In the following paper, we provide a stochastic analogue to work of Shea and Wainger by showing that when the measure and state-independent diffusion coefficient of a linear Itô–Volterra equation are in appropriate Lp– weighted spaces, the solution lies in a weighted Lp–space in both an almost sure and moment sense.

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In this paper we develop a geometric theory for quasilinear parabolic problems in weighted Lp-spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the ω-limit sets and the long-time behaviour of the solutions. These techniques are applied to a...

An example is given of a unilateral weighted shift on complex Hubert space which is strictly cyclic but not strongly strictly cyclic. Similar examples of weighted shifts on the sequence spaces lp, for 1 < p < oo, are indicated.

In [18] and [19] we have studied compact embeddings of weighted function spaces on Rn, id : Hs q (w(x),Rn) −→ Lp(R), s > 0, 1 < q ≤ p < ∞, s − n/q + n/p > 0, with, for example, w(x) = 〈x〉α, α > 0, or w(x) = logβ〈x〉, β > 0, and 〈x〉 = (2 + |x|2)1/2. We have determined the behaviour of their entropy numbers ek(id). Now we are interested in the limiting case 1/q = 1/p + s/n. Let w(x) = logβ〈x〉, β >...

We obtain Fourier inequalities in the weighted Lp spaces for any 1<p<∞ involving Hardy–Cesàro and Hardy–Bellman operators. extend these results to product Hardy p⩽1. Moreover, boundedness of Hardy-Cesàro Hardy-Bellman operators various (Lebesgue, Hardy, BMO) is discussed. One our main tools an appropriate version Hardy–Littlewood–Paley inequality ‖fˆ‖Lp′,q≲‖f‖Lp,q.

LetX be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem u̇(t)+ Au(t) = f (t), t > 0, u(0) = 0, enjoys maximal regularity in weighted Lp-spaces with weights ω(t) = tp(1−μ), where 1/p < μ, if and only if it has the property of maximal Lp-regularity. Moreover, it is also shown that the derivation operator D = d/dt admits anH∞-calculus in weigh...

Two uniqueness results for C0 semigroups on weighted Lp spaces over Rn generated by operators of type + r with singular drift are proven. A key ingredient in the proofs is the veri cation of some kind of \weak Kato inequality" which seems to break down exactly for those drift singularities where Lp uniqueness breaks down as well. 3 Lp uniqueness of non{symmetric di usion operators with singular...