We find the conditions on the pair which ensures the boundedness of the maximal operator and Calderón-Zygmund singular integral operators from one generalized Morrey space to another,, and from the space to the weak space.
By (Wmathcal{M}^{p,lambda}({mathbb{R}^{n}})) we denote the weak Morrey space defined as the set of functions f in the local weak space (WL^{p}_{mathrm{loc}}({mathbb{R}^{n}}) ) for which Vert fVert _{Wmathcal{M}^{p,lambda}} = sup_{xin{mathbb{R}^{n}}, r>0} r^{-frac{lambda}{p}} |f|_{WL^{p}(B x,r))} < infty.
We find the sufficient conditions on the pair ((varphi_{1},varphi_{2})) which ensure the boundedness of the operators (mu_{j}^{L}) from one vanishing generalized Morrey space (mathit{VM}_{p,varphi_{1}}^{alpha,V}) to another (mathit{VM}_{p,varphi_{2}}^{alpha,V}), (1< p<infty) and from the space (mathit{VM}_{1,varphi_{1}}^{alpha,V}) to the weak space (VWM_{1,varphi _{2}}^{alpha,V}).
Then the weighted weak space (aleph^{p} _{alpha }(Gamma ) ) on cylinders can be defined by Vert u Vert _{aleph^{p}_{alpha }(Gamma )}:= biggl( int_{ Gamma }biglvert u y) bigrvert ^{p},dwp_{alpha } y) biggr) ^{frac{1}{q}}< infty, where u are weak solutions of (1.1) on cylinders, (dwp_{alpha } y)=operatorname{dist} y,partial Gamma )^{alpha },dy) and (1/p+1/q=1).
We find a new expression for the norm of a function in the weighted Loreritz space, with respect to the distribution funetion, and obtain as a simple consequence a generalization of the classical embeddings Lp, 1 ⊂ · · · ⊂ Lp ⊂ · · · ⊂ Lp, ∞ and a new definition of the weak space Λp, ∞u (w).
In this paper, we prove the existence of global attractor and exponential attractor in some stronger spaces for the strongly damped nonlinear wave equation when the nonlinear term (f u,u_{t})) depends on (u_{t}) and contains a critical exponent with respect to u and the external forcing term g merely belongs to the weak space (H^{-1}(Omega)).
Corals coexist with CCA in healthy coral reefs despite appearing to be weaker space competitors than inhibitor CCA.
The composite shows weaker space charge accumulation, lower complex permittivity and conduction current than EP above Tg.
This, in turn, allows us to assume additional regularity in order to show consistency in a weaker space X.
Moreover, with a smaller molecular structure, resveratrol has a weaker space steric, which may be the reason for its somewhat stronger inhibition activity of FAS.
