共查询到20条相似文献,搜索用时 578 毫秒
1.
Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet formIntegrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).
相似文献
$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$
2.
This paper investigates the existence and uniqueness of positive and nondecreasing solution for nonlinear boundary value problem with fractional q-derivative where \(D_{q}^{\alpha }\) denotes the Riemann–Liouville q-derivative of order \(\alpha \), \(0<\eta <1\) and \(1-\beta \eta ^{\alpha -3}>0\). Our analysis relies a fixed point theorem in partially ordered sets. An example to illustrate our results is given.
相似文献
$$\begin{aligned}&D_{q}^{\alpha }u(t)+f(t,u(t))=0, \quad {0<t<1, ~3<\alpha \le 4,} \\&u(0)= D_{q}u(0)=D_{q}^{2}u(0)=0, \quad D_{q}^{2}u(1)=\beta D_{q}^{2}u(\eta ), \end{aligned}$$
3.
Let \(\vec {P}=(p_{1},\dotsc ,p_{m})\) with 1 < p 1, …, p m < ∞, 1/p 1+?+1/p m =1/p and \(\vec {w}=(w_{1},\dotsc ,w_{m})\in A_{\vec {P}}\). In this paper, we investigate the weighted bounds with dependence on aperture α for multilinear square functions \(S_{\alpha ,\psi }(\vec {f})\). We show that This result extends the result in the linear case which was obtained by Lerner in 2014. Our proof is based on the local mean oscillation technique presented firstly to find the weighted bounds for Calderón–Zygmund operators. This method helps us avoiding intrinsic square functions in the proof of our main result. 相似文献
$$\|S_{\alpha,\psi}(\vec{f})\|_{L^{p}(\nu_{\vec{w}})} \leq C_{n,m,\psi,\vec{P}} \alpha^{mn}[\vec{w}]_{A_{\vec{P}}}^{\max(\frac{1}{2},\tfrac{p_{1}^{\prime}}{p},\dotsc,\tfrac{p_{m}^{\prime}}{p})} \prod\limits_{i=1}^{m} \|f_{i}\|_{L^{p_{i}}(w_{i})}. $$
4.
F. E. Lomovtsev 《Differential Equations》2008,44(6):866-871
We prove the well-posed solvability in the strong sense of the boundary value Problems where the unbounded operators A s (t), s > 0, in a Hilbert space H have domains D(A s (t)) depending on t, are subordinate to the powers A 1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A 0(t) and the symmetric operators A s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that , where the smoothing operators are defined by .
相似文献
$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$
$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$
$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$
5.
Huixue Lao 《Acta Appl Math》2010,110(3):1127-1136
Let L(sym j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that and
相似文献
$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$
$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$
6.
Let M Ω be the maximal operator with homogeneous kernel Ω. In the present paper, we show that if Ω satisfies the L 1-Dini condition on ?? n?1, then the following weak type (1,1) behaviors
hold for the maximal operator M Ω and \(f\in L^{1}(\mathbb {R}^{n})\), here \(\tilde {\omega }_{1}\) denotes the L 1 integral modulus of continuity of Ω defined by translation in \(\mathbb {R}^{n}\). 相似文献
$$\lim\limits _{\lambda \rightarrow 0_{+}}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})=\frac {1}{n} \|\Omega \|_{1} \|f\|_{1},$$
$$\sup\limits_{\lambda >0}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})\lesssim {\bigg ((\log n)\|\Omega \|_{1}+{\int }_{0}^{1/n}\frac {\tilde {\omega }_{1}(\delta )}{\delta }d\delta \bigg )}\|f\|_{1}$$
7.
V. G. Zhuravlev 《Journal of Mathematical Sciences》2012,182(4):472-483
For the number n s (α, β; X) of points (x 1 , x 2) in the two-dimensional Fibonacci quasilattices \( \mathcal{F}_m^2 \) of level m?=?0, 1, 2,… lying on the hyperbola x 1 2 ? ??αx 2 2 ?=?β and such that 0?≤?x 1? ≤?X, x 2? ≥?0, the asymptotic formulais established, and the coefficient c s (α, β) is calculated exactly. Using this, we obtain the following result. Let F m be the Fibonacci numbers, A i ∈ \( \mathbb{N} \), i?=?1, 2, and let \( \overleftarrow {{A_i}} \) be the shift of A i in the Fibonacci numeral system. Then the number n s (X) of all solutions (A 1 , A 2) of the Diophantine system0?≤?A 1? ≤?X, A 2? ≥?0, satisfies the asymptotic formulaHere τ?=?(?1?+?√5)/2 is the golden ratio, and c s ?=?1/2 or 1 for s?=?0 or s?≥?1, respectively.
相似文献
$ {n_s}\left( {\alpha, \beta; X} \right)\sim {c_s}\left( {\alpha, \beta } \right)\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty $
$ \left\{ {\begin{array}{*{20}{c}} {A_1^2 + \overleftarrow {A_1^2} - 2{A_2}{{\overleftarrow A }_2} + \overleftarrow {A_2^2} = {F_{2s}},} \\ {\overleftarrow {A_1^2} - 2{A_1}{{\overleftarrow A }_1} + A_2^2 - 2{A_2}{{\overleftarrow A }_2} + 2\overleftarrow {A_2^2} = {F_{2s - 1}},} \\ \end{array} } \right. $
$ {n_s}(X)\sim \frac{{{c_s}}}{{{\text{ar}}\cosh \left( {{{1} \left/ {\tau } \right.}} \right)}}\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty . $
8.
We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms:
Hence the Lie algebras \(\chi (\sinh {u})\) and χ(eu + e??2u) are slowly linearly growing Lie algebras with average growth rates \(\frac {3}{2}\) and \(\frac {4}{3}\) respectively. 相似文献
- 1)the first one is between the characteristic Lie algebra \(\chi (\sinh {u})\) of the sinh-Gordon equation \(u_{xy}=\sinh {u}\) and the non-negative part \({\mathcal {L}}({\mathfrak {sl}}(2,{\mathbb {C}}))^{\ge 0}\) of the loop algebra of \({\mathfrak {sl}}(2,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{1}^{(1)}\)$$\chi(\sinh{u})\cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(2,{\mathbb{C}}))^{\ge 0}={\mathfrak{s}\mathfrak{l}}(2, {\mathbb{C}}) \otimes {\mathbb{C}}[t]. $$
- 2)the second isomorphism is for the Tzitzeica equation uxy = eu + e??2uwhere \({\mathcal {L}}({\mathfrak {sl}}(3,{\mathbb {C}}), \mu )=\bigoplus _{j \in {\mathbb {Z}}}{\mathfrak {g}}_{j (\text {mod} \; 2)} \otimes t^{j}\) is the twisted loop algebra of the simple Lie algebra \({\mathfrak {sl}}(3,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{2}^{(2)}\).$$\chi(e^{u}{+}e^{-2u}) \cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(3,{\mathbb{C}}), \mu)^{\ge0}=\bigoplus_{j = 0}^{+\infty}{\mathfrak{g}}_{j (\text{mod} \; 2)} \otimes t^{j}, $$
9.
Jay Taylor 《Israel Journal of Mathematics》2017,217(1):435-475
In this paper we establish the following estimate: where ω ≥ 0, 0 < ε < 1 and Φ(t) = t(1 + log+(t)). This inequality relies upon the following sharp L p estimate: where 1 < p < ∞, ω ≥ 0 and 0 < δ < 1. As a consequence we recover the following estimate essentially contained in [18]: We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.
相似文献
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant \frac{{{c_T}}}{{{\varepsilon ^2}}}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right){M_{L{{\left( {\log L} \right)}^{1 + \varepsilon }}}}} \omega \left( x \right)dx$$
$${\left\| {\left[ {b,T} \right]f} \right\|_{{L^p}\left( \omega \right)}} \leqslant {c_T}{\left( {p'} \right)^2}{p^2}{\left( {\frac{{p - 1}}{\delta }} \right)^{\frac{1}{{p'}}}}{\left\| b \right\|_{BMO}}{\left\| f \right\|_{{L^p}\left( {{M_{L{{\left( {{{\log }_L}} \right)}^{2p - 1 + {\delta ^\omega }}}}}} \right)}}$$
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant {c_T}{\left[ \omega \right]_{{A_\infty }}}{\left( {1 + {{\log }^ + }{{\left[ \omega \right]}_{{A_\infty }}}} \right)^2}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)M} \omega \left( x \right)dx.$$
10.
Wolfhard Hansen 《Potential Analysis》2017,46(1):1-21
We consider the stochastic differential equation (SDE) of the form where \(\sigma :{\mathbb {R}}^ d\to {\mathbb {R}}^ d\) is globally Lipschitz continuous and L={L(t):t≥0} is a Lévy process. Under this condition on σ it is well known that the above problem has a unique solution X. Let \((\mathcal {P}_{t})_{t\ge 0}\) be the Markovian semigroup associated to X defined by \(\left ({\mathcal {P}}_{t} f\right ) (x) := \mathbb {E} \left [ f(X^ x(t))\right ]\), t≥0, \(x\in {\mathbb {R}}^{d}\), \(f\in \mathcal {B}_{b}({\mathbb {R}}^{d})\). Let B be a pseudo–differential operator characterized by its symbol q. Fix \(\rho \in \mathbb {R}\). In this article we investigate under which conditions on σ, L and q there exist two constants γ>0 and C>0 such that
相似文献
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{rcl} dX^ x(t) &=& \sigma(X(t-)) dL(t) \\ X^ x(0)&=&x,\quad x\in{\mathbb{R}}^ d, \end{array}\right. \end{array} $$
$$\left| B {\mathcal{P}}_{t} u \right|_{H^{\rho}_{2}} \le C \, t^{-\gamma} \,\left| u \right|_{H^{\rho}_{2}}, \quad \forall u \in {H^{\rho}_{2}}(\mathbb{R}^{d} ),\, t>0. $$
11.
We investigate generalised Piterbarg constants determined in terms of a fractional Brownian motion B α with Hurst index α/2∈(0,1], the non-negative constant δ and a continuous function h. We show that these constants, similarly to generalised Pickands constants, appear naturally in the tail asymptotic behaviour of supremum of Gaussian processes. Further, we derive several bounds for \(\mathcal {P}_{\alpha , \delta }^{h}\) and in special cases explicit formulas are obtained.
相似文献
$$\mathcal{P}_{\alpha, \delta}^{h}=\lim\limits_{T \rightarrow \infty} \mathbb{E}\left\{ \sup\limits_{t\in \delta \mathbb{Z} \cap [0,T]} e^{\sqrt{2}B_{\alpha}(t)-|t|^{\alpha}- h(t)}\right\} $$
12.
Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated towhere \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)
相似文献
$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$
$$\mathcal{L}f + qf = 0 $$
13.
Let \(\mathcal{M} =\{m_{j}\}_{j=1}^{\infty}\) be a family of Marcinkiewicz multipliers of sufficient uniform smoothness in \(\mathbb{R}^{n}\). We show that the L p norm, 1<p<∞, of the related maximal operator is at most C(log(N+2)) n/2. We show that this bound is sharp.
相似文献
$$M_Nf(x)= \sup_{1\leq j \leq N} |\mathcal{F}^{-1} ( m_j \mathcal{F} f)|(x) $$
14.
We consider the existence of single and multi-peak solutions of the following nonlinear elliptic Neumann problem where \(\lambda \) is a large number, \(p\in (2,\frac{2N}{N-2})\) for \(N\ge 3\), f(x, u) is subcritical about u and Q is positive and has some non-degenerate critical points in \(\mathbb {R}^{N}_{+}\). For \(\lambda \) large, we can get solutions which have peaks near the non-degenerate critical points of Q.
相似文献
$$\begin{aligned} \left\{ \begin{aligned} -\Delta u+\lambda ^{2} u&=Q(x)|u|^{p-2}u \qquad&\text {in} ~~~~\mathbb {R}^{N}_{+}, \\ \frac{\partial u }{\partial n}&=f(x,u) \qquad&\text {on}~~\partial \mathbb {R}^{N}_{+}, \end{aligned}\right. \end{aligned}$$
15.
In the space L 2(?2), we consider the operator . We study the spectrum of H and, for V ∈ C 0 2 (?2), prove the trace formula where c 0 = π ?1 \(\smallint _{\mathbb{R}^2 } \) V(x) dx and the µ k (i) are the eigenvalues of H.
相似文献
$H = \left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }} - x_2 } \right)^2 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }} + x_1 } \right)^2 + V,V = V(x) \in L_2 (\mathbb{R}^2 ).$
$\sum\limits_{k = 0}^\infty {\left( {\sum\limits_{i = - k}^\infty {(4k + 2 - \mu _k^{(i)} ) + c_0 } } \right)} = \frac{1}{{8\pi }}\int\limits_{\mathbb{R}^2 } {V^2 (x)dx,} $
16.
MOUSOMI BHAKTA 《Proceedings Mathematical Sciences》2017,127(2):337-347
We study the existence and multiplicity of sign-changing solutions of the following equation where Ω is a bounded domain in \(\mathbb {R}^{N}\), 0∈?Ω, all the principal curvatures of ?Ω at 0 are negative and μ≥0, a>0, N≥7, 0<t<2, \(2^{\star }=\frac {2N}{N-2}\) and \(2^{\star }(t)=\frac {2(N-t)}{N-2}\).
相似文献
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllllll} -{\Delta} u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^{t}}+a(x)u \quad\text{in}\, {\Omega}, \\ u=0 \quad\text{on}\quad\partial{\Omega}, \end{array}\right. \end{array} $$
17.
This paper is concerned with the following Kirchhoff-type equations: where \(a>0,~b,~\mu \ge 0\) are constants, \(\alpha \in (0,3)\), \(p\in [2,3+2\alpha )\), the potential V(x) may be unbounded from below and \(\phi |u|^{p-2}u\) is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrödinger equations, the Kirchhoff equations and the Schrödinger–Poisson system.
相似文献
$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$
18.
In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov space \(\dot B_{\infty ,\infty }^{ - r}\left( {{\mathbb{R}^2}} \right)\). The result shows that if θ is a weak solutions satisfies then θ is regular at t = T. In view of the embedding \({L^{\frac{2}{r}}} \subset M_{\frac{2}{r}}^p \subset \dot B_{\infty ,\infty }^{ - r}\) with \(2 \leqslant p < \frac{2}{r}\) and 0 ≤ r < 1, we see that our result extends the results due to [20] and [31].
相似文献
$$\int_0^T {\frac{{\left\| {\nabla \theta ( \cdot ,s)} \right\|_{\dot B_{\infty ,\infty }^{ - r} }^{\tfrac{\alpha }{{\alpha - r}}} }}{{1 + \ln \left( {e + \left\| {\nabla ^ \bot \theta ( \cdot ,s)} \right\|_{L^{\tfrac{2}{r}} } } \right)!}}ds < \infty for some 0 < r < \alpha and 0 < \alpha < 1,}$$
19.
Given a continuous strictly monotone function \(\varphi \) defined on an open real interval I and a probability measure \(\mu \) on the Borel subsets of [0, 1], the Makó–Páles mean is defined by Under some conditions on the functions \(\varphi \) and \(\psi \) defined on I, the quotient mean is given by In this paper, we study some invariance of the quotient mean with respect to Makó–Páles means.
相似文献
$$\begin{aligned} {\mathcal {M}}_{\varphi ,\mu }(x,y):=\varphi ^{-1}\left( \int ^1_0\varphi (tx+(1-t)y)\, d\mu (t)\right) ,\quad x,y\in I. \end{aligned}$$
$$\begin{aligned} Q_{\varphi ,\psi }(x,y):=\left( \frac{\varphi }{\psi }\right) ^{-1}\left( \frac{\varphi (x)}{\psi (y)}\right) , \quad x,y\in I. \end{aligned}$$