On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space $\mathbb{X}$ and $f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}$ is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}_(t,s)∈Δ such that for every s∈?₊ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈?₊, and f(t,0)=0 for all t∈?₊) we can generate a (nonlinear) evolution family {X(t,s)}_(t,s)∈Δ , in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}$ is the unique solution of Eq. (4), for every $x\in\mathbb{X}$ and s∈?₊. Considering the Green’s operator $(\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds$ we prove that if the following conditions hold

• the map $\mathbb{G}{f}$ lies in $L^{q}(\mathbb{R}_{+},\mathbb{X})$ for all $f\in L^{p}(\mathbb{R}_{+},\mathbb{X})$ , and
• $\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})$ is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.     

On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators

We consider processes of the form [s,T]?t?u(t,X _t ), where (X,P _s,x ) is a multidimensional diffusion corresponding to a uniformly elliptic divergence form operator. We show that if $u\in{\mathbb{L}}_{2}(0,T;H_{\rho }^{1})$ with $\frac{\partial u}{\partial t} \in{\mathbb{L}}_{2}(0,T;H_{\rho }^{-1})$ then there is a quasi-continuous version $\tilde{u}$ of u such that $\tilde{u}(t,X_{t})$ is a P _s,x -Dirichlet process for quasi-every (s,x)∈[0,T)×? ^d with respect to parabolic capacity, and we describe the martingale and the zero-quadratic variation parts of its decomposition. We also give conditions on u ensuring that $\tilde{u}(t,X_{t})$ is a semimartingale.     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

VMO spaces associated with divergence form elliptic operators

Consider the second order divergence form elliptic operator L with complex bounded coefficients. In this paper we introduce and develop a function space VMO _L of vanishing mean oscillation associated with the operator L. Using the theory of tent spaces and the Littlewood-Paley theory, we prove that the Hardy space ${H_{L}^1}$ of Hofmann and Mayboroda introduced in Hofmann and Mayboroda (Math Ann 344:37?C116, 2009) is the dual of our ${{\rm VMO}_{L^{\ast}}}$ , in which L* is the adjoint operator of L. We also give an equivalent characterization of the space VMO _L in the context of the theory of tent spaces.     

Democracy functions and optimal embeddings for approximation spaces

We prove optimal embeddings for nonlinear approximation spaces $\mathcal{A}^{\alpha}_q$ , in terms of weighted Lorentz sequence spaces, with the weights depending on the democracy functions of the basis. As applications we recover known embeddings for N-term wavelet approximation in L ^p , Orlicz, and Lorentz norms. We also study the ??greedy classes?? ${\mathcal{G}_{q}^{\alpha}}$ introduced by Gribonval and Nielsen, obtaining new counterexamples which show that ${\mathcal{G}_{q}^{\alpha}}\not=\mathcal{A}^{\alpha}_q$ for most non-democratic unconditional bases.     

Solutions of second order degenerate integro-differential equations in vector-valued function spaces

We study the well-posedness of the second order degenerate integro-differential equations(P2):(Mu)(t)+α(Mu)(t) = Au(t)+ft-∞ a(ts)Au(s)ds + f(t),0t2π,with periodic boundary conditions M u(0)=Mu(2π),(Mu)(0) =(M u)(2π),in periodic Lebesgue-Bochner spaces Lp(T,X),periodic Besov spaces B s p,q(T,X) and periodic Triebel-Lizorkin spaces F s p,q(T,X),where A and M are closed linear operators on a Banach space X satisfying D(A) D(M),a∈L1(R+) and α is a scalar number.Using known operatorvalued Fourier multiplier theorems,we completely characterize the well-posedness of(P2) in the above three function spaces.     

The integral representation of the exponential product of stochastic semigroups

An integral representation is obtained for the exponential product of stochastic semigroups $$X_s^t \otimes Z_s^t = X_s^t + \mathop \smallint \limits_{s< u< t} X_u^t dV_u X_s^u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} X_{u_2 }^t dV_{u_2 } X_{u_1 }^{u_2 } dV_{u_1 } X_s^{u_1 } + \cdots ,$$ whereV _t is the generating process of the semigroupZ _s ^t and the integrals are understood in the sense of mean-square limits of the Riemann-Stieltjes sums. This representation is different from the traditional representation $$X_s^t \otimes Z_s^t = E + \mathop \smallint \limits_{s< u< t} dW_u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} dW_{u_2 } dW_{u_1 } + \cdots ,$$ in which the integration extends over the processW _t=Y_t+V_t that is the generating process of the exponential productX _s ^t ?Z _s ^t andY _t is the generator of the semigroupX _s ^t .     

Exponential Stability and Uniform Boundedness of Solutions for Nonautonomous Periodic Abstract Cauchy Problems. An Evolution Semigroup Approach

Let u _{μ, x, s} (., 0) be the solution of the following well-posed inhomogeneous Cauchy Problem on a complex Banach space X $$\left \{\begin{array}{ll}\dot{u}(t) = A(t)u(t) + e^{i\mu t}x, \quad t > s \\ u(s) = 0. \end{array} \right.$$ Here, x is a vector in X,?μ is a real number, q is a positive real number and A(·) is a q-periodic linear operator valued function. Under some natural assumptions on the evolution family ${\mathcal{U} = \{U(t, s): t \geq s\}}$ generated by the family {A(t)}, we prove that if for each?μ, each s?≥ 0 and every x the solution u _{μ, x, s} (·, 0) is bounded on R ₊ by a positive constant, depending only on x, then the family ${\mathcal{U}}$ is uniformly exponentially stable. The approach is based on the theory of evolution semigroups.     

Weighted Hardy Spaces Associated to Discrete Laplacians on Graphs and Applications

Let Γ be a infinite graph with a weight μ and let d and m be the distance and the measure associated with μ such that (Γ,d,m) is a space of homogeneous type. Let p(·,·) be the natural reversible Markov kernel on (Γ,d,m) and its associated operator be defined by \(Pf(x) = \sum _{y} p(x, y)f(y)\) . Then the discrete Laplacian on L ²(Γ) is defined by L=I?P. In this paper we investigate the theory of weighted Hardy spaces \({H^{p}_{L}}(\Gamma , w)\) associated to the discrete Laplacian L for 0<p≤1 and \(w\in A_{\infty }\) . Like the classical results, we prove that the weighted Hardy spaces \({H^{p}_{L}}(\Gamma , w)\) can be characterized in terms of discrete area operators and atomic decompositions as well. As applications, we study the boundedness of singular integrals on (Γ,d,m) such as square functions, spectral multipliers and Riesz transforms on these weighted Hardy spaces \({H^{p}_{L}}(\Gamma ,w)\) .     

Multipliers and Wiener-Hopf operators on weighted L p spaces

We study multipliers M (bounded operators commuting with translations) on weighted spaces L _ω ^p (?), and establish the existence of a symbol µ _M for M, and some spectral results for translations S _t and multipliers. We also study operators T on the weighted space L _ω ^p (?⁺) commuting either with the right translations S _t , t ∈ ?⁺, or left translations P ⁺ S _?t , t ∈ ?⁺, and establish the existence of a symbol µ of T. We characterize completely the spectrum σ(S _t ) of the operator S _t proving that $\sigma (S_t ) = \{ z \in \mathbb{C}:|z| \leqslant e^{t\alpha _0 } \} ,$ where α ₀ is the growth bound of (S _t ) _t≥0. A similar result is obtained for the spectrum of (P ⁺ S _?t ), t ≥ 0. Moreover, for an operator T commuting with S _t , t ≥ 0, we establish the inclusion , where $\mathcal{O}$ = {z ∈ ?: Im z < α ₀}.     

Bounds of Singular Integrals on Weighted Hardy Spaces and Discrete Littlewood–Paley Analysis

We apply the discrete version of Calderón??s reproducing formula and Littlewood?CPaley theory with weights to establish the $H^{p}_{w} \to H^{p}_{w}$ (0<p<??) and $H^{p}_{w}\to L^{p}_{w}$ (0<p??1) boundedness for singular integral operators and derive some explicit bounds for the operator norms of singular integrals acting on these weighted Hardy spaces when we only assume w??A _??. The bounds will be expressed in terms of the A _q constant of w if q>q _w =inf?{s:w??A _s }. Our results can be regarded as a natural extension of the results about the growth of the A _p constant of singular integral operators on classical weighted Lebesgue spaces $L^{p}_{w}$ in Hytonen et al. (arXiv:1006.2530, 2010; arXiv:0911.0713, 2009), Lerner (Ill.?J.?Math. 52:653?C666, 2008; Proc. Am. Math. Soc. 136(8):2829?C2833, 2008), Lerner et?al. (Int.?Math. Res. Notes 2008:rnm 126, 2008; Math. Res. Lett. 16:149?C156, 2009), Lacey et?al. (arXiv:0905.3839v2, 2009; arXiv:0906.1941, 2009), Petermichl (Am. J. Math. 129(5):1355?C1375, 2007; Proc. Am. Math. Soc. 136(4):1237?C1249, 2008), and Petermichl and Volberg (Duke Math. J. 112(2):281?C305, 2002). Our main result is stated in Theorem?1.1. Our method avoids the atomic decomposition which was usually used in proving boundedness of singular integral operators on Hardy spaces.     

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$ with a general function ${\varphi(x, r)}$ defining the Morrey-type norm. Here ${\Pi \subseteq \Omega}$ is an arbitrary subset in Ω including the extremal cases ${\Pi = \{x_0\}, x_0 \in \Omega}$ and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$ -theorem for the potential operator I ^α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${\varphi(x, r)}$ . No monotonicity type condition is imposed on ${\varphi(x, r)}$ . In case ${\varphi}$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ${\varphi}$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces     

Weighted Local Estimates for Fractional Type Operators

In this note we prove the estimate \(M^{\sharp }_{0,s}(Tf)(x) \le c\,M_{\gamma } f(x)\) for general fractional type operators T, where \(M^{\sharp }_{0,s}\) is the local sharp maximal function and M _γ the fractional maximal function, as well as a local version of this estimate. This allows us to express the local weighted control of T f by M _γ f. Similar estimates hold for T replaced by fractional type operators with kernels satisfying Hörmander-type conditions or integral operators with homogeneous kernels, and M _γ replaced by an appropriate maximal function M _T . We also prove two-weight, \({L^{p}_{\text {\textit {v}}}}\) - \({L^{q}_{\text {\textit {w}}}}\) estimates for the fractional type operators described above for 1 < p < q < ∞ and a range of q. The local nature of the estimates leads to results involving generalized Orlicz-Campanato and Orlicz-Morrey spaces.     

Generalised Gagliardo–Nirenberg Inequalities Using Weak Lebesgue Spaces and BMO

Using elementary arguments based on the Fourier transform we prove that for ${1 \leq q < p < \infty}$ and ${s \geq 0}$ with s > n(1/2 ? 1/p), if ${f \in L^{q,\infty} (\mathbb{R}^n) \cap \dot{H}^s (\mathbb{R}^n)}$ , then ${f \in L^p(\mathbb{R}^n)}$ and there exists a constant c _p,q,s such that $$\| f \|_{L^{p}} \leq c_{p,q,s} \| f \|^\theta _{L^{q,\infty}} \| f \|^{1-\theta}_{\dot{H}^s},$$ where 1/p = θ/q + (1?θ)(1/2?s/n). In particular, in ${\mathbb{R}^2}$ we obtain the generalised Ladyzhenskaya inequality ${\| f \| _{L^4} \leq c \| f \|^{1/2}_{L^{2,\infty}} \| f \|^{1/2}_{\dot{H}^1}}$ .We also show that for s = n/2 and q > 1 the norm in ${\| f \|_{\dot{H}^{n/2}}}$ can be replaced by the norm in BMO. As well as giving relatively simple proofs of these inequalities, this paper provides a brief primer of some basic concepts in harmonic analysis, including weak spaces, the Fourier transform, the Lebesgue Differentiation Theorem, and Calderon–Zygmund decompositions.     

GI-graphs: a new class of graphs with many symmetries

The class of generalized Petersen graphs was introduced by Coxeter in the 1950s. Frucht, Graver and Watkins determined the automorphism groups of generalized Petersen graphs in 1971, and much later, Nedela and ?koviera and (independently) Lovre?i?-Sara?in characterised those which are Cayley graphs. In this paper we extend the class of generalized Petersen graphs to a class of GI-graphs. For any positive integer n and any sequence j ₀,j ₁,…,j _t?1 of integers mod n, the GI-graph GI(n;j ₀,j ₁,…,j _t?1) is a (t+1)-valent graph on the vertex set \(\mathbb{Z}_{t} \times\mathbb{Z}_{n}\) , with edges of two kinds:

• an edge from (s,v) to (s′,v), for all distinct \(s,s' \in \mathbb{Z}_{t}\) and all \(v \in\mathbb{Z}_{n}\) ,
• edges from (s,v) to (s,v+j _s ) and (s,v?j _s ), for all \(s \in\mathbb{Z}_{t}\) and \(v \in\mathbb{Z}_{n}\) .

By classifying different kinds of automorphisms, we describe the automorphism group of each GI-graph, and determine which GI-graphs are vertex-transitive and which are Cayley graphs. A GI-graph can be edge-transitive only when t≤3, or equivalently, for valence at most 4. We present a unit-distance drawing of a remarkable GI(7;1,2,3).     

Essential norm of composition operators on the Hardy space H 1 and the weighted Bergman spaces {A_{\alpha}^{p}} on the ball

We estimate the essential norm of a composition operator acting on the Hardy space H ¹ and the weighted Bergman spaces ${A_{\alpha}^{p}}$ on the unit ball. In passing, we recover (and somehow simplify the proof of) parts of the recent article by Demazeux, dealing with the same question for H ¹ of the unit disc. We also estimate the essential norm of a composition operator acting on ${A_{\alpha}^{p}}$ in terms of the angular derivatives of ${\phi}$ , under a mild condition on ${\phi}$ .     

Global Morrey estimates for a class of Ornstein-Uhlenbeck operators

In this paper, we consider a class of hypoelliptic Ornstein-Uhlenbeck operators in ? ^N given by $\mathcal{A} = \sum\limits_{i,j = 1}^{p_0 } {a_{ij} \partial _{x_i x_j }^2 + } \sum\limits_{i,j = 1}^N {b_{ij} x_i \partial _x } ,$ where (a _ij ), (b _ij ) are N × N constant matrices, and (a _ij ) is symmetric and positive semidefinite. We deduce global Morrey estimates forA from similar estimates of its evolution operator L on a strip domain S = ? ^N × [?1, 1].     

The Two-Weighted Inequalities for Sublinear Operators Generated by B Singular Integrals in Weighted Lebesgue Spaces

In this paper, the authors establish several general theorems for the boundedness of sublinear operators (B sublinear operators) satisfies the condition (1.2), generated by B singular integrals on a weighted Lebesgue spaces $L_{p,\omega,\gamma}(\mathbb{R}_{k,+}^{n})$ , where $B=\sum_{i=1}^{k} (\frac{\partial^{2}}{\partial x_{k}^{2}} + \frac{\gamma_{i}}{x_{i}}\frac{\partial}{\partial x_{i}} )$ . The condition (1.2) are satisfied by many important operators in analysis, including B maximal operator and B singular integral operators. Sufficient conditions on weighted functions ω and ω ₁ are given so that B sublinear operators satisfies the condition (1.2) are bounded from $L_{p,\omega,\gamma}(\mathbb{R}_{k,+}^{n})$ to $L_{p,\omega_{1},\gamma}(\mathbb{R}_{k,+}^{n})$ .