Composition operators in generalized Morrey spaces

We study a backward stochastic differential equation driven by fractional Brownian motion whose generator is Lipschitz continuous in y and uniformly continuous in z, we give a result of existence and uniqueness for the solution to this equation. The main idea consists in quasi-conditional expectation for fractional Brownian motion.

In the paper the authors find conditions on the pair \( (\varphi_{1},\varphi_{2}) \) which ensure the Spanne type boundedness of the fractional maximal operator \( M_{\alpha} \) and the Riesz potential operator \( I_{\alpha} \) from one generalized Morrey spaces \( M_{p,{\varphi_{1}}} \) to another \( M_{q,{\varphi_{2}}}, 1 < p < q < \infty, 1/p-1/q = \alpha/n, \) and from \( M_{1,{\varphi_{1}}} \) to the weak space W \( M_{q,{\varphi_{2}}}, 1 < p < q < \infty, 1- 1/q = \alpha/n, \) We also find conditions on \( \varphi \) which ensure the Adams type boundedness of the \( M_{\alpha}\; {\rm and}\; I_{\alpha}\; {\rm from} \; M_{p,{\varphi}^{\frac {1}{p}}}\; \rm{to}\; M_{q,{\varphi}^{\frac {1}{q}}}\;\rm {for 1 < p < q < \infty \; and\; from\; M_{1,{\varphi}}\; to \;W\;M_{q,{\varphi}^{\frac{1}{p}}} \; for \; 1 < q < \infty.}\) As applications of those results, the boundeness of the commutators of operators \( I_{\alpha} and I_{\alpha} \) on generalized Morrey spaces is also obtained. In the case \( b \in BMO{\mathbb{(R)}^{n}}\; \rm and \;1 < p < q < \infty,\) we find the sufficient conditions on the pair \( (\varphi_{1},\varphi_{2}) \) which ensures the boundedness of the operators \( {M_{b,\alpha}}\; \rm {and \;[b,I_{\alpha}] \; from \; M_{p,\varphi_{1}}\; to \; M_{q,\varphi_{2}}\; with\; 1/p - 1/q = \alpha/n.} \) We also find the sufficient conditions on \( \varphi \) which ensures the boundedness of the operators \( {M_{b,\alpha}}\; \rm {and \;[b,I_{\alpha}] \; from \; M_{p,{\varphi^{\frac{1}{p}}}}\; to \; M_{q,\varphi^{\frac{1}{p}}}\; for\; 1 < p < q < \infty.} \) In all cases conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \( \rm {(\varphi_{1},\varphi_{2}) \;and \;\varphi} ,\)which do not assume any assumption on monotonicity of \( \rm {\varphi_{1},\varphi_{2} \;and \;\varphi} \;\rm{in\; r} ,\) As applications, we get some estimates for Marcinkiewicz operator and fractional powers of the some analytic semigroups on generalized Morrey spaces.KeywordsFractional maximal operatorRiesz potential operatorgeneralized Morrey spacecommutatorBMO space.

The boundedness of maximal Bochner-Riesz operator B*δ and that of maximal commutator Bδ,*b generated by this operator and Lipschitz function on the classical Morrey space and generalized Morrey space are established.

In this paper, we define some new hyperbolic classes of functions, which will be called Bα,log* and Flog*(p,q,s) classes. Then, we introduce D-metrics in these defined hyperbolic classes. Also, these classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, necessary and sufficient conditions are given for a composition operator Cφ to be bounded and compact from Bα,log* to Flog*(p,q,s).

A necessary and sufficient condition is obtained for M‐quasi‐hyponormal composition operators. It has also been proved that the class of M‐quasi‐hyponormal composition operators coincides with the class of M‐paranormal composition operators. Existence of M‐hyponormal composition operators which are not hyponormal; and M‐quasihyponormal composition operators which are not M‐hyponormal and quasi‐hyponormal are also shown.

The Sufficient Condition for the Approximate Controllability of Nonlinear Abstract Control System

We consider the Riesz potential operator , on the Heisenberg group $$ {{\mathbb{H}}_n} $$ in generalized Morrey spaces $$ {M_{{p,\varphi }}}\left( {{{\mathbb{H}}_n}} \right) $$ and find conditions for the boundedness of as an operator from $$ {M_{{p,\varphi 1}}}\left( {{{\mathbb{H}}_n}} \right) $$ to $$ {M_{{p,\varphi 2}}}\left( {{{\mathbb{H}}_n}} \right) $$ , 1 < p < ∞, and from $$ {M_{{1,\varphi 1}}}\left( {{{\mathbb{H}}_n}} \right) $$ to a weak Morrey space $$ W{M_{{1,\varphi 2}}}\left( {{{\mathbb{H}}_n}} \right) $$ . The boundedness conditions are formulated it terms of Zygmund type integral inequalities. Based on the properties of the fundamental solution of the sub-Laplacian on $$ {{\mathbb{H}}_n} $$ , we prove two Sobolev–Stein embedding theorems for generalized Morrey and Besov–Morrey spaces. Bibliography: 40 titles.

Strong uniform continuity

In this paper, we study boundedness of integral operators on generalized Morrey spaces and its application to estimates in Morrey spaces for the Schrödinger operator L 2 = − Δ + V ( x ) + W ( x ) L_2=-\Delta +V(x)+W(x) with nonnegative V ∈ ( R H ) ∞ V\in (RH)_{\infty } (reverse Hölder class) and small perturbed potentials W W .

In this paper, we study quasicompact and Riesz composition operators on Banach spaces of Lipschitz–Holder functions on pointed metric spaces. For a composition operator T on these spaces, we give an upper bound for $$r_e(T)$$ , the essential spectral radius of T, and establish a formula for $$r_e(T)$$ whenever metric spaces are compact. We also give some necessary and some sufficient conditions that a composition operator T on these spaces to be quasicompact or Riesz. Finally, we get a relation for the set of eigenvalues and the spectrum of a quasicompact and Riesz composition operator on these spaces.

Suppose $\\varphi$ is an analytic self-map of open unit disk $\\mathbf{D}$ and $\\psi$ is an analytic function on $\\mathbf{D}$. Then a weighted composition operator induced by $\\varphi$ with weight $\\psi$ is given by $(W_{\\psi, \\varphi}f)(z)= \\psi(z)f(\\varphi(z))$, for $z \\in \\mathbf{D}$ and $f$ analytic on $\\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\\psi, \\varphi}$. In terms of fixed points in the closed unit disk $\\overline{\\mathbf{D}}$, conditions under which $W_{\\psi, \\varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\\varphi}$ are given in terms of the angular derivative $\\varphi^{\\prime}(\\zeta)$ where $\\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\\beta_1), H^q(\\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

This paper investigates global asymptotic stability (GAS) and global exponential stability (GES) of a class of continuous-time recurrent neural networks. First, we introduce a necessary and sufficient condition for the existence and uniqueness of equilibrium of the neural networks with Lipschitz continuous activation functions. Next, we present two sufficient conditions to ascertain the GAS of the neural networks with globally Lipschitz continuous and monotone nondecreasing activation functions. We then give two GES conditions for the neural networks whose activation functions may not be monotone nondecreasing. We also provide a Lyapunov diagonal stability condition, without the nonsingularity requirement for the connection weight matrices, to ascertain the GES of the neural networks with globally Lipschitz continuous and monotone nondecreasing activation functions. This Lyapunov diagonal stability condition generalizes and unifies many of the existing GAS and GES results. Moreover, two higher exponential convergence rates are estimated.

In this paper we investigate the boundedness of Mikhlin perators on generalized Morrey spaces. Theresults show that the operators are bounded on generalized Morrey spaces under some assumptions.

In this paper, we consider the commutators $[b,\mathcal{I}_\alpha]_i$ $(i=1,2)$, where $\mathcal{I}_\alpha$ is a bilinear fractional integral operator, and $b$ is a function in generalized Campanato spaces with the variable growth condition. We give necessary and sufficient 条件 for the boundedness of $~[b,\mathcal{I}_\alpha]_i$ $(i=1,2)$ on generalized Morrey spaces with the variable growth condition.

ABSTRACTLet μ be a normal function on . Suppose ϕ is a holomorphic self-map on the unit ball B of . In this paper, the authors characterize the conditions that the composition operator is bounded or compact on the normal weight Zygmund space when n>1. At the same time, the authors give the sufficient and necessary conditions for some special cases.