=1¥(1-qdn)rd=?n=0¥a(n)qn\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n} and three positive integers m,t,p, and which returns true if a(mn+t) o 0 mod p,n 3 0a(mn+t)\equiv0\pmod{p},n\geq0, or false otherwise. Our method builds on work by Rademacher (Trans. Am. Math. Soc. 51(3):609–636, 1942), Kolberg (Math. Scand. 5:77–92, 1957), Sturm (Lecture Notes in Mathematics, pp. 275–280, Springer, Berlin/Heidelberg, 1987), Eichhorn and Ono (Proceedings for a Conference in Honor of Heini Halberstam, pp. 309–321, 1996). 相似文献
9.
D. Walsh 《Archiv der Mathematik》1999,73(6):442-458
Suppose that $1 < p < \infty $1 < p < \infty , q=p/(p-1)q=p/(p-1), and for non-negative f ? Lp(-¥ ,¥)f\in L^p(-\infty\! ,\infty ) and any real x we let F(x)-F(0)=ò0xf(t) dtF(x)-F(0)=\int _0^xf(t)\ dt; suppose in addition that ò-¥¥ F(t)exp(-|t|) dt=0\int\limits _{-\infty }^\infty F(t)\exp (-|t|)\ dt=0. Moser's second one-dimensional inequality states that there is a constant CpC_p, such that ò-¥¥ exp[a |F(x)|q-|x|] dx £ Cp\int\limits _{-\infty }^\infty \exp [a |F(x)|^q-|x|] \ dx\le C_p for each f with ||f||p £ 1||f||_p\le 1 and every a £ 1a\le 1. Moreover the value a = 1 is sharp. We replace the operation connecting f with F by a more general integral operation; specifically we consider non-negative kernels K(t,x) with the property that xK(t,x) is homogeneous of degree 0 in t, x. We state an analogue of the inequality above for this situation, discuss some applications and consider the sharpness of the constant which replaces a. 相似文献
10.
Damir Z. Arov Mikael Kurula Olof J. Staffans 《Complex Analysis and Operator Theory》2011,5(2):331-402
This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary
passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift
invariant subspace of the Kreĭn space L2([0,¥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L2([0,¥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C1([0,¥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,¥);W){w \in C([0,\infty);\mathcal W)}, and
([(x)\dot](t),x(t),w(t)) ? V, t ? [0,¥), (\dot x(t),x(t),w(t))\in V,\quad t \in [0,\infty), 相似文献
11.
We consider Dirichlet series zg,a(s)=?n=1¥ g(na) e-ln s{\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ
n
= n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ?n=1¥ g(na) zn{\sum_{n=1}^{\infty} g(n\alpha) z^n}. We prove that a Dirichlet series zg,a(s) = ?n=1¥ g(n a)/ns{\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s} has an abscissa of convergence σ
0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ
0 satisfies σ
0 ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ
g,α
(s) has an analytic continuation to the entire complex plane. 相似文献
12.
Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system
Jihong Zhao Qiao Liu Shangbin Cui 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):1-18
In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye–Hückel system. We prove
that if the initial data belong to the critical Lebesgue space
L\fracn2(\mathbbRn){L^{\frac{n}{2}}(\mathbb{R}^{n})} , then the L
q
-norm (
\fracn2 £ q £ ¥{\frac{n}{2} \leq q \leq \infty}) of the βth order spatial derivative of mild solutions are majorized by
K1(K2|b|)|b|t-\frac|b|2-1+\fracn2q{K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2}-1+\frac{n}{2q}}} for some constants K
1 and K
2. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in
the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild
solutions whose initial data belong to the critical homogeneous Besov space
[(B)\dot]-2+\fracnpp,¥(\mathbbRn){\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)} (
\fracn2 < p < n{\frac{n}{2} < p < n}). 相似文献
13.
T. Nakazi 《Archiv der Mathematik》1999,73(6):439-441
Let a\alpha and b\beta be bounded measurable functions on the unit circle T. The singular integral operator Sa, bS_{\alpha ,\,\beta } is defined by Sa, b f = aPf + bQf(f ? L2 (T))S_{\alpha ,\,\beta } f = \alpha Pf + \beta Qf(f \in L^2 (T)) where P is an analytic projection and Q is a co-analytic projection. In the previous paper, the norm of Sa, bS_{\alpha ,\,\beta } was calculated in general, using a,b\alpha ,\beta and a[`(b)] + H¥\alpha \bar {\beta } + H^\infty where H¥H^\infty is a Hardy space in L¥ (T).L^\infty (T). In this paper, the essential norm ||Sa, b ||e\Vert S_{\alpha ,\,\beta } \Vert _e of Sa, bS_{\alpha ,\,\beta } is calculated in general, using a[`(b)] + H¥ + C\alpha \bar {\beta } + H^\infty + C where C is a set of all continuous functions on T. Hence if a[`(b)]\alpha \bar {\beta } is in H¥ + CH^\infty + C then ||Sa, b ||e = max(||a||¥ , ||b||¥ ).\Vert S_{\alpha ,\,\beta } \Vert _e = \max (\Vert \alpha \Vert _\infty , \Vert \beta \Vert _\infty ). This gives a known result when a, b\alpha , \beta are in C. 相似文献
14.
Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of
hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured
that a(n) = 0 if and only if b(n) = 0, where integers a(n) and b(n) are defined by
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