New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces

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, t_p, 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, t_p, 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, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, 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, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, 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, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]^-s, t_p￠, q￠(\mathbbRⁿ){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .     

On Li’s coefficients for the Rankin–Selberg <Emphasis Type="Italic">L</Emphasis>-functions

We define a generalized Li coefficient for the L-functions attached to the Rankin–Selberg convolution of two cuspidal unitary automorphic representations π and π ^′ of GL_m(\mathbbA_F)GL_{m}(\mathbb{A}_{F}) and GL_m^￠(\mathbbA_F)GL_{m^{\prime }}(\mathbb{A}_{F}) . Using the explicit formula, we obtain an arithmetic representation of the n th Li coefficient l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) attached to L(s,p_f×[(p)\tilde]_f^￠)L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime }) . Then, we deduce a full asymptotic expansion of the archimedean contribution to l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) and investigate the contribution of the finite (non-archimedean) term. Under the generalized Riemann hypothesis (GRH) on non-trivial zeros of L(s,p_f×[(p)\tilde]_f^￠)L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime }) , the nth Li coefficient l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) is evaluated in a different way and it is shown that GRH implies the bound towards a generalized Ramanujan conjecture for the archimedean Langlands parameters μ _π (v,j) of π. Namely, we prove that under GRH for L(s,p_f×[(p)\tilde]_f)L(s,\pi _{f}\times \widetilde{\pi}_{f}) one has |Rem_p(v,j)| ￡ \frac14|\mathop {\mathrm {Re}}\mu_{\pi}(v,j)|\leq \frac{1}{4} for all archimedean places v at which π is unramified and all j=1,…,m.     

On Linear Fractional Transformations Associated with Generalized J-Inner Matrix Functions

A class U_k1 (J){\mathcal{U}}_{\kappa 1} (J) of generalized J-inner mvf’s (matrix valued functions) W(λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of p ×q  mvf￠s S_k^{p ×q}p \times q \, {\rm mvf's}\, {\mathcal{S}}_{\kappa}^{p \times q} and some associated reproducing kernel Pontryagin spaces are studied. These spaces are used to describe the range of the linear fractional transformation T_W based on W and applied to S_k2^{p ×q}{\mathcal{S}}_{\kappa 2}^{p \times q}. Factorization formulas for mvf’s W in a subclass U^°_k1 (J) of U_k1(J){\mathcal{U}^{\circ}_{\kappa 1}} (J)\, {\rm of}\, {\mathcal{U}}_{\kappa 1}(J) found and then used to parametrize the set S_k1+k2^{p ×q} ?T_W [ S_k2^{p ×q} ]{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} \cap T_{W} \left[ {\mathcal{S}}_{\kappa 2}^{p \times q} \right]. Applications to bitangential interpolation problems in the class S_k1+k2^{p ×q}{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} will be presented elsewhere.     

End-point maximal regularity and its application to two-dimensional Keller–Segel system

We prove the maximal L ^ρ regularity of the Cauchy problem of the heat equation in the Besov space [(B)\dot]_1,r⁰(\mathbbRⁿ){\dot{B}_{1,\rho}^0(\mathbb{R}^n)}, 1 < ρ ≤ ∞, which is not UMD space. And as its application, we establish the time local well-posedness of the solution of two dimensional nonlinear parabolic system with the Poisson equation in [(B)\dot]_1,2⁰(\mathbbR²){\dot{B}_{1,2}^0(\mathbb{R}^2)} , where the equation is considered in the space invariant by a scaling and particularly the natural free energy is well defined from the initial time. The small data global existence is also obtained in the same class.     

Schlicht envelopes of holomorphy and topology

Let Ω be a domain in ${\mathbb{C}^{2}}Let Ω be a domain in \mathbbC²{\mathbb{C}^{2}}, and let p: [(W)\tilde]? \mathbbC²{\pi: \tilde{\Omega}\rightarrow \mathbb{C}^{2}} be its envelope of holomorphy. Also let W￠=p([(W)\tilde]){\Omega'=\pi(\tilde{\Omega})} with i: W\hookrightarrow W￠{i: \Omega \hookrightarrow \Omega'} the inclusion. We prove the following: if the induced map on fundamental groups i_*:p₁(W) ? p₁(W￠){i_{*}:\pi_{1}(\Omega) \rightarrow \pi_{1}(\Omega')} is a surjection, and if π is a covering map, then Ω has a schlicht envelope of holomorphy. We then relate this to earlier work of Fornaess and Zame.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

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.     

From Ginzburg-Landau to Gross-Pitaevskii

(w, c) ? R², u ? L_loc³ (R^N, C)\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}(\omega, c)\in {\Op R}^2, {\upsilon} \in L_{\rm loc}^3 ({\Op R}^N, {\bf C}) and x||j||_{L^￥(R^N×R)}² ￡ max{0, 1-w+[(c²)/4]}.\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}\Vert\varphi\Vert_{L^\infty({\Op R}^N\times{\Op R})}^2 \le \max\bigg\{0, 1-\omega+{c^2\over 4}\bigg\}.     

On Singular Integral Operators with Rough Kernel Along Surface

In this note we give a simple method to transfer the effect of the surface to the radial function in the kernel of singular integral along surface. Using this idea, we give some continuity of the singular integrals along surface with Hardy space function kernels on some function spaces, such as L^p(\mathbb Rⁿ),L^p(\mathbb Rⁿ,w){L^p({\mathbb R}^n),L^p({\mathbb R}^n,\omega)}, Triebel–Lizorkin spaces [(F)\dot]_p^s,q(\mathbb Rⁿ){{\dot F}_{p}^{s,q}({\mathbb R}^n)}, Besov spaces [(B)\dot]_p^s,q(\mathbb Rⁿ){{\dot B}_{p}^{s,q}({\mathbb R}^n)}, generalized Morrey spaces L^p,f(\mathbb Rⁿ){L^{p,\phi}({\mathbb R}^n)} and Herz spaces [(K)\dot]_p^{a, q}(\mathbb Rⁿ){\dot K_p^{\alpha, q}({\mathbb R}^n)}. Our results improve and extend substantially some known results on the singular integral operators along surface.     

Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system

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+\fracnp_p,￥(\mathbbRⁿ){\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)} ( \fracn2 < p < n{\frac{n}{2} < p < n}).     

Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity

We consider the global attractor for the weakly damped forced KdV equation in Sobolev spaces [(H)\dot]^s(T){\dot{H}^s({\mathbf T})}for s < 0. Under the assumption that the external forcing term belongs to [(L)\dot]²(T),{\dot{L}^2({\mathbf T}),} we prove the existence of the global attractor in [(H)\dot]^s(T){\dot{H}^s({\mathbf T})} for −1/2 ≤ s < 0, which is identical to the one in [(L)\dot]²(T){\dot{L}^2({\mathbf T})} and thus is compact in H ³(T). The argument is a combination of the I-method and decomposing the solution into two parts, one of which is uniformly bounded in [(L)\dot]²(T){\dot{L}^2({\mathbf T})} and the other decays exponentially in [(H)\dot]^s(T){\dot{H}^s({\mathbf T})}.     

O’Neil Inequality for Multilinear Convolutions and Some Applications

In this paper we prove the O’Neil inequality for the k-linear convolution f ⊗ g. By using the O’Neil inequality for rearrangements we obtain a pointwise rearrangement estimate of the k-linear convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the k-sublinear fractional maximal operator M _{Ω, α} and k-linear fractional integral operator I _{Ω, α} with rough kernels from the spaces V.S. Guliyev partially supported by the grant of INTAS (project 05-1000008-8157).     

New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets

In this paper, the authors establish new characterizations of the recently introduced Besov-type spaces $\dot{B}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ and Triebel-Lizorkin-type spaces $\dot{F}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ with p∈(0,∞], s∈?, τ∈[0,∞), and q∈(0,∞], as well as their preduals, the Besov-Hausdorff spaces $B\!\dot{H}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ and Triebel-Lizorkin-Hausdorff spaces $F\!\dot{H}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ , in terms of the local means, the Peetre maximal function of local means, and the tent space (the Lusin area function) in both discrete and continuous types. As applications, the authors then obtain interpretations as coorbits in the sense of Rauhut (Stud. Math. 180:237–253, 2007) and discretizations via biorthogonal wavelet bases for the full range of parameters of these function spaces. Even for some special cases of this setting such as $\dot{F}^{s}_{\infty,q}({\mathbb{R}}^{n})$ for s∈?, q∈(0,∞] (including ?BMO(? ⁿ ) when s=0 and q=2), the Q space Q _α (? ⁿ ), the Hardy-Hausdorff space HH _?α (? ⁿ ) for α∈(0,min{n/2,1}), the Morrey space ${\mathcal{M}}^{u}_{p}({\mathbb{R}}^{n})$ for 1<p≤u<∞, and the Triebel-Lizorkin-Morrey space $\dot{\mathcal{E}}^{s}_{upq}({\mathbb{R}}^{n})$ for 0<p≤u<∞, s∈? and q∈(0,∞], some of these results are new.     

Optimal Cubature Formulas in Weighted Besov Spaces with A ∞ Weights on Multivariate Domains

For the unit ball $\mathit{UB}_{\tau}^{\alpha}(L_{p,w})$ of the weighted Besov space $B_{\tau}^{\alpha}(L_{p,w})$ with an A _∞ weight w on the domain Ω, which denotes either the unit sphere, or the unit ball, or the standard simplex of the Euclidean space ? ^d , the sharp asymptotic order of the quantity $$\mathop{\inf}_{{\lambda}_1, \ldots, {\lambda}_n \in \mathbb{R}\atop{\xi_1,\ldots, \xi_n \in\varOmega }} \sup_{f\in \mathit{UB}_\tau^\alpha(L_{p,w})} \biggl|\int _{\varOmega } f(x) w(x)\, dx-\sum_{j=1}^n{\lambda}_j f(\xi_j) \biggr|$$ is obtained as n→∞. A similar result is also established on unweighted spherical caps.     

Minimal rotational hypersurfaces in Minkowski (<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">β</Emphasis>)-space

In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal rotational hypersurfaces generated by plane curves rotating around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski (α, β)-space (\mathbbVⁿ⁺¹,[(F_b)\tilde]){(\mathbb{V}^{n+1},\tilde{F_b})} , where \mathbbVⁿ⁺¹{\mathbb{V}^{n+1}} is an (n+1)-dimensional real vector space, [(F_b)\tilde]=[(a)\tilde]f([(b)\tilde]/[(a)\tilde]), [(a)\tilde]{\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}} is the Euclidean metric, [(b)\tilde]{\tilde{\beta}} is a one form of constant length b:=||[(b)\tilde]||_[(a)\tilde], [(b)\tilde]^\sharp{b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}} is the dual vector of [(b)\tilde]{\tilde{\beta}} with respect to [(a)\tilde]{\tilde{\alpha}} . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski Randers 3-space (\mathbbV³,[(a)\tilde]+[(b)\tilde]){(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})} .     

On estimation of a density and its derivatives

Summary Letf _n ^(p) be a recursive kernel estimate off ^(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of and show that the rate of almost sure convergence of to zero isO(n ^−α), α<(r−p)/(2r+1), iff ^(r),r>p≧0, is a continuousL ₂(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of to zero under different conditions onf. This work was supported in part by the Research Foundation of SUNY.     

An Lp-Lq-Version of Morgan’s Theorem for the Dunkl-Bessel Transform

In this paper, we give an L^p-L^q-version of Morgans theorem for the Dunkl-Bessel transform on More precisely, we prove that for all and then for all measurable function f on the conditions and imply f = 0, if and only if where are the Lebesgue spaces associated with the Dunkl-Bessel transform.Received: November 21, 2003 Revised: April 26, 2004 Accepted: May 28, 2004     

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾ (?,q) of Sobolev type $$W_1^{(2)} (\mathbb{R},q) = \left\{ {y \in A_{loc}^{(1)} (\mathbb{R}):\left\| {y''} \right\|_{L_1 (\mathbb{R})} + \left\| {qy} \right\|_{L_1 (\mathbb{R})} < \infty } \right\} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x \in \mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$

     

Uniqueness of a solution to the problem of atmosphere dynamics

We consider the model of atmosphere dynamics and prove the uniqueness of a solution in a bounded domain W ì \mathbbR³ \Omega \subset {\mathbb{R}^3} in the space V(Q) of weak solutions equipped with the finite norm

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Iterated Logarithm Law for Anticipating Stochastic Differential Equations

We prove a functional law of iterated logarithm for the following kind of anticipating stochastic differential equations